Appendix 1: Proof of Theorem 10
Let \(\varepsilon = \{\varepsilon _\ell \}_{\ell \in \Omega }\) be a sequence of complex unit-magnitude numbers. The goal is to construct a function \(c_0 \in L^\infty (\mathbb {R}^2)\) such that \((\mathcal {A}_g^*c_0)(t_\ell ) = \varepsilon _\ell \), for all \(\ell \in \Omega \), and \(\left| (\mathcal {A}_g^*c_0)(t)\right| < 1\) for all \(t \in \mathbb {R}\setminus T\), where \(T= \{t_\ell \}_{\ell \in \Omega }\) is the support set of \(\mu \). Inspired by [12], we take \(c_0\) to be of the form
$$\begin{aligned} \forall (\tau , f) \in \mathbb {R}^2, \quad c_0(\tau , f) :=\frac{1}{2f_c}\sum _{\ell \in \Omega } \left( \alpha _\ell \, g(t_\ell - \tau )e^{-2\pi ift_\ell } + \beta _\ell \, g'(t_\ell - \tau )e^{-2\pi ift_\ell }\right) , \end{aligned}$$
(75)
where \(\alpha _\ell , \beta _\ell \in \mathbb {C}\), for all \(\ell \in \Omega \), and we proceed as follows:
-
1.
We first verify that \(\alpha :=\{\alpha _\ell \}_{\ell \in \Omega }\) and \(\beta :=\{\beta _\ell \}_{\ell \in \Omega }\) both in \( \ell ^\infty (\Omega )\) implies \(c_0 \in L^\infty (\mathbb {R}^2)\).
-
2.
Next, we show that one can find \(\alpha , \beta \in \ell ^\infty (\Omega )\) such that the interpolation conditions \((\mathcal {A}_g^*c_0)(t_\ell ) = \varepsilon _\ell \), for all \(\ell \in \Omega \), are satisfied and \(\left| \mathcal {A}_g^*c_0\right| \) has a local extremum at every \(t_\ell \), \(\ell \in \Omega \).
-
3.
Then, we verify, with \(\alpha , \beta \in \ell ^\infty (\Omega )\) chosen as in 2., that the magnitude of \(\mathcal {A}_g^*c_0\) is indeed strictly smaller than 1 outside the support set \(T= \{t_\ell \}_{\ell \in \Omega }\) of \(\mu \). This will be accomplished in two stages. First, we show that \(\left| \mathcal {A}_g^*c_0\right| \) is strictly smaller than 1 “away” from each point \(t_\ell \), \(\ell \in \Omega \), specifically, on \(\mathbb {R}\setminus \bigcup _{\ell \in \Omega } [t_\ell - \frac{1}{7f_c}, t_\ell + \frac{1}{7f_c}]\). We then complete the proof by establishing that \(\left| \mathcal {A}_g^*c_0\right| \) is strictly concave on each set \([t_\ell - \frac{1}{7f_c}, t_\ell + \frac{1}{7f_c}]\), \(\ell \in \Omega \), which, combined with the fact that \(\left| (\mathcal {A}_g^*c_0)(t_\ell )\right| = 1\), for every \(\ell \in \Omega \), implies that \(\left| \mathcal {A}_g^*c_0\right| \) is also strictly smaller than 1 on each of these sets.
The main conceptual components in our proof are due to Candès and Fernandez-Granda [12]. Although [12] considers recovery of measures on \(\mathbb {T}\) only and from pure Fourier measurements, we can still borrow technical ingredients from the proof of [12, Thm. 1.2]. However, the different nature of the measurements and, in particular, the case \(G= \mathbb {R}\), pose additional technical challenges relative to the proof of [12, Thm. 1.2]. Specifically, the sum corresponding to (75) in [12] is always finite, whereas here it can be infinite, which presents us with delicate convergence issues that need to be addressed properly. Further fundamental differences between the proof in [12] for the pure Fourier case and our proof stem from the choice of the interpolation kernel, which here is given by \(t \mapsto R(t){{\mathrm{sinc}}}(2\pi f_c t)\). Specifically, we do not have to impose a bandwidth constraint on the interpolation kernel. For pure Fourier measurements, on the other hand, the interpolation kernel has to be band-limited to \([-f_c, f_c]\) (Candès and Fernandez-Granda [12] use the square of the Fejér kernel which offers a good trade-off between localization in time and frequency). As already mentioned in the main body, this leads to a factor-of-two improvement in the minimum spacing condition for STFT measurements over pure Fourier measurements. Note, however, that STFT measurements, owing to their redundancy, provide more information than pure Fourier measurements. We finally note that our proof also borrows a number of technical results from [24].
1.1
\(\alpha , \beta \in \ell ^\infty (\Omega )\) implies \(c_0 \in L^\infty (\mathbb {R}^2)\)
Let \(\alpha :=\{\alpha _\ell \}_{\ell \in \Omega } \in \ell ^\infty (\Omega )\) and \(\beta :=\{\beta _\ell \}_{\ell \in \Omega } \in \ell ^\infty (\Omega )\). Take \((\tau , f) \in \mathbb {R}^2\) and define \(\ell _0, \ell _1 \in \Omega \) to be the indices of the points in \(T\), that are closest and second closest, respectively, to \(\tau \), that is,
$$\begin{aligned} \ell _0 :=\mathop {\mathrm{arg \, min}}\limits _{\ell \in \Omega } \left| t_\ell - \tau \right| \qquad \text {and} \qquad \ell _1 :=\mathop {\mathrm{arg \, min}}\limits _{\ell \in \Omega \setminus \{\ell _0\}} \left| t_\ell - \tau \right| . \end{aligned}$$
For brevity of exposition, we detail the case \(t_{\ell _0} \le \tau \le t_{\ell _1}\) only, the cases \(t_{\ell _1} \le \tau \le t_{\ell _0}\), \(t_{\ell _1} \le t_{\ell _0} \le \tau \), and \(\tau \le t_{\ell _0} \le t_{\ell _1}\) are all dealt with similarly. For all \(\ell \in \Omega _\tau ^{-} :=\{m\in \Omega :t_m< t_{\ell _0}\}\), it then holds that
$$\begin{aligned} \tau - t_\ell \ge t_{\ell _0} - t_\ell \ge \Delta , \end{aligned}$$
(76)
and for all \(\ell \in \Omega _\tau ^{+} :=\{m\in \Omega :t_m> t_{\ell _1}\}\), we have
$$\begin{aligned} t_\ell - \tau \ge t_\ell - t_{\ell _1} \ge \Delta . \end{aligned}$$
(77)
Hence, we get the following:
$$\begin{aligned} \left| c_0(\tau , f)\right|&\le \frac{1}{2f_c}\left( \left\| \alpha \right\| _{\ell ^\infty }\sum _{\ell \in \Omega }\left| g(t_\ell - \tau )\right| + \left\| \beta \right\| _{\ell ^\infty }\sum _{\ell \in \Omega }\left| g'(t_\ell - \tau )\right| \right) \nonumber \\&= \frac{\left\| \alpha \right\| _{\ell ^\infty }}{2f_c}\left| g(t_{\ell _0} - \tau )\right| + \frac{\left\| \alpha \right\| _{\ell ^\infty }}{2f_c}\left| g(t_{\ell _1} - \tau )\right| + \frac{\left\| \beta \right\| _{\ell ^\infty }}{2f_c}\left| g'(t_{\ell _0} - \tau )\right| \end{aligned}$$
(78)
$$\begin{aligned}&\quad + \frac{\left\| \beta \right\| _{\ell ^\infty }}{2f_c}\left| g'(t_{\ell _1} - \tau )\right| + \frac{\left\| \alpha \right\| _{\ell ^\infty }}{2f_c}\sum _{\ell \in \Omega \setminus \{\ell _0, \ell _1\}} \left| g(t_{\ell } - \tau )\right| \nonumber \\&\quad + \frac{\left\| \beta \right\| _{\ell ^\infty }}{2f_c}\sum _{\ell \in \Omega \setminus \{\ell _0, \ell _1\}} \left| g'(t_\ell - \tau )\right| \nonumber \\&\le \frac{\left\| \alpha \right\| _{\ell ^\infty }}{f_c}\left\| g\right\| _{\infty } + \frac{\left\| \beta \right\| _{\ell ^\infty }}{f_c}\left\| g'\right\| _{\infty } + \frac{\left\| \alpha \right\| _{\ell ^\infty }}{2f_c}\sum _{\ell \in \Omega _\tau ^-} \left| g(t_{\ell _0} - t_\ell )\right| \end{aligned}$$
(79)
$$\begin{aligned}&\quad + \frac{\left\| \alpha \right\| _{\ell ^\infty }}{2f_c}\sum _{\ell \in \Omega _\tau ^+} \left| g(t_{\ell } - t_{\ell _1})\right| + \frac{\left\| \beta \right\| _{\ell ^\infty }}{2f_c}\sum _{\ell \in \Omega _\tau ^-} \left| g'(t_{\ell _0} - t_\ell )\right| \nonumber \\&\quad + \frac{\left\| \beta \right\| _{\ell ^\infty }}{2f_c}\sum _{\ell \in \Omega _\tau ^+} \left| g'(t_\ell - t_{\ell _1})\right| , \end{aligned}$$
(80)
where the step from (78) to (79)–(80) follows from \(g, g' \in C_b(\mathbb {R})\), (76), (77), and the fact that \(\left| g\right| \) and \(\left| g'\right| \) are both symmetric and non-increasing on \([\Delta , \infty )\), which is by the assumption \(\Delta > 4\sigma \). Note that we eliminated the dependence of the upper bound in (79)–(80) on \((\tau , f)\). It remains to establish that every sum in the upper bound (79)–(80) is finite. The minimum separation between pairs of points of \(T= \{t_\ell \}_{\ell \in \Omega }\) is \(\Delta \), by assumption. Consequently, since \(\left| g\right| \) and \(\left| g'\right| \) are both symmetric and non-increasing on \([\Delta , \infty )\), the sums in (79) and (80) take on their maxima when the points \(t_\ell \), \(\ell \in \Omega \), are equispaced on \(\mathbb {R}\) with spacing \(\Delta \), i.e., when
$$\begin{aligned} \big \{t_{\ell _0} - t_\ell :\ell \in \Omega _\tau ^-\big \}&\subseteq \big \{n\Delta :n \in \mathbb {N}\setminus \{0\}\big \} \\ \big \{t_\ell - t_{\ell _1} :\ell \in \Omega _\tau ^+\big \}&\subseteq \big \{n\Delta :n \in \mathbb {N}\setminus \{0\}\big \}. \end{aligned}$$
It therefore follows that
$$\begin{aligned} \left| c_0(\tau , f)\right|\le & {} \frac{\left\| \alpha \right\| _{\ell ^\infty }}{f_c}\left\| g\right\| _{\infty } + \frac{\left\| \beta \right\| _{\ell ^\infty }}{f_c}\left\| g'\right\| _{\infty } + \frac{\left\| \alpha \right\| _{\ell ^\infty }}{f_c}\sum _{n = 1}^\infty \left| g(n\Delta )\right| \nonumber \\&+ \frac{\left\| \beta \right\| _{\ell ^\infty }}{f_c}\sum _{n = 1}^\infty \left| g'(n\Delta )\right| , \end{aligned}$$
(81)
where
$$\begin{aligned} \sum _{n = 1}^\infty \left| g(n\Delta )\right|&= \frac{1}{\sqrt{\sigma }}\sum _{n = 1}^\infty \exp \!\left( -\frac{\pi n^2\Delta ^2}{2\sigma ^2}\right)< \infty \\ \sum _{n = 1}^\infty \left| g'(n\Delta )\right|&= \frac{\pi }{\sigma ^2\sqrt{\sigma }} \sum _{n = 1}^\infty n\Delta \exp \!\left( -\frac{\pi n^2\Delta ^2}{2\sigma ^2}\right) < \infty , \end{aligned}$$
which establishes that \(c_0 \in L^\infty (\mathbb {R}^2)\).
1.2 Existence of \(\alpha , \beta \in \ell ^\infty (\Omega )\) such that \((\mathcal {A}_g^*c_0)(t_\ell ) = \varepsilon _\ell \) and \(\left| \mathcal {A}_g^*c_0\right| \) has a local extremum at every \(t_\ell \), \(\ell \in \Omega \)
Using (75) in (18), we get
$$\begin{aligned} \forall t&\in \mathbb {R}, \quad (\mathcal {A}_g^* c_0)(t) = \int _{-f_c}^{f_c}\int _\mathbb {R}c_0(\tau , f)g(t - \tau )e^{2\pi ift}\mathrm {d}\tau \mathrm {d}f\nonumber \\&= \int _{-f_c}^{f_c} \int _\mathbb {R}\frac{1}{2f_c}\sum _{\ell \in \Omega } \Big (\alpha _\ell g(t_\ell - \tau )e^{-2\pi ift_\ell } + \beta _\ell g'(t_\ell - \tau )e^{-2\pi ift_\ell }\Big )\nonumber \\&\quad \times g(t - \tau ) e^{2\pi ift} \mathrm {d}\tau \mathrm {d}f \end{aligned}$$
(82)
$$\begin{aligned}&= \sum _{\ell \in \Omega } \left[ \alpha _\ell \left( \int _\mathbb {R}g(t_\ell - \tau )g(t - \tau )\mathrm {d}\tau \right) \left( \frac{1}{2f_c}\int _{-f_c}^{f_c} e^{2\pi if(t - t_\ell )}\mathrm {d}f\right) \right. \end{aligned}$$
(83)
$$\begin{aligned}&\quad \left. + \beta _\ell \left( \int _\mathbb {R}g'(t_\ell - \tau )g(t - \tau )\mathrm {d}\tau \right) \left( \frac{1}{2f_c}\int _{-f_c}^{f_c} e^{2\pi if(t - t_\ell )}\mathrm {d}f\right) \right] \end{aligned}$$
(84)
$$\begin{aligned}&= \sum _{\ell \in \Omega } \left( \alpha _\ell \underbrace{R(t - t_\ell ){{\mathrm{sinc}}}(2\pi f_c(t - t_\ell ))}_{:=u(t - t_\ell )} + \beta _\ell \underbrace{R'(t_\ell - t){{\mathrm{sinc}}}(2\pi f_c(t - t_\ell ))}_{:=v(t - t_\ell )}\right) , \end{aligned}$$
(85)
where we set
$$\begin{aligned} \forall t \in \mathbb {R}, \quad u(t)&:=R(t){{\mathrm{sinc}}}(2\pi f_c t) \end{aligned}$$
(86)
$$\begin{aligned} \forall t \in \mathbb {R}, \quad v(t)&:=R'(-t){{\mathrm{sinc}}}(2\pi f_c t) = \frac{\pi t}{2\sigma ^2}R(t){{\mathrm{sinc}}}(2\pi f_c t) \end{aligned}$$
(87)
with
$$\begin{aligned} \forall t \in \mathbb {R}, \quad R(t) = \exp \!\left( -\frac{\pi t^2}{4\sigma ^2}\right) \end{aligned}$$
as defined in (15). The conditions for Fubini’s Theorem, applied in the step from (82)–(83) to (84), can be verified as follows:
$$\begin{aligned} \sum _{\ell \in \Omega } \int _{-f_c}^{f_c}&\int _\mathbb {R}\left| \alpha _\ell g(t_\ell - \tau )g(t - \tau )e^{2\pi if(t - t_\ell )}\right| \mathrm {d}\tau \mathrm {d}f\nonumber \\&\le \sum _{\ell \in \Omega } \left| \alpha _\ell \right| \left( \int _\mathbb {R}\left| g(t_\ell - \tau )g(t - \tau )\right| \mathrm {d}\tau \right) \left( \frac{1}{2f_c}\int _{-f_c}^{f_c} \left| e^{2\pi ift}\right| \mathrm {d}f\right) \nonumber \\&=\sum _{\ell \in \Omega } \left| \alpha _\ell \right| R(t - t_\ell ) \le 2\left\| \alpha \right\| _{\ell ^\infty }\left( \left\| R\right\| _{\infty } + \sum _{n = 1}^\infty R(n\Delta )\right) \end{aligned}$$
(88)
and
$$\begin{aligned} \sum _{\ell \in \Omega } \int _{-f_c}^{f_c}&\int _\mathbb {R}\left| \beta _\ell g'(t_\ell - \tau )g(t - \tau )e^{2\pi if(t - t_\ell )}\right| \mathrm {d}\tau \mathrm {d}f\nonumber \\&\le \sum _{\ell \in \Omega } \left| \beta _\ell \right| \left( \int _\mathbb {R}\left| g'(t_\ell - \tau )g(t - \tau )\right| \mathrm {d}\tau \right) \left( \frac{1}{2f_c}\int _{-f_c}^{f_c} \left| e^{2\pi ift}\right| \mathrm {d}f\right) \nonumber \\&\le \sum _{\ell \in \Omega } \left| \beta _\ell \right| \widetilde{R}(t_\ell -t) \le 2\left\| \beta \right\| _{\ell ^\infty }\left( \big \Vert \widetilde{R}\big \Vert _\infty + \sum _{n = 1}^\infty \widetilde{R}(n\Delta )\right) , \end{aligned}$$
(89)
where we used
$$\begin{aligned} \forall t \in \mathbb {R}, \quad \int _\mathbb {R}g(\tau )\left| g'(t + \tau )\right| \mathrm {d}\tau&= \frac{1}{\sigma }\exp \!\left( -\frac{\pi t^2}{2\sigma ^2}\right) - R'(t){{\mathrm{erf}}}\left( \frac{\sqrt{\pi }}{2\sigma }t\right) \\&\le \frac{1}{\sigma }\exp \!\left( -\frac{\pi t^2}{2\sigma ^2}\right) + \left| R'(t)\right| =:\widetilde{R}(t) \end{aligned}$$
and the fact that \(\widetilde{R}\) is bounded, symmetric, and non-decreasing on \([\Delta , \infty )\) as a consequence of \(\Delta > 4\sigma \). The upper bounds in (88) and (89) are both finite as the series \(\sum _{n \ge 1} R(n\Delta )\) and \(\sum _{n \ge 1} \widetilde{R}(n\Delta )\) converge.
We have shown in Lemma 3 that for \(c_0 \in L^\infty (\mathbb {R}^2)\), the function \(\mathcal {A}_g^*c_0\) is in \(C_b(\mathbb {R})\). With \(c_0\) taken as in (75), \(\mathcal {A}_g^*c_0\) is not only in \(C_b(\mathbb {R})\), but also differentiable, as we show next. We start by noting that the functions u and v defined in (86) and (87) are differentiable on \(\mathbb {R}\), and their derivatives are given by
$$\begin{aligned} \forall t \in \mathbb {R}, \quad u'(t)&= R'(t){{\mathrm{sinc}}}(2\pi f_c t) + 2\pi f_c\,R(t){{\mathrm{sinc}}}'(2\pi f_c t) \\ \forall t \in \mathbb {R}, \quad v'(t)&= -R''(-t){{\mathrm{sinc}}}(2\pi f_c t) + 2\pi f_c\,R'(-t){{\mathrm{sinc}}}'(2\pi f_c t) \\&=\left[ \left( \frac{\pi }{2\sigma ^2} - \frac{\pi ^2t^2}{4\sigma ^4}\right) {{\mathrm{sinc}}}(2\pi f_c t) + \frac{\pi ^2f_c t}{\sigma ^2}{{\mathrm{sinc}}}'(2\pi f_c t)\right] R(t). \end{aligned}$$
Then, using
$$\begin{aligned} \forall t \in \mathbb {R}\setminus \{0\}, \quad \left| {{\mathrm{sinc}}}(t)\right| \le \frac{1}{\left| t\right| } \qquad \text {and} \qquad \left| {{\mathrm{sinc}}}'(t)\right| = \left| \frac{\cos (t)}{t} - \frac{\sin (t)}{t^2}\right| \le \frac{1}{\left| t\right| } + \frac{1}{\left| t\right| ^2}, \end{aligned}$$
(90)
we obtain the following upper bounds on \(u'\) and \(v'\):
$$\begin{aligned} \forall t \in \mathbb {R}\setminus \{0\}, \quad \left| u'(t)\right|&\le \left( \frac{1}{4\sigma ^2f_c} + \frac{1}{\left| t\right| } + \frac{1}{2\pi f_c\left| t\right| ^2}\right) R(t) =:U(t) \end{aligned}$$
(91)
$$\begin{aligned} \forall t \in \mathbb {R}\setminus \{0\}, \quad \left| v'(t)\right|&\le \left( \frac{1}{2\sigma ^2f_c\left| t\right| } + \frac{\pi }{2\sigma ^2} + \frac{\pi \left| t\right| }{8\sigma ^4f_c}\right) R(t) =:V(t). \end{aligned}$$
(92)
Next, we establish that \(\sum _{\ell \in \Omega } \left( \alpha _\ell u'(t - t_\ell ) + \beta _\ell v'(t - t_\ell )\right) \) converges uniformly on every compact set \([-r, r]\), \(r > 0\), so that we can apply [13, Thm. V.2.14] to show that the series in (85) can be differentiated term by term. For \(r > 0\), we have
$$\begin{aligned}&\sum _{\ell \in \Omega } \sup _{t \in [-r, r]} \left| \alpha _\ell u'(t - t_\ell ) + \beta _\ell v'(t - t_\ell )\right| \\&\quad \le \sum _{\ell \in \Omega } \left( \left\| \alpha \right\| _{\ell ^\infty }\sup _{t \in [-r, r]} \!\!\left| u'(t - t_\ell )\right| + \left\| \beta \right\| _{\ell ^\infty }\sup _{t \in [-r, r]}\!\!\left| v'(t - t_\ell )\right| \right) \\&\quad = \left\| \alpha \right\| _{\ell ^\infty }\left( \sum _{\ell \in \Omega _r} \sup _{t \in [-r, r]} \left| u'(t - t_\ell )\right| \right. \\&\qquad \left. + \sum _{\ell \in \Omega _r^+} \sup _{t \in [-r, r]} \left| u'(t - t_\ell )\right| + \sum _{\ell \in \Omega _r^-} \sup _{t \in [-r, r]} \left| u'(t - t_\ell )\right| \right) \\&\qquad + \left\| \beta \right\| _{\ell ^\infty }\left( \sum _{\ell \in \Omega _r} \sup _{t \in [-r, r]} \left| v'(t - t_\ell )\right| \right. \\&\qquad \left. + \sum _{\ell \in \Omega _r^+} \sup _{t \in [-r, r]} \left| v'(t - t_\ell )\right| + \sum _{\ell \in \Omega _r^-} \sup _{t \in [-r, r]} \left| v'(t - t_\ell )\right| \right) , \end{aligned}$$
where we defined the sets \(\Omega _r :=\left\{ \ell \in \Omega :t_\ell \in [-r, r]\right\} \), \(\Omega _r^+ :=\left\{ \ell \in \Omega :t_\ell > r\right\} \), and \(\Omega _r^- :=\left\{ \ell \in \Omega :t_\ell < -r\right\} \). The functions U and V are both positive and symmetric, U is non-increasing on \((0, \infty )\), and V is non-increasing on \((0, \infty )\) as
$$\begin{aligned} \forall t \in (0, \infty ), \quad V'(t)&= \left( -\frac{1}{2\sigma ^2f_c t^2} + \frac{\pi }{8\sigma ^4f_c}\right) R(t) \\&\quad + \left( \frac{1}{2\sigma ^2f_c t} + \frac{\pi }{2\sigma ^2} + \frac{\pi t}{8\sigma ^4f_c}\right) R'(t) \\&= -\left( \frac{1}{2\sigma ^2f_c t^2} + \frac{\pi }{8\sigma ^4f_c} + \left( \frac{\pi }{2\sigma ^2}\right) ^2t + \frac{\pi ^2 t^2}{16\sigma ^6f_c}\right) R(t) \le 0. \end{aligned}$$
It therefore follows that
$$\begin{aligned} \sup _{t \in [-r, r]} \left| u'(t- t_\ell )\right| \le {\left\{ \begin{array}{ll} \left\| u'\right\| _{\infty }, &{} t_\ell \in [-r, r] \\ U(r - t_\ell ), &{} t_\ell > r \\ U(-r - t_\ell ), &{} t_\ell < -r \end{array}\right. } \end{aligned}$$
and
$$\begin{aligned} \sup _{t \in [-r, r]} \left| v'(t-t_\ell )\right| \le {\left\{ \begin{array}{ll} \left\| v'\right\| _{\infty }, &{} t_\ell \in [-r, r] \\ V(r - t_\ell ), &{} t_\ell > r \\ V(-r - t_\ell ), &{} t_\ell < -r. \end{array}\right. } \end{aligned}$$
Since the support set \(T= \{t_\ell \}_{\ell \in \Omega }\) is closed and uniformly discrete, by assumption, and \([-r, r]\) is compact, the set \(T\cap [-r, r]\), and thereby the index set \(\Omega _r\), contains a finite number of elements, say \(L_r \in \mathbb {N}\). We thus have the following
$$\begin{aligned}&\sum _{\ell \in \Omega } \sup _{t \in [-r, r]} \left| \alpha _\ell u'(t - t_\ell ) + \beta _\ell v'(t - t_\ell )\right| \\&\quad \le \left\| \alpha \right\| _{\ell ^\infty }\!\!\left( \!L_r\left\| u'\right\| _{\infty } + \sum _{\ell \in \Omega _r^+} U(r - t_\ell ) + \sum _{\ell \in \Omega _r^-} U(-r - t_\ell ) \!\!\right) \\&\qquad + \left\| \beta \right\| _{\ell ^\infty }\!\!\left( L_r\left\| v'\right\| _{\infty } + \sum _{\ell \in \Omega _r^+} V(r - t_\ell ) + \sum _{\ell \in \Omega _r^-} V(-r - t_\ell ) \right) \\&\quad \le \left\| \alpha \right\| _{\ell ^\infty }\!\!\left( \!L_r\left\| u'\right\| _{\infty } + 2\left\| U\right\| _{\infty } + 2\sum _{n = 1}^\infty U(n\Delta )\!\!\right) \\&\qquad + \left\| \beta \right\| _{\ell ^\infty }\!\!\left( \!L_r\left\| v'\right\| _{\infty } + 2\left\| V\right\| _{\infty } + 2\sum _{n = 1}^\infty V(n\Delta )\!\!\right) \!, \end{aligned}$$
where we isolated the points in \(T\) that are closest to r and \(-r\) as in (79)–(80) and we used the fact that a regular spacing of the \(t_\ell \), \(\ell \in \Omega \), maximizes the sum as in (81). Since \(\sum _{n \ge 1} U(n\Delta ) < \infty \) and \(\sum _{n \ge 1} V(n\Delta ) < \infty \), the Weierstrass M-test tells us that \(\sum _{\ell \in \Omega } \left( \alpha _\ell u'(t - t_\ell ) + \beta _\ell v'(t - t_\ell )\right) \) converges uniformly on every compact set \([-r, r]\), \(r > 0\). Thanks to [13, Thm. V.2.14] this implies that the function \(\mathcal {A}_g^*c_0\) is differentiable on \(\mathbb {R}\), and that its derivative equals
$$\begin{aligned} \forall t \in \mathbb {R}, \quad (\mathcal {A}_g^*c_0)'(t) = \sum _{\ell \in \Omega } \big (\alpha _\ell u'(t - t_\ell ) + \beta _\ell v'(t - t_\ell )\big ) \end{aligned}$$
for \(\alpha , \beta \in \ell ^\infty (\Omega )\). We next show that there exist \(\alpha , \beta \in \ell ^\infty (\Omega )\) such that \((\mathcal {A}_g^*c_0)(t_\ell ) = \varepsilon _\ell \), for all \(\ell \in \Omega \), and \(\left| (\mathcal {A}_g^*c_0)(t)\right| < 1\) for all \(t \in \mathbb {R}\setminus T\). To this end, we seek \(\alpha , \beta \in \ell ^\infty (\Omega )\) such that
$$\begin{aligned} \begin{array}{l} (\mathcal {A}_g^*c_0)(t_\ell ) = \varepsilon _\ell \\ (\mathcal {A}_g^*c_0)'(t_\ell ) = 0, \end{array} \end{aligned}$$
(93)
for all \(\ell \in \Omega \). In developing an approach to solving the equation system (93), it will turn out convenient to define the operators
$$\begin{aligned} \begin{array}{llll} \mathcal {U}_p :&{} \ell ^\infty (\Omega ) &{} \longrightarrow &{} \ell ^\infty (\Omega ) \\ &{} \alpha = \{\alpha _\ell \}_{\ell \in \Omega } &{} \longmapsto &{} \left\{ \sum \limits _{m\in \Omega } \alpha _mu^{(p)}(t_\ell - t_m)\right\} _{\ell \in \Omega } \end{array} \end{aligned}$$
and
$$\begin{aligned} \begin{array}{llll} \mathcal {V}_p :&{} \ell ^\infty (\Omega ) &{} \longrightarrow &{} \ell ^\infty (\Omega ) \\ &{} \beta = \{\beta _\ell \}_{\ell \in \Omega } &{} \longmapsto &{} \left\{ \sum \limits _{m\in \Omega } \beta _mv^{(p)}(t_\ell - t_m)\right\} _{\ell \in \Omega }, \end{array} \end{aligned}$$
where \(p \in \{0, 1\}\). We defer the proof of \(\mathcal {U}_p\) and \(\mathcal {V}_p\), \(p \in \{0, 1\}\), mapping \(\ell ^\infty (\Omega )\) into \(\ell ^\infty (\Omega )\) to later. The equation system (93) can now be expressed as
$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {U}_0\alpha + \mathcal {V}_0\beta = \varepsilon \\ \mathcal {U}_1\alpha + \mathcal {V}_1\beta = 0. \end{array}\right. \end{aligned}$$
(94)
If both \(\mathcal {V}_1\) and \(\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1\) are invertible, then, as in [12], one can choose \(\alpha = (\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1)^{-1}\varepsilon \) and \(\beta = -\mathcal {V}_1^{-1}\mathcal {U}_1\alpha \) to satisfy (94). The Neumann expansion theorem [25, Thm. 1.3, p. 5] now says that \(\left\| \mathcal {I} - (v'(0))^{-1}\mathcal {V}_1\right\| < 1\) and \(\left\| \mathcal {I} - (\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1)\right\| < 1\) are sufficient conditions for \(\mathcal {V}_1\) and \(\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1\) to be invertible. We next verify these conditions.
1.2.1
\(\mathcal {V}_1\) is invertible
Fix a sequence \(\beta \in \ell ^\infty (\Omega )\), define \(\zeta = (\mathcal {I} - (v'(0))^{-1}\mathcal {V}_1)\beta \), and let \(\ell \in \Omega \). We then have
$$\begin{aligned} \zeta _\ell&= \beta _\ell - (v'(0))^{-1}\sum _{m\in \Omega } \beta _mv'(t_\ell - t_m) \\&= - (v'(0))^{-1}\sum _{m\in \Omega \setminus \{\ell \}} \beta _mv'(t_\ell - t_m) \\&= (R''(0))^{-1}\sum _{m\in \Omega \setminus \{\ell \}} \beta _mv'(t_\ell - t_m), \end{aligned}$$
where we used \(v'(0) = -R''(0) > 0\). With (92) and \(R''(0) = -\frac{\pi }{2\sigma ^2}\), we obtain
$$\begin{aligned} \left| \zeta _\ell \right|&\le \sum _{m\in \Omega \setminus \{\ell \}} \frac{2\sigma ^2}{\pi }\left| \beta _m\right| \left| v'(t_\ell - t_m)\right| \nonumber \\&\le \frac{2\sigma ^2}{\pi }\left\| \beta \right\| _{\ell ^\infty } \sum _{m\in \Omega \setminus \{\ell \}} V(t_\ell - t_m) \end{aligned}$$
(95)
$$\begin{aligned}&= \left\| \beta \right\| _{\ell ^\infty } \sum _{m\in \Omega \setminus \{\ell \}} \left( \frac{1}{\pi f_c \left| t_\ell - t_m\right| } + \frac{\left| t_\ell - t_m\right| }{4\sigma ^2f_c} + 1\right) R(t_\ell - t_m). \end{aligned}$$
(96)
We further upper-bound (96) using the same line of reasoning that led to (81). Specifically, we make use of the fact that V is non-increasing on \((0, \infty )\) and that the minimum distance between points in \(T\) is \(\Delta \). This implies that \(\sum _{m\in \Omega \setminus \{\ell \}} V(t_\ell - t_m)\) is maximized for
$$\begin{aligned} \big \{t_\ell - t_m:m\in \Omega \setminus \{\ell \}\big \} \subseteq \big \{n\Delta :n \in \mathbb {Z}\setminus \{0\}\big \}. \end{aligned}$$
With (92) this gives
$$\begin{aligned} \left| \zeta _\ell \right| \le 2\left\| \beta \right\| _{\ell ^\infty }\left( \sum _{n = 1}^\infty \frac{R(n\Delta )}{\pi f_c n\Delta } + \sum _{n = 1}^\infty \frac{n\Delta }{4\sigma ^2f_c}R(n\Delta ) + \sum _{n = 1}^\infty R(n\Delta )\right) . \end{aligned}$$
(97)
As \(n \ge 1\), we have \(R(n\Delta ) = \exp \!\left( -\frac{\pi n^2 \Delta ^2}{4\sigma ^2}\right) \le \exp \!\left( -\frac{\pi n \Delta ^2}{4\sigma ^2}\right) \), which when used in (97) leads to a further upper bound in terms of the following power series
$$\begin{aligned} \forall x \in (-1, 1), \quad \ln (1 - x) = -\sum _{n = 1}^\infty \frac{x^n}{n}, \qquad \frac{x}{(1 - x)^2} = \sum _{n = 1}^\infty nx^n, \qquad \frac{x}{1 - x} = \sum _{n = 1}^{\infty } x^n, \end{aligned}$$
all evaluated at \(x = \exp (-\frac{\pi \Delta ^2}{4\sigma ^2}) < 1\). Putting things together, we obtain
$$\begin{aligned} \left\| \mathcal {I} - (v'(0))^{-1}\mathcal {V}_1\right\| =&\sup _{\beta \ne 0} \frac{\left\| \zeta \right\| _{\ell ^\infty }}{\left\| \beta \right\| _{\ell ^\infty }} \le -\frac{2}{\pi f_c\Delta }\ln \left( 1 - \exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) \right) \\&+ \frac{\Delta \exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) }{2\sigma ^2f_c\left( 1 - \exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) \right) ^2} + \frac{2\exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) }{1 - \exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) }. \end{aligned}$$
Defining the functions
$$\begin{aligned} \forall x> 0, \quad \varphi (x)&:=-\ln \left( 1 - \exp \!\left( -\pi x^2\right) \right) \\ \forall x> 0, \quad \psi (x)&:=\frac{x^2 \exp \!\left( -\pi x^2\right) }{\left( 1 - \exp \!\left( -\pi x^2\right) \right) ^2} \\ \forall x > 0, \quad \xi (x)&= \frac{\exp \!\left( -\pi x^2\right) }{1 - \exp \!\left( -\pi x^2\right) }, \end{aligned}$$
we can then write
$$\begin{aligned} \left\| \mathcal {I} - (v'(0))^{-1}\mathcal {V}_1\right\| \le \frac{\frac{2}{\pi }\varphi \!\left( \frac{\Delta }{2\sigma }\right) + 2\psi \!\left( \frac{\Delta }{2\sigma }\right) }{f_c\Delta } + 2\xi \!\left( \frac{\Delta }{2\sigma }\right) . \end{aligned}$$
The functions \(\varphi \) and \(\xi \) are non-increasing on \((0, \infty )\), as their derivatives satisfy
$$\begin{aligned} \forall x > 0, \qquad \varphi '(x) = \frac{-2\pi x \exp \!\left( -\pi x^2\right) }{1 - \exp \!\left( -\pi x^2\right) } \le 0 \end{aligned}$$
$$\begin{aligned} \forall x > 0, \quad \xi '(x) = -\frac{2\pi x\exp (-\pi x^2)}{1 - \exp \!\left( -\pi x^2\right) } - \frac{2\pi x\exp \!\left( -2\pi x^2\right) }{\left( 1 - \exp \!\left( -\pi x^2\right) \right) ^2} \le 0. \end{aligned}$$
As for \(\psi \), we first write
$$\begin{aligned} \forall x > 0, \qquad \psi (x) = \left( \frac{x\exp (-\pi x^2/2)}{1 - \exp (-\pi x^2)}\right) ^2 = \left( \frac{x}{2\sinh (\pi x^2/2)}\right) ^2, \end{aligned}$$
and then show that the function
$$\begin{aligned} \forall x > 0, \quad \eta (x) :=\frac{x}{2\sinh (\pi x^2/2)} \end{aligned}$$
is non-increasing on \((0,\infty )\) by computing its first derivative:
$$\begin{aligned} \forall x > 0, \quad \eta '(x)&= \frac{2\sinh (\pi x^2/2) - 2\pi x^2\cosh (\pi x^2/2)}{\left( 2\sinh (\pi x^2/2)\right) ^2} \\&= \frac{\cosh (\pi x^2/2)\left( \tanh (\pi x^2/2) - \pi x^2\right) }{2\left( \sinh (\pi x^2/2)\right) ^2} \le 0, \end{aligned}$$
where the inequality is thanks to \(\tanh (\pi x^2/2) \le \pi x^2/2 \le \pi x^2\), for all \(x > 0\). Therefore, the function \(\psi \) is also non-increasing on \((0, \infty )\). Since by assumption \(\Delta > 4\sigma \) and \(\Delta > 1/f_c\), we get
$$\begin{aligned} \left\| \mathcal {I} - (v'(0))^{-1}\mathcal {V}_1\right\|&\le \frac{\frac{2}{\pi }\varphi (2) + 2\psi (2)}{f_c\Delta } + 2\xi (2) \\&< \frac{2\varphi (2)}{\pi } + 2\psi (2) + 2\xi (2) \le 3.71 \cdot 10^{-5} < 1. \end{aligned}$$
It therefore follows that \(\mathcal {V}_1\) is invertible. Furthermore, according to the Neumann expansion theorem, the operator norm of \(\mathcal {V}_1^{-1}\) satisfies
$$\begin{aligned} \left\| \mathcal {V}_1^{-1}\right\| \le \frac{\left| v'(0)\right| ^{-1}}{1 - \left\| \mathcal {I} - (v'(0))^{-1}\mathcal {V}_1\right\| } \le \frac{2\sigma ^2}{\pi - 2(\varphi (2) + \pi \psi (2) + \pi \xi (2))}. \end{aligned}$$
(98)
1.2.2
\(\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1\) is invertible
We start by noting that thanks to the triangle inequality,
$$\begin{aligned} \left\| \mathcal {I} - (\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1)\right\|&\le \left\| \mathcal {I} - \mathcal {U}_0\right\| + \left\| \mathcal {V}_0\right\| \left\| \mathcal {V}_1^{-1}\right\| \left\| \mathcal {U}_1\right\| \nonumber \\&\le \left\| \mathcal {I} - \mathcal {U}_0\right\| + \frac{\left| v'(0)\right| ^{-1} \left\| \mathcal {V}_0\right\| \left\| \mathcal {U}_1\right\| }{1 - \left\| \mathcal {I} - (v'(0))^{-1}\mathcal {V}_1\right\| }. \end{aligned}$$
(99)
An upper bound on \(\left\| \mathcal {I} - \mathcal {U}_0\right\| \) can easily be derived using arguments similar to those employed in Appendix Section “\(\mathcal {V}_1\) is invertible” to get an upper bound on \(\left\| \mathcal {I} - (v'(0))^{-1}\mathcal {V}_1\right\| \). Specifically, for \(\alpha \in \ell ^\infty (\Omega )\), the sequence \(\zeta = (\mathcal {I} - \mathcal {U}_0)\alpha \) obeys
$$\begin{aligned} \left| \zeta _\ell \right|&\le \sum _{m\in \Omega \setminus \{\ell \}} \!\!\left| \alpha _m\right| \left| u(t_\ell - t_m)\right| \le \left\| \alpha \right\| _{\ell ^\infty }\!\!\!\sum _{m\in \Omega \setminus \{\ell \}} \frac{1}{2\pi f_c\left| t_\ell - t_m\right| }R(t_\ell - t_m) \\&\le 2\left\| \alpha \right\| _{\ell ^\infty } \sum _{n = 1}^\infty \frac{R(n\Delta )}{2\pi f_c n\Delta } \\&\le 2\left\| \alpha \right\| _{\ell ^\infty } \sum _{n = 1}^\infty \frac{1}{2\pi f_c n\Delta }\exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) = -\frac{\left\| \alpha \right\| _{\ell ^\infty }}{\pi f_c\Delta } \ln \left( 1 - \exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) \right) , \end{aligned}$$
for all \(\ell \in \Omega \), where we used
$$\begin{aligned} \forall t \in \mathbb {R}{\setminus } \{0\}, \quad \left| u(t)\right| \le \frac{R(t)}{2\pi f_c\left| t\right| }. \end{aligned}$$
This implies that
$$\begin{aligned} \left\| \mathcal {I} - \mathcal {U}_0\right\| \le \frac{\varphi \!\left( \frac{\Delta }{2\sigma }\right) }{\pi f_c\Delta } \le \frac{\varphi (2)}{\pi f_c\Delta } < \frac{\varphi (2)}{\pi }. \end{aligned}$$
(100)
Next, we compute an upper bound on \(\left\| \mathcal {U}_1\right\| \). To this end, we fix \(\alpha \in \ell ^\infty (\Omega )\) and set \(\zeta = \mathcal {U}_1\alpha \). Since \(R'(0) = 0\) and \({{\mathrm{sinc}}}'(0) = 0\), we have \(u'(0) = 0\), which, combined with (91) gives, for all \(\ell \in \Omega \),
$$\begin{aligned} \left| \zeta _\ell \right|&\le \sum _{m\in \Omega } \left| \alpha _m\right| \left| u'(t_\ell - t_m)\right| = \sum _{m\in \Omega \setminus \{\ell \}} \left| \alpha _m\right| \left| u'(t_\ell - t_m)\right| \nonumber \\&\le \left\| \alpha \right\| _{\ell ^\infty } \sum _{m\in \Omega \setminus \{\ell \}} \left( \frac{1}{4\sigma ^2f_c} + \frac{1}{\left| t_\ell - t_m\right| } + \frac{1}{2\pi f_c\left| t_\ell - t_m\right| ^2}\right) R(t_\ell - t_m) \nonumber \\&\le 2\left\| \alpha \right\| _{\ell ^\infty } \left( \frac{1}{4f_c\sigma ^2}\sum _{n = 1}^{\infty }R(n\Delta ) + \sum _{n = 1}^{\infty } \frac{R(n\Delta )}{n\Delta } + \sum _{n = 1}^{\infty } \frac{R(n\Delta )}{2\pi f_cn^2\Delta ^2} \right) \nonumber \\&\le \left\| \alpha \right\| _{\ell ^\infty } \left[ \frac{1}{2f_c\sigma ^2} \sum _{n = 1}^{\infty } \exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) + \left( \frac{2}{\Delta }+ \frac{1}{\pi f_c\Delta ^2}\right) \sum _{n = 1}^{\infty } \frac{1}{n}\exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) \right] \\&= \left\| \alpha \right\| _{\ell ^\infty }\left[ \frac{\exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) }{2f_c\sigma ^2\left( 1 - \exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) \right) } - \left( \frac{2}{\Delta }+ \frac{1}{\pi f_c\Delta ^2}\right) \ln \left( 1 - \exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) \right) \right] , \nonumber \end{aligned}$$
(101)
where in (101) we used the fact that for \(n \ge 1\), \(1/n^2 \le 1/n\) and \(R(n\Delta ) = \exp \!\left( -\frac{\pi n^2\Delta ^2}{4\sigma ^2}\right) \le \exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) \). Based on the upper bound (101) we can now conclude that
$$\begin{aligned} \left\| \mathcal {U}_1\right\| \le \frac{\xi \!\left( \frac{\Delta }{2\sigma }\right) }{2f_c\sigma ^2} + \varphi \!\left( \frac{\Delta }{2\sigma ^2}\right) \left( \frac{2}{\Delta }+ \frac{1}{\pi f_c\Delta ^2}\right) . \end{aligned}$$
Setting
$$\begin{aligned} \forall x > 0, \quad \rho (x) :=x^2\xi (x) = \frac{x^2\exp \!\left( -\pi x^2\right) }{1 - \exp \!\left( -\pi x^2\right) }, \end{aligned}$$
(102)
we can rewrite (102) as
$$\begin{aligned} \left\| \mathcal {U}_1\right\| \le \frac{2\rho \left( \frac{\Delta }{2\sigma }\right) }{f_c\Delta ^2} + \varphi \!\left( \frac{\Delta }{2\sigma ^2}\right) \left( \frac{2}{\Delta }+ \frac{1}{\pi f_c\Delta ^2}\right) . \end{aligned}$$
We can verify that \(\rho \) is non-increasing on \((0, \infty )\), which finally yields
$$\begin{aligned} \left\| \mathcal {U}_1\right\| \le \frac{2\rho \left( 2\right) }{f_c\Delta ^2} + \varphi \!\left( 2\right) \left( \frac{2}{\Delta }+ \frac{1}{\pi f_c\Delta ^2}\right) \le \frac{2\rho (2) + (2 + 1/\pi )\varphi (2)}{4\sigma }. \end{aligned}$$
(103)
It remains to upper-bound \(\left\| \mathcal {V}_0\right\| \). To this end, we fix \(\beta \in \ell ^\infty (\Omega )\) and define \(\zeta = \mathcal {V}_0\beta \). As \(v(0) = 0\) and
$$\begin{aligned} \forall t \in \mathbb {R}\!\setminus \!\{0\}, \quad \left| v(t)\right| \le \frac{R(t)}{4\sigma ^2f_c}, \end{aligned}$$
we get
$$\begin{aligned} \left| \zeta _\ell \right|&\le \sum _{m\in \Omega } \left| \beta _m\right| \left| v(t_\ell - t_m)\right| = \sum _{m\in \Omega \setminus \{\ell \}} \left| \beta _m\right| \left| v(t_\ell - t_m)\right| \\&\le \left\| \beta \right\| _{\ell ^\infty }\sum _{m\in \Omega \setminus \{\ell \}} \frac{R(t_\ell - t_m)}{4\sigma ^2f_c} \le \frac{\left\| \beta \right\| _{\ell ^\infty }}{2\sigma ^2f_c}\sum _{n = 1}^\infty R(n\Delta ) \\&\le \frac{\left\| \beta \right\| _{\ell ^\infty }}{2f_c\sigma ^2}\sum _{n = 1}^\infty \exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) = \frac{\left\| \beta \right\| _{\ell ^\infty }}{2f_c\sigma ^2} \frac{\exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) }{1 - \exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) }, \end{aligned}$$
for all \(\ell \in \Omega \). This yields
$$\begin{aligned} \left\| \mathcal {V}_0\right\| \le \frac{\xi \!\left( \frac{\Delta }{2\sigma }\right) }{2f_c\sigma ^2} = \frac{\rho \!\left( \frac{\Delta }{2\sigma }\right) }{2\sigma } \le \frac{\rho (2)}{2\sigma }, \end{aligned}$$
(104)
where the last inequality follows from \(\Delta > 1/f_c\), \(\Delta > 4\sigma \), and the fact that \(\rho \) is non-increasing on \((0, \infty )\). Finally, using (98), (100), (103), and (104) in (99), we obtain
$$\begin{aligned} \left\| \mathcal {I} - \mathcal {W}\right\| \le \frac{\varphi (2)}{\pi } + \frac{\rho (2)\left[ \rho (2) + (1 + 1/(2\pi ))\varphi (2)\right] }{2\pi - 4(\varphi (2) + \pi \psi (2) + \pi \xi (2))} \le 1.12 \cdot 10^{-6} < 1, \end{aligned}$$
where \(\mathcal {W} :=\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1\). Again, applying the Neumann expansion theorem, we can conclude that the operator \(\mathcal {W} = \mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1\) is invertible and that its inverse satisfies
$$\begin{aligned} \left\| \mathcal {W}^{-1}\right\| \le \frac{1}{1 - \left\| \mathcal {I} - \mathcal {W}\right\| } \le \left( 1 - \frac{\varphi (2)}{\pi } - \frac{\rho (2)\left[ \rho (2) + (1 + 1/(2\pi ))\varphi (2)\right] }{2\pi - 4(\varphi (2) + \pi \psi (2) + \pi \xi (2))}\right) ^{-1}. \end{aligned}$$
(105)
For later use, we record that for the choices \(\alpha = (\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1)^{-1}\varepsilon \) and \(\beta = -\mathcal {V}_1^{-1}\mathcal {U}_1\alpha \), we have
$$\begin{aligned} \left\| \alpha \right\| _{\ell ^\infty }&\le \left\| \mathcal {W}^{-1}\right\| \left\| \varepsilon \right\| _{\ell ^\infty } = \left\| \mathcal {W}^{-1}\right\| \nonumber \\&\le \left( 1 - \frac{\varphi (2)}{\pi } - \frac{\rho (2)\left[ \rho (2) + (1 + 1/(2\pi ))\varphi (2)\right] }{2\pi - 4(\varphi (2) + \pi \psi (2) + \pi \xi (2))}\right) ^{-1} \le 1.01 \end{aligned}$$
(106)
and
$$\begin{aligned} \left\| \beta \right\| _{\ell ^\infty }&\le \left\| \mathcal {V}_1^{-1}\right\| \left\| \mathcal {U}_1\right\| \left\| \alpha \right\| _{\ell ^\infty } \le \frac{2\sigma ^2}{\pi - 2(\varphi (2) + \pi \psi (2) + \pi \xi (2))}\nonumber \\&\cdot \frac{2\rho (2) + (2 + 1/\pi )\varphi (2)}{4\sigma } \nonumber \\& \cdot \left( 1 - \frac{\varphi (2)}{\pi } - \frac{\rho (2)\left[ \rho (2) + (1 + 1/(2\pi ))\varphi (2)\right] }{2\pi - 4(\varphi (2) + \pi \psi (2) + \pi \xi (2))}\right) ^{-1} \le 5.73 \cdot 10^{-6}\sigma . \end{aligned}$$
(107)
In the remainder of the proof, we exclusively consider \(c_0\) with \(\alpha = (\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1)^{-1}\varepsilon \) and \(\beta = -\mathcal {V}_1^{-1}\mathcal {U}_1\alpha \).
1.3
\(\left| (\mathcal {A}_g^*c_0)(t)\right| < 1\) for all \(t \in \mathbb {R}{\setminus } T\)
1.3.1
\(\left| (\mathcal {A}_g^*c_0)(t)\right| < 1\) for all \(t \in \mathbb {R}{\setminus } \bigcup _{\ell \in T} \left[ t_\ell - \frac{1}{7f_c}, t_\ell + \frac{1}{7f_c}\right] \)
Take \(\ell _0 \in \Omega \) and let \(\ell _1 \in \Omega \) be the index of the point in \(T\) that is closest to \(t_{\ell _0}\) and satisfies \(t_{\ell _1} > t_{\ell _0}\). Take \(t \in \left( t_{\ell _0} + \frac{1}{7f_c}, t_{\ell _1} - \frac{1}{7f_c}\right) \) and note that the interval \(\left( t_{\ell _0} + \frac{1}{7f_c}, t_{\ell _1} - \frac{1}{7f_c}\right) \) is non-empty because \(\left| t_{\ell _0} - t_{\ell _1}\right| \ge \Delta> \frac{1}{f_c} > \frac{2}{7f_c}\). Without loss of generality, we assume that \(\left| t - t_{\ell _0}\right| \le \left| t - t_{\ell _1}\right| \), which implies \(\left| t - t_{\ell _1}\right| \ge \left| t_{\ell _1} - t_{\ell _0}\right| /2 \ge \Delta /2\). We set \(h :=\left| t - t_{\ell _0}\right| \) and note that \(h \ge \frac{1}{7f_c}\). The following holds
$$\begin{aligned} \left| (\mathcal {A}_g^*c_0)(t)\right|&\le \sum _{\ell \in \Omega } \Big (\left\| \alpha \right\| _{\ell ^\infty } \left| u(t - t_\ell )\right| + \left\| \beta \right\| _{\ell ^\infty } \left| v(t - t_\ell )\right| \Big ) \nonumber \\&\le \sum _{\ell \in \Omega } \left( \left\| \alpha \right\| _{\ell ^\infty } \frac{R(t - t_\ell )}{2\pi f_c \left| t - t_\ell \right| } + \left\| \beta \right\| _{\ell ^\infty } \frac{R(t - t_\ell )}{4\sigma ^2 f_c} \right) \nonumber \\&\le \left\| \alpha \right\| _{\ell ^\infty } \frac{R(h)}{2\pi f_c h} + \left\| \beta \right\| _{\ell ^\infty } \frac{R(h)}{4\sigma ^2f_c} + \left\| \alpha \right\| _{\ell ^\infty } \frac{R\left( \frac{\Delta }{2}\right) }{\pi f_c\Delta } + \left\| \beta \right\| _{\ell ^\infty }\frac{R\left( \frac{\Delta }{2}\right) }{4\sigma ^2f_c} \nonumber \\&\qquad \quad + \sum _{\ell \in \Omega \setminus \{\ell _0, \ell _1\}} \left( \left\| \alpha \right\| _{\ell ^\infty } \frac{R(t - t_\ell )}{2\pi f_c \left| t - t_\ell \right| } + \left\| \beta \right\| _{\ell ^\infty } \frac{R(t - t_\ell )}{4\sigma ^2 f_c} \right) \nonumber \\&\le \left\| \alpha \right\| _{\ell ^\infty } \frac{R(h)}{2\pi f_c h} + \left\| \beta \right\| _{\ell ^\infty } \frac{R(h)}{4\sigma ^2f_c} + \left\| \alpha \right\| _{\ell ^\infty } \frac{R\left( \frac{\Delta }{2}\right) }{\pi f_c\Delta } + \left\| \beta \right\| _{\ell ^\infty }\frac{R\left( \frac{\Delta }{2}\right) }{4\sigma ^2f_c} \nonumber \\&\qquad \quad + 2\sum _{n =1}^\infty \left( \left\| \alpha \right\| _{\ell ^\infty } \frac{R(n\Delta )}{2\pi f_c n\Delta } + \left\| \beta \right\| _{\ell ^\infty } \frac{R(n\Delta )}{4\sigma ^2 f_c}\right) \nonumber \\&\le \left\| \alpha \right\| _{\ell ^\infty }\frac{R\left( \frac{1}{7f_c}\right) }{2\pi /7} + \left\| \beta \right\| _{\ell ^\infty }\frac{R\left( \frac{1}{7f_c}\right) }{4\sigma ^2f_c} + \left\| \alpha \right\| _{\ell ^\infty } \frac{R\left( \frac{\Delta }{2}\right) }{\pi f_c\Delta } \nonumber \\&\qquad \quad + \left\| \beta \right\| _{\ell ^\infty }\frac{R\left( \frac{\Delta }{2}\right) }{4\sigma ^2f_c}\end{aligned}$$
(108)
$$\begin{aligned}&\qquad \quad + \frac{\left\| \alpha \right\| _{\ell ^\infty }}{\pi f_c\Delta }\varphi \!\left( \frac{\Delta }{2\sigma }\right) + \frac{\left\| \beta \right\| _{\ell ^\infty }}{2\sigma ^2f_c}\xi \!\left( \frac{\Delta }{2\sigma }\right) \end{aligned}$$
(109)
$$\begin{aligned} &\le \left\| \alpha \right\| _{\ell ^\infty } \frac{7 \exp \!\left( -\frac{4\pi }{49}\right) }{2\pi }+ \left\| \beta \right\| _{\ell ^\infty }\frac{\exp \!\left( -\frac{4\pi }{49}\right) }{\sigma } + \left\| \alpha \right\| _{\ell ^\infty }\frac{\exp (-\pi )}{\pi } \end{aligned}$$
(110)
$$\begin{aligned}&\qquad \quad + \left\| \beta \right\| _{\ell ^\infty } \frac{\exp (-\pi )}{\sigma } + \left\| \alpha \right\| _{\ell ^\infty }\frac{\varphi (2)}{\pi } + \left\| \beta \right\| _{\ell ^\infty }\frac{2\xi (2)}{\sigma } \le 0.876 < 1, \end{aligned}$$
(111)
where (108) and (109) follow from \(h \ge \frac{1}{7f_c}\), and (110) and (111) can be derived invoking the assumptions \(\Delta > 1/f_c\) and \(\sigma = \frac{1}{4f_c}\).
1.4
\(\left| \mathcal {A}_g^*c_0\right| \) is concave on \(\bigcup _{\ell \in \Omega } \left[ t_\ell - \frac{1}{7f_c}, t_\ell + \frac{1}{7f_c}\right] \)
Let \(\ell \in \Omega \). We show that \(t \mapsto A(t) :=\left| (\mathcal {A}_g^*c_0)(t)\right| \) is strictly concave on \(\left[ t_\ell - \frac{1}{7f_c}, t_\ell + \frac{1}{7f_c}\right] \). Since \(\left| R\right| \), \(\left| {{\mathrm{sinc}}}\right| \), \(\left| R'\right| \), and \(\left| {{\mathrm{sinc}}}'\right| \) are all symmetric, A is symmetric as well, and therefore, it suffices to show that \(A''(t) < 0\) for \(t \in \left[ t_\ell , t_\ell + \frac{1}{7f_c}\right] \). Since \(\left| \varepsilon _\ell \right| = 1\), we can write
$$\begin{aligned} \forall t \in \mathbb {R}, \quad A(t) = \left| (\mathcal {A}_g^*c_0)(t)\right| = \left| \frac{(\mathcal {A}_g^*c_0)(t)}{\varepsilon _\ell }\right| = \sqrt{(A_R(t))^2 + (A_I(t))^2} \end{aligned}$$
where \(A_R(t) :={{\mathrm{Re}}}\left\{ \frac{(\mathcal {A}_g^*c_0)(t)}{\varepsilon _\ell }\right\} \), \(A_I(t) :=\hbox {Im}\left\{ \frac{(\mathcal {A}_g^*c_0)(t)}{\varepsilon _\ell }\right\} \), for \(t \in \mathbb {R}\). With \(\Lambda :=\{t \in \mathbb {R}:A(t) \ne 0\}\) we have
$$\begin{aligned}&\forall t \in \Lambda , \quad A''(t) = \frac{A_R''(t)A_R(t) + A_I''(t)A_I(t) + \left| (\mathcal {A}_g^*c_0)'(t)\right| ^2}{A(t)}\\&\quad - \frac{(A_R'(t)A_R(t) + A_I'(t)A_I(t))^2}{(A(t))^3}. \end{aligned}$$
For A to be concave on \(\left[ t_\ell , t_\ell + \frac{1}{7f_c}\right] \), it therefore suffices to show that
$$\begin{aligned} \forall t \in \left[ t_\ell , t_\ell + \frac{1}{7f_c}\right] , \quad A_R''(t)A_R(t) + A_I''(t)A_I(t) + \left| (\mathcal {A}_g^*c_0)'(t)\right| ^2 < 0. \end{aligned}$$
Let \(t \in \left[ t_\ell , t_\ell + \frac{1}{7f_c}\right] \). We have the following
$$\begin{aligned} A_R(t)&= \sum _{m\in \Omega } \left( {{\mathrm{Re}}}\left\{ \frac{\alpha _m}{\varepsilon _\ell }\right\} u(t - t_m) + {{\mathrm{Re}}}\left\{ \frac{\beta _m}{\varepsilon _\ell }\right\} v(t - t_m)\right) \\&= {{\mathrm{Re}}}\left\{ \frac{\alpha _\ell }{\varepsilon _\ell }\right\} u(t - t_\ell ) + {{\mathrm{Re}}}\left\{ \frac{\beta _\ell }{\varepsilon _\ell }\right\} v(t - t_\ell ) + \sum _{m\in \Omega \setminus \{\ell \}} \left( {{\mathrm{Re}}}\left\{ \frac{\alpha _m}{\varepsilon _\ell }\right\} u(t - t_m) \right. \\& \left. + {{\mathrm{Re}}}\left\{ \frac{\beta _m}{\varepsilon _\ell }\right\} v(t - t_m) \right) \\ &\ge {{\mathrm{Re}}}\left\{ \frac{\alpha _\ell }{\varepsilon _\ell }\right\} u(t - t_\ell ) - \left\| \beta \right\| _{\ell ^\infty }\left| v(t - t_\ell )\right| - \sum _{m\in \Omega \setminus \{\ell \}} \Big (\left\| \alpha \right\| _{\ell ^\infty } \left| u(t - t_m)\right| \\&+ \left\| \beta \right\| _{\ell ^\infty } \left| v(t - t_m)\right| \Big ). \end{aligned}$$
With \(\alpha = \mathcal {W}^{-1}\varepsilon = \varepsilon - (\mathcal {I} - \mathcal {W}^{-1})\varepsilon \), it follows that
$$\begin{aligned} {{\mathrm{Re}}}\left\{ \frac{\alpha _\ell }{\varepsilon _\ell }\right\}&= 1 - {{\mathrm{Re}}}\left\{ \frac{\left[ (\mathcal {I} - \mathcal {W}^{-1})\varepsilon \right] _\ell }{\varepsilon _\ell }\right\} \ge 1 - \left| \frac{\left[ (\mathcal {I} - \mathcal {W}^{-1})\varepsilon \right] _\ell }{\varepsilon _\ell }\right| \nonumber \\&\ge 1 - \left\| \mathcal {I} - \mathcal {W}^{-1}\right\| \nonumber \\&= 1 - \left\| \mathcal {W}^{-1}(\mathcal {I} - \mathcal {W})\right\| \ge 1 - \left\| \mathcal {W}^{-1}\right\| \left\| \mathcal {I} - \mathcal {W}\right\| \ge 0.999998, \end{aligned}$$
(112)
where (112) is due to (105). Next, it follows from \(t - t_\ell \le \frac{1}{7f_c}\) and \(\sigma = \frac{1}{4f_c}\) that
$$\begin{aligned} u(t - t_\ell ) = R(t - t_\ell ){{\mathrm{sinc}}}(2\pi f_c(t - t_\ell ))&= \exp \!\left( -\frac{\pi (t - t_\ell )^2}{4\sigma ^2}\right) {{\mathrm{sinc}}}(2\pi f_c(t - t_\ell )) \nonumber \\&\ge \exp \!\left( -\frac{\pi }{49f_c^2 \cdot 4\sigma ^2}\right) {{\mathrm{sinc}}}\!\left( \frac{2\pi }{7}\right) \nonumber \\&= \exp \!\left( -\frac{4\pi }{49}\right) {{\mathrm{sinc}}}\!\left( \frac{2\pi }{7}\right) . \end{aligned}$$
(113)
Since \(\left| {{\mathrm{sinc}}}\right| \le 1\), we have
$$\begin{aligned} \left\| \beta \right\| _{\ell ^\infty }\left| v(t - t_\ell )\right| \le \left\| \beta \right\| _{\ell ^\infty }|R'(t_\ell - t)|\left| {{\mathrm{sinc}}}(2\pi f_c(t - t_\ell ))\right| \le \left\| \beta \right\| _{\ell ^\infty }\left| R'(t - t_\ell )\right| . \end{aligned}$$
As \(\left| R'\right| \) has its maxima at the points \(\pm \sigma \sqrt{\frac{2}{\pi }}\) with corresponding maximum values \(\frac{1}{\sigma }\exp \!\left( -\frac{1}{2}\right) \sqrt{\frac{\pi }{2}}\), we get
$$\begin{aligned} \left\| \beta \right\| _{\ell ^\infty }\left| v(t - t_\ell )\right| \le \frac{\left\| \beta \right\| _{\ell ^\infty }}{\sigma }\exp \!\left( -\frac{1}{2}\right) \sqrt{\frac{\pi }{2}}. \end{aligned}$$
(114)
As for every \(m\in \Omega \setminus \{\ell \}\), we have \(\left| t - t_m\right| \ge \left| t_\ell - t_m\right| - \left| t - t_\ell \right| \ge \Delta - \frac{1}{7f_c} > \frac{6}{7f_c}\), it holds that
$$\begin{aligned} \sum _{m\in \Omega \setminus \{\ell \}}&\Big (\left\| \alpha \right\| _{\ell ^\infty }\left| u(t - t_m)\right| + \left\| \beta \right\| _{\ell ^\infty }\left| v(t - t_m)\right| \Big ) \le \left\| \alpha \right\| _{\ell ^\infty }\frac{7R\!\left( \frac{6}{7f_c}\right) }{12\pi }\nonumber \\&\quad + \left\| \beta \right\| _{\ell ^\infty }\frac{R\!\left( \frac{6}{7f_c}\right) }{4\sigma ^2f_c} + 2\sum _{n = 1}^\infty \left( \left\| \alpha \right\| _{\ell ^\infty } \frac{R(n\Delta )}{2\pi f_c n\Delta } + \left\| \beta \right\| _{\ell ^\infty } \frac{R(n\Delta )}{4\sigma ^2f_c}\right) \nonumber \\&\le \left\| \alpha \right\| _{\ell ^\infty } \frac{7\exp \!\left( -\pi \left( \frac{12}{7}\right) ^2\right) }{12\pi } \nonumber \\&\quad \quad \ + \frac{\left\| \beta \right\| _{\ell ^\infty }}{\sigma }\exp \!\left( -\pi \left( \frac{12}{7}\right) ^2\right) + \left\| \alpha \right\| _{\ell ^\infty }\frac{\varphi (2)}{\pi } + \frac{\left\| \beta \right\| _{\ell ^\infty }}{\sigma } 2\xi (2). \end{aligned}$$
(115)
Combining (106), (107), (112), (113), (114), and (115) yields
$$\begin{aligned} A_R(t) \ge 0.673. \end{aligned}$$
(116)
Next, we derive an upper bound on \(A_R''(t)\):
$$\begin{aligned} A_R''(t)&= \sum _{m\in \Omega } \left( {{\mathrm{Re}}}\left\{ \frac{\alpha _m}{\varepsilon _m}\right\} u''(t - t_m) + {{\mathrm{Re}}}\left\{ \frac{\beta _m}{\varepsilon _m}\right\} v''(t - t_m)\right) \\&\le {{\mathrm{Re}}}\left\{ \frac{\alpha _\ell }{\varepsilon _\ell }\right\} u''(t - t_\ell ) + \left\| \beta \right\| _{\ell ^\infty }\left| v''(t - t_\ell )\right| \\&\quad + \sum _{m\in \Omega \setminus \{\ell \}} \Big (\left\| \alpha \right\| _{\ell ^\infty }\left| u''(t - t_m)\right| + \left\| \beta \right\| _{\ell ^\infty } \left| v''(t - t_m)\right| \Big ). \end{aligned}$$
For all \(t \in \mathbb {R}\), we have
$$\begin{aligned} u''(t) = R''(t){{\mathrm{sinc}}}(2\pi f_c t) + 4\pi f_cR'(t){{\mathrm{sinc}}}'(2\pi f_c t) + (2\pi f_c)^2R(t){{\mathrm{sinc}}}''(2\pi f_c t). \end{aligned}$$
(117)
The function \(t \mapsto R''(t){{\mathrm{sinc}}}(2\pi f_c t)\) is non-decreasing on \(\big [0, \frac{1}{7f_c}\big ]\), since, on this interval, \(R''\) is negative and non-decreasing and \(t \mapsto {{\mathrm{sinc}}}(2\pi f_c t)\) is positive and non-increasing. The function \(t \mapsto 4\pi f_cR'(t){{\mathrm{sinc}}}'(2\pi f_c t)\) is non-decreasing on \(\big [0, \frac{1}{7f_c}\big ]\), as both \(R'\) and \(t \mapsto {{\mathrm{sinc}}}'(2\pi f_c t)\) are negative and non-increasing on this interval. The function \(t \mapsto (2\pi f_c)^2R(t){{\mathrm{sinc}}}''(2\pi f_c t)\) is non-decreasing on \(\big [0, \frac{1}{7f_c}\big ]\), as, on this interval, \(R\) is positive and non-increasing, and \(t \mapsto {{\mathrm{sinc}}}''(2\pi f_c t)\) is negative and non-decreasing. Taken together, it follows that \(u''\) is non-decreasing on \(\big [0, \frac{1}{7f_c}\big ]\). Since \(t - t_\ell \in \big [0, \frac{1}{7f_c}\big ]\), we then have
$$\begin{aligned} u''(t - t_\ell )&\le u''\!\left( \frac{1}{7f_c}\right) = f_c^2\left[ 8\pi \left( \frac{8\pi }{49} - 1\right) {{\mathrm{sinc}}}\left( \frac{2\pi }{7}\right) - \frac{32\pi ^2}{7}{{\mathrm{sinc}}}'\left( \frac{2\pi }{7}\right) \right. \\& \left. + 4\pi ^2{{\mathrm{sinc}}}''\left( \frac{2\pi }{7}\right) \right] \exp \!\left( -\frac{4\pi }{49}\right) , \end{aligned}$$
where we used \(\sigma = \frac{1}{4f_c}\). Combined with (112), this yields
$$\begin{aligned} {{\mathrm{Re}}}\left\{ \frac{\alpha _\ell }{\varepsilon _\ell }\right\} u''(t - t_\ell ) \le -6.46 f_c^2. \end{aligned}$$
Since
$$\begin{aligned}&\forall x \in \mathbb {R}\setminus \{0\}, \quad {{\mathrm{sinc}}}'(x) = \frac{\cos (x)}{x} - \frac{\sin (x)}{x^2} \quad \text { and }\qquad \\&{{\mathrm{sinc}}}''(x) = -\frac{\sin (x)}{x} - \frac{2\cos (x)}{x^2} + \frac{2\sin (x)}{x^3}, \end{aligned}$$
we get the following from (117):
$$\begin{aligned} \left| u''(t)\right|&\le \left( \frac{\pi }{2\sigma ^2} + \frac{\pi ^2\left| t\right| ^2}{4\sigma ^4}\right) \frac{1}{2\pi f_c \left| t\right| }R(t)+ 4\pi f_c\frac{\pi \left| t\right| }{2\sigma ^2}\left( \frac{1}{2\pi f_c\left| t\right| } + \frac{1}{(2\pi f_c)^2\left| t\right| ^2}\right) R(t) \\&\quad + (2\pi f_c)^2 \left( \frac{1}{2\pi f_c\left| t\right| } + \frac{2}{(2\pi f_c)^2\left| t\right| ^2} + \frac{2}{(2\pi f_c)^3\left| t\right| ^3} \right) R(t) \\&= \left[ \frac{\pi \left| t\right| }{8f_c\sigma ^4} + \frac{\pi }{\sigma ^2} + \left( 2\pi f_c + \frac{3}{4\sigma ^2f_c}\right) \frac{1}{\left| t\right| } + \frac{2}{\left| t\right| ^2} + \frac{1}{\pi f_c\left| t\right| ^3}\right] R(t). \end{aligned}$$
As a result, we have the following chain of inequalities
$$\begin{aligned} \sum _{m\in \Omega \setminus \{\ell \}} \left| u''(t - t_m)\right|&\le 2\sum _{n = 1}^\infty \left[ \pi \frac{n\Delta }{8f_c\sigma ^4} + \frac{\pi }{\sigma ^2} + \left( 2\pi f_c + \frac{3}{4\sigma ^2f_c}\right) \frac{1}{n\Delta }\right. \\&\quad \left. + \frac{2}{n^2\Delta ^2} + \frac{1}{\pi f_c n^3\Delta ^3}\right] R(n\Delta ) \\&\le \frac{\pi \Delta }{4\sigma ^4f_c}\sum _{n = 1}^\infty n\exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) + \frac{2\pi }{\sigma ^2}\sum _{n = 1}^\infty \exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) \\&\quad + \left( \frac{4\pi f_c}{\Delta } + \frac{3}{2\sigma ^2f_c\Delta }\right) \sum _{n = 1}^\infty \frac{1}{n}\exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) \nonumber \\&\quad + \frac{4}{\Delta ^2}\sum _{n = 1}^\infty \frac{1}{n}\exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) + \frac{2}{\pi f_c\Delta ^3}\sum _{n = 1}^\infty \frac{1}{n}\exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) \\&\!\!\le \frac{\pi \Delta }{4\sigma ^4f_c}\psi \!\left( \frac{\Delta }{2\sigma }\!\right) + \frac{2\pi }{\sigma ^2}\xi \!\left( \frac{\Delta }{2\sigma }\!\right) + \left( \!\frac{4\pi f_c}{\Delta } + \frac{3}{2\sigma ^2f_c\Delta }\!\right) \varphi \!\left( \frac{\Delta }{2\sigma }\!\right) \\&\quad + \frac{4}{\Delta ^2}\varphi \!\left( \frac{\Delta }{2\sigma }\!\right) + \frac{2}{\pi f_c\Delta ^3}\varphi \!\left( \frac{\Delta }{2\sigma }\!\right) . \end{aligned}$$
We can now define \(\eta (x) :=x^2\psi (x)\), for all \(x > 0\), which can be shown to be non-increasing on \((0, \infty )\). This yields
$$\begin{aligned} \sum _{m\in \Omega \setminus \{\ell \}} \left| u''(t - t_m)\right|&\le \frac{\pi }{\Delta \sigma ^2f_c}\eta \!\left( \frac{\Delta }{2\sigma }\!\right) + \frac{2\pi }{\sigma ^2}\xi \!\left( \frac{\Delta }{2\sigma }\!\right) + \left( \!\frac{4\pi f_c}{\Delta } + \frac{3}{2\sigma ^2f_c\Delta }\!\right) \varphi \!\left( \frac{\Delta }{2\sigma }\!\right) \\& + \frac{4}{\Delta ^2}\varphi \!\left( \frac{\Delta }{2\sigma }\!\right) + \frac{2}{\pi f_c\Delta ^3}\varphi \!\left( \frac{\Delta }{2\sigma }\!\right) \\&\le \left[ 16\pi \eta (2) + 32\pi \xi (2) + \left( 4\pi + 28 + \frac{2}{\pi }\right) \varphi (2)\right] f_c^2. \end{aligned}$$
Combined with (106), we get
$$\begin{aligned} \left\| \alpha \right\| _{\ell ^\infty } \sum _{m\in \Omega \setminus \{\ell \}} \left| u''(t - t_m)\right| \le 1.196 \cdot 10^{-3}f_c^2. \end{aligned}$$
We have, for all \(t \in \mathbb {R}\), that
$$\begin{aligned} v''(t)= & {} R'''(-t){{\mathrm{sinc}}}(2\pi f_c t) - 4\pi f_cR''(-t){{\mathrm{sinc}}}'(2\pi f_c t) \\&+ (2\pi f_c)^2R'(-t){{\mathrm{sinc}}}''(2\pi f_c t). \end{aligned}$$
Therefore, we get
$$\begin{aligned} \left| v''(t)\right|&\le \left( \frac{3\pi ^2}{4\sigma ^4}\left| t\right| + \frac{\pi ^3}{8\sigma ^6}\left| t\right| ^3\right) \frac{1}{2\pi f_c\left| t\right| }R\left( t\right) \\&\quad + 4\pi f_c\left( \frac{\pi ^2\left| t\right| ^2}{4\sigma ^4} + \frac{\pi }{2\sigma ^2}\right) \left( \frac{1}{2\pi f_c\left| t\right| } + \frac{1}{(2\pi f_c)^2\left| t\right| ^2}\right) R(t) \\&\quad + (2\pi f_c)^2\frac{\pi }{2\sigma ^2}\left| t\right| \left( \frac{1}{2\pi f_c\left| t\right| } + \frac{2}{(2\pi f_c)^2\left| t\right| ^2} + \frac{2}{(2\pi f_c)^3\left| t\right| ^3}\right) \\&\le \left( \frac{\pi ^2\left| t\right| ^2}{16\sigma ^6f_c} + \frac{\pi ^2\left| t\right| }{2\sigma ^4} + \frac{5\pi }{8f_c\sigma ^4} + \frac{\pi ^2f_c}{\sigma ^2} + \frac{2\pi }{\sigma ^2\left| t\right| } + \frac{1}{f_c\sigma ^2\left| t\right| ^2}\right) R(t), \end{aligned}$$
which leads to the following chain of inequalities
$$\begin{aligned}&\sum _{m\in \Omega \setminus \{\ell \}} \!\!\!\left| v''(t_\ell - t_m)\right| \nonumber \\&\quad \le 2\sum _{n = 1}^\infty \!\left( \frac{\pi ^2n^2\Delta ^2}{16\sigma ^6f_c} + \frac{\pi ^2 n\Delta }{2\sigma ^4} + \frac{5\pi }{8f_c\sigma ^4} + \frac{\pi ^2f_c}{\sigma ^2} + \frac{2\pi }{\sigma ^2 n\Delta } + \frac{1}{f_c\sigma ^2 n^2\Delta ^2}\right) \!R(n\Delta ) \nonumber \\&\quad \le \frac{\pi ^2\Delta ^2}{8\sigma ^6f_c}\sum _{n = 1}^\infty n^2\exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) + \frac{\pi ^2\Delta }{\sigma ^4}\sum _{n = 1}^\infty n\exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) \\&\qquad + \frac{5\pi }{4f_c\sigma ^4}\sum _{n = 1}^\infty \exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) + \frac{2\pi ^2f_c}{\sigma ^2}\sum _{n = 1}^\infty \exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) \nonumber \\&\qquad + \frac{4\pi }{\sigma ^2\Delta }\sum _{n = 1}^\infty \frac{1}{n}\exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) + \frac{1}{f_c\sigma ^2\Delta ^2}\sum _{n = 1}^\infty \frac{1}{n}\exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) . \nonumber \end{aligned}$$
(118)
Now, in (118), we recognize the power series
$$\begin{aligned} \forall x \in (-1, 1), \quad \sum _{n = 1}^\infty n^2 x^n = \frac{x(x+1)}{(1 - x)^3} \end{aligned}$$
evaluated at \(x = \exp \!\left( -\frac{\pi \Delta ^2}{4\sigma ^2}\right) \), which leads us to set
$$\begin{aligned} \forall x > 0, \quad \Gamma (x) :=\frac{\exp (-\pi x^2)(\exp (-\pi x^2) + 1)}{(1 - \exp (-\pi x^2))^3}. \end{aligned}$$
This yields
$$\begin{aligned} \sum _{m\in \Omega \setminus \{\ell \}} \left| v''(t_\ell - t_m)\right|&\le \frac{\pi ^2\Delta ^2}{8\sigma ^6f_c}\Gamma \!\left( \frac{\Delta }{2\sigma }\right) + \frac{\pi ^2\Delta }{\sigma ^4}\psi \!\left( \frac{\Delta }{2\sigma }\right) \\&\quad + \left( \frac{5\pi }{4f_c\sigma ^4} + \frac{2\pi ^2f_c}{\sigma ^2}\right) \xi \!\left( \frac{\Delta }{2\sigma }\right) \\&\quad + \left( \frac{4\pi }{\sigma ^2\Delta } + \frac{1}{f_c\sigma ^2\Delta ^2}\right) \varphi \!\left( \frac{\Delta }{2\sigma }\right) \\&\le \frac{\pi ^2}{2f_c\sigma ^4}\gamma \!\left( \frac{\Delta }{2\sigma }\right) + \frac{2\pi ^2}{3\sigma ^2\Delta }\eta \!\left( \frac{\Delta }{2\sigma }\right) \\&\quad + \left( \frac{5\pi }{4f_c\sigma ^4} + \frac{2\pi ^2f_c}{\sigma ^2}\right) \xi \!\left( \frac{\Delta }{2\sigma }\right) \\&\quad + \left( \frac{4\pi }{\sigma ^2\Delta } + \frac{1}{f_c\sigma ^2\Delta ^2}\right) \varphi \!\left( \frac{\Delta }{2\sigma }\right) \\&\le \Big [128\pi ^2 \gamma (2) + 64\pi ^2\eta (2) + (320\pi + 32\pi ^2)\xi (2)\\&\qquad + (64\pi +16)\varphi (2)\Big ]f_c^3, \end{aligned}$$
where we set \(\gamma (x) :=x^2\Gamma (x)\), for \(x > 0\), and used the fact that \(\gamma \) is non-increasing on \((0, \infty )\). Combined with (107), this results in
$$\begin{aligned} \left\| \beta \right\| _{\ell ^\infty }\sum _{m\in \Omega \setminus \{\ell \}} \left| v''(t_\ell - t_m)\right| \le 8.34 \cdot 10^{-8}f_c^2. \end{aligned}$$
(119)
Finally, we have \(A_R''(t) \le -22.1f_c^2\). Multiplying (119) with (116) leads to \(A_R(t)A_R''(t) \le -14.6f_c^2\). Exactly the same line of reasoning can be applied to get \(A_I(t)A_I''(t) \le -14.6f_c^2\), and therefore,
$$\begin{aligned} A_R(t)A_R''(t) + A_I(t)A_I''(t) \le -29.1f_c^2. \end{aligned}$$
(120)
It remains to find an upper bound on \(\left| (\mathcal {A}_g^*c_0)'(t)\right| ^2\). We have
$$\begin{aligned} \left| (\mathcal {A}_g^*c_0)'(t)\right|&\le \sum _{m\in \Omega } \Big (\left\| \alpha \right\| _{\ell ^\infty } \left| u'(t - t_m)\right| + \left\| \beta \right\| _{\ell ^\infty } \left| v'(t - t_m)\right| \Big ) \nonumber \\&\le \left\| \alpha \right\| _{\ell ^\infty } \left| u'(t - t_\ell )\right| + \left\| \beta \right\| _{\ell ^\infty } \left| v'(t - t_\ell )\right| \end{aligned}$$
(121)
$$\begin{aligned}&\quad + \sum _{m\in \Omega \setminus \{\ell \}} \Big (\left\| \alpha \right\| _{\ell ^\infty } \left| u'(t - t_m)\right| + \left\| \beta \right\| _{\ell ^\infty } \left| v'(t - t_m)\right| \Big ). \end{aligned}$$
(122)
We can derive upper bounds for the terms in (121) by noting that
$$\begin{aligned} \left| u'(t)\right| \le u'\!\left( \frac{1}{7f_c}\right)&= R'\!\left( \frac{1}{7f_c}\right) {{\mathrm{sinc}}}\!\left( \frac{2\pi }{7}\right) + 2\pi f_cR\left( \frac{1}{7f_c}\right) {{\mathrm{sinc}}}'\left( \frac{2\pi }{7}\right) \\&= \exp \!\left( -\frac{4\pi }{49}\right) {{\mathrm{sinc}}}\!\left( \frac{2\pi }{7}\right) + 2\pi f_c \exp \!\left( -\frac{4\pi }{49}\right) {{\mathrm{sinc}}}'\left( \frac{2\pi }{7}\right) \nonumber \end{aligned}$$
(123)
and
$$\begin{aligned} \left| v'(t)\right|&\le v'(0) = -R''(0) = \frac{\pi }{2\sigma ^2} \end{aligned}$$
(124)
for all \(t \in \left[ 0, \frac{1}{7f_c}\right] \). Indeed, we have seen that \(u''(t) \le 0\) for all \(t \in \left[ 0, \frac{1}{7f_c}\right] \), which implies that \(u'\) is non-increasing on \(\left[ 0, \frac{1}{7f_c}\right] \). As \(u'(0) = 0\), this means that \(u'\) is non-positive on \(\left[ 0, \frac{1}{7f_c}\right] \). Therefore, \(\left| u'\right| \) is non-decreasing on \(\left[ 0, \frac{1}{7f_c}\right] \), which results in (123). The inequality in (124) follows from the fact that \(\left| v'\right| \) is decreasing on \(\left[ 0, \frac{1}{7f_c}\right] \), as we show next. We have
$$\begin{aligned} \forall t \in \mathbb {R}, \quad v'(t)&= -R''(-t){{\mathrm{sinc}}}(2\pi f_c t) + 2\pi f_c R'(-t){{\mathrm{sinc}}}'(2\pi f_c t) \\&= -R''(t){{\mathrm{sinc}}}(2\pi f_c t) - 2\pi f_c R'(t){{\mathrm{sinc}}}'(2\pi f_c t). \end{aligned}$$
As the functions \(t \mapsto R''(t){{\mathrm{sinc}}}(2\pi f_c t)\) and \(t \mapsto 2\pi f_cR'(t){{\mathrm{sinc}}}'(2\pi f_c t)\) were shown to both be non-decreasing on \(\left[ 0, \frac{1}{7f_c}\right] \), we get that \(v'\) is non-increasing on \(\left[ 0, \frac{1}{7f_c}\right] \). Moreover, we have
$$\begin{aligned} v'\!\left( \frac{1}{7f_c}\right) \ge 3.43f_c^2 \ge 0. \end{aligned}$$
Hence, \(v'\) is non-negative on \(\left[ 0, \frac{1}{7f_c}\right) \). This allows us to conclude that \(\left| v'\right| \) is non-increasing on \(\left[ 0, \frac{1}{7f_c}\right] \), which establishes (124). It remains to upper-bound the term in (122), which is done as follows:
$$\begin{aligned} \sum _{m\in \Omega \setminus \{\ell \}}&\Big (\left\| \alpha \right\| _{\ell ^\infty } \left| u'(t - t_m)\right| + \left\| \beta \right\| _{\ell ^\infty } \left| v'(t - t_m)\right| \Big ) \nonumber \\&\le \sum _{m\in \Omega \setminus \{\ell \}} \Big (\left\| \alpha \right\| _{\ell ^\infty } U(t - t_m) + \left\| \beta \right\| _{\ell ^\infty } V(t - t_m) \Big ) \nonumber \\&\le \left\| \alpha \right\| _{\ell ^\infty } U\!\left( \frac{6}{7f_c}\right) + \left\| \beta \right\| _{\ell ^\infty }V\!\left( \frac{6}{7f_c}\right) + \left\| \alpha \right\| _{\ell ^\infty } \left[ \frac{2\rho (2) + (2+1/\pi )\varphi (2)}{\Delta }\right] \nonumber \\& +\left\| \beta \right\| _{\ell ^\infty } \left[ \frac{\pi }{2\sigma ^2} \left( \frac{2}{\pi } \varphi (2) + 2\psi (2) + 2\xi (2)\right) \right] \nonumber \\&\le \left\| \alpha \right\| _{\ell ^\infty } \left( \frac{1}{4\sigma ^2f_c} + \frac{7f_c}{6} + \frac{49f_c}{72\pi }\right) R\!\left( \frac{6}{7f_c}\right) \nonumber \\&\quad + \left\| \beta \right\| _{\ell ^\infty }\left( \frac{7}{12\sigma ^2} + \frac{\pi }{2\sigma ^2} + \frac{3\pi }{28\sigma ^4f_c^2}\right) R\!\left( \frac{6}{7f_c}\right) \nonumber \\& + \left\| \alpha \right\| _{\ell ^\infty } \big [2\rho (2) + (2+1/\pi )\varphi (2)\big ]f_c\nonumber \\&\quad + \frac{\left\| \beta \right\| _{\ell ^\infty }}{\sigma ^2} \big (\varphi (2) + \pi \psi (2) + \pi \xi (2)\big ) \nonumber \\&\le \left\| \alpha \right\| _{\ell ^\infty }\left( 4+\frac{7}{6}+\frac{49}{72\pi }\right) f_c\exp \!\left( -\frac{576\pi }{49}\right) \nonumber \\&\quad + \frac{\left\| \beta \right\| _{\ell ^\infty }}{\sigma }\left( \frac{7}{3} + 2\pi + \frac{48\pi }{7}\right) f_c\exp \!\left( -\frac{576\pi }{49}\right) \nonumber \\& + \left\| \alpha \right\| _{\ell ^\infty } \big [2\rho (2) + (2+1/\pi )\varphi (2)\big ]f_c\nonumber \\&\quad + \frac{\left\| \beta \right\| _{\ell ^\infty }}{\sigma } \big (4\varphi (2) + 4\pi \psi (2) + 4\pi \xi (2)\big )f_c \nonumber \\&\le 4.05f_c. \end{aligned}$$
(125)
Putting (120) and (125) together yields
$$\begin{aligned} A_R(t)A_R''(t) + A_I(t)A_I''(t) + \left| (\mathcal {A}_g^*c_0)'(t)\right| ^2 \le -12.68f_c^2 < 0, \end{aligned}$$
which completes the proof.
Appendix 2: Proof of Theorem 11
We could prove Theorem 11 following similar arguments as in the proof of Theorem 10, namely by choosing a function \(c_0 \in L^\infty (\mathbb {T}\times \mathbb {Z})\) of the form
$$\begin{aligned}&\forall \tau \in \mathbb {T},\ \forall k \in \{-K, \ldots , K\}, \\&\qquad c_0(\tau , k) = \sum _{\ell = 1}^L \Big (\alpha _\ell g(t_\ell - \tau )e^{-2\pi i kt_\ell } + \beta _\ell g'(t_\ell - \tau )e^{-2\pi ikt_\ell }\Big ) \end{aligned}$$
and determining \(\alpha :=\{\alpha _\ell \}_{\ell = 1}^L\) and \(\beta :=\{\beta _\ell \}_{\ell = 1}^L\) such that the uniqueness conditions (41) and (42) are met. It turns out, however, that a more direct path is possible, namely by choosing a function \(c_0 \in L^\infty (\mathbb {T}\times \mathbb {Z})\) of slightly different form and then reducing to a case already treated in the proof of Theorem 10; this approach leads to a substantially shorter proof. We start by defining this function \(c_0 \in L^\infty (\mathbb {T}\times \mathbb {Z})\) as
$$\begin{aligned}&\forall \tau \in \mathbb {T},\ \forall k \in \{-K, \ldots , K\}, \\&\quad c_0(\tau , k) :=\frac{1}{2K+1}\sum _{\ell = 1}^L \Big (\alpha _\ell p(\tau - t_\ell )e^{-2\pi i kt_\ell } + \beta _\ell q(\tau - t_\ell ) e^{-2\pi ikt_\ell }\Big ), \end{aligned}$$
where \(p :\mathbb {T}\rightarrow \mathbb {C}\) and \(q :\mathbb {T}\rightarrow \mathbb {C}\) are defined (for reasons that will become clear later) as
$$\begin{aligned} p(\tau ) :=\sum _{n \in \mathbb {Z}} p_n e^{2\pi in\tau } \qquad \text {and} \qquad q(\tau ) :=\sum _{n \in \mathbb {Z}} q_n e^{2\pi in\tau }, \end{aligned}$$
for \(\tau \in \mathbb {T}\), with
$$\begin{aligned} p_n&:=\sqrt{2\sigma }\exp \left( -2\pi \sigma ^2 n^2\right) \int _{-1/2}^{1/2} \exp \left( -4\pi \sigma ^2 u^2\right) \exp \left( -8\pi \sigma ^2 nu\right) \mathrm {d}u \\ q_n&:=-2\pi i\sigma \sqrt{2\sigma }\exp \left( -2\pi \sigma ^2 n^2\right) \int _{-1/2}^{1/2} (u+n)\exp \left( -4\pi \sigma ^2 u^2\right) \\&\qquad \times \exp \left( -8\pi \sigma ^2 nu\right) \mathrm {d}u, \end{aligned}$$
for \(n \in \mathbb {Z}\). We first verify that the resulting function \(c_0\) is, indeed, in \(L^\infty (\mathbb {T}\times \mathbb {Z})\). This is accomplished by showing that the functions p and q are well-defined and are in \(L^\infty (\mathbb {T})\), that is, by verifying that \(\sum _{n \in \mathbb {Z}} \left| p_n\right| < \infty \) and \(\sum _{n \in \mathbb {Z}} \left| q_n\right| < \infty \). Indeed, we have
$$\begin{aligned} \sum _{n \in \mathbb {Z}} \left| p_n\right|&= \sqrt{2\sigma }\sum _{n \in \mathbb {Z}} \exp \left( -2\pi \sigma ^2 n^2\right) \int _{-1/2}^{1/2} \exp \left( -4\pi \sigma ^2 u^2\right) \exp \left( -8\pi \sigma ^2 nu\right) \mathrm {d}u \nonumber \\&\le \sqrt{2\sigma } \sum _{n \in \mathbb {Z}} \int _{-1/2}^{1/2} \exp \left( -2\pi \sigma ^2(4u + n)n\right) \mathrm {d}u \end{aligned}$$
(126)
$$\begin{aligned}&= C\ + \sqrt{2\sigma }\sum _{\begin{array}{c} n \in \mathbb {Z}\\ \left| n\right| \ge 3 \end{array}} \int _{-1/2}^{1/2} \exp \left( -2\pi \sigma ^2(4u + n)n\right) \mathrm {d}u, \end{aligned}$$
(127)
where (126) follows from \(\exp (-4\pi \sigma ^2 u^2) \le 1\), for all \(u \in [-1/2, 1/2]\), and we set
$$\begin{aligned} C :=\sqrt{2\sigma }\sum _{n = -2}^2 \int _{-1/2}^{1/2} \exp \left( -2\pi \sigma ^2(4u + n)n\right) \mathrm {d}u < \infty . \end{aligned}$$
To see that the sum in (127) is finite, first note that for \(n \ge 3\) and \(u \in [-1/2, 1/2]\), we have
$$\begin{aligned} (4u + n)n = (4u + n)\!\left| n\right| \ge (-2 + n)\!\left| n\right| \ge \left| n\right| . \end{aligned}$$
Similarly, for \(n \le -3\), we get
$$\begin{aligned} (4u + n)n = -(4u + n)\!\left| n\right| \ge -(2 + n)\!\left| n\right| \ge \left| n\right| . \end{aligned}$$
It therefore follows that
$$\begin{aligned} \sum _{\begin{array}{c} n \in \mathbb {Z}\\ \left| n\right| \ge 3 \end{array}} \int _{-1/2}^{1/2}\exp \left( -2\pi \sigma ^2(4u + n)n\right) \mathrm {d}u&\le \sum _{\begin{array}{c} n \in \mathbb {Z}\\ \left| n\right| \ge 3 \end{array}} \int _{-1/2}^{1/2}\exp (-2\pi \sigma ^2\!\left| n\right| ) \mathrm {d}u \nonumber \\&= 2\sum _{n = 3}^\infty \exp (-2\pi \sigma ^2n) < \infty . \end{aligned}$$
This concludes the proof of \(\sum _{n \in \mathbb {Z}} \left| p_n\right| < \infty \). Similar reasoning shows that \(\sum _{n \in \mathbb {Z}} \left| q_n\right| < \infty \). For \(t \in \mathbb {T}\), we then have
$$\begin{aligned} (\mathcal {A}_g^*c_0)(t)&= \sum _{k = -K}^K \int _{-1/2}^{1/2} c_0(\tau , k)g(t - \tau )e^{2\pi i kt}\mathrm {d}\tau \\&= \frac{1}{2K+1}\sum _{\ell = 1}^L \left[ \alpha _\ell \left( \int _{-1/2}^{1/2} p(\tau - t_\ell )g(t - \tau )\mathrm {d}\tau \right) D_K(t - t_\ell ) \right. \nonumber \\&\quad \left. + \beta _\ell \left( \int _{-1/2}^{1/2} q(\tau - t_\ell )g(t - \tau )\mathrm {d}\tau \right) D_K(t - t_\ell ) \right] \nonumber \\&= \frac{1}{2K+1}\sum _{\ell = 1}^L \Big (\alpha _\ell P(t - t_\ell )D_K(t - t_\ell )+ \beta _\ell Q(t - t_\ell )D_K(t - t_\ell )\Big ), \nonumber \end{aligned}$$
(128)
where \(D_K\) is the Dirichlet kernel, that is,
$$\begin{aligned} \forall t \in \mathbb {T}, \quad D_K(t) :=\frac{\sin ((2K+1)\pi t)}{\sin (\pi t)}, \end{aligned}$$
and P and Q designate the cross-correlation between the functions p and \(g\), and q and \(g\), respectively, that is,
$$\begin{aligned} \forall t \in \mathbb {T}, \quad P(t)&:=\int _{-1/2}^{1/2} p(\tau )g(t - \tau )\mathrm {d}\tau \\ \forall t \in \mathbb {T}, \quad Q(t)&:=\int _{-1/2}^{1/2} q(\tau )g(t - \tau )\mathrm {d}\tau . \end{aligned}$$
Note that since \(g\) and \(\tau \mapsto c_0(\tau , k)\), \(k \in \{-K, \ldots , K\}\), are all 1-periodic, we can integrate over the interval \([-1/2, 1/2]\) in (128) (instead of [0, 1] as done in (18)) and in the remainder of the proof. We next derive an alternative expression for the function P. As in (54), we have
$$\begin{aligned} \forall t \in \mathbb {T}, \quad g(t) = \sum _{n \in \mathbb {Z}} g_n e^{2\pi i nt}, \end{aligned}$$
where \(g_n :=\sqrt{2\sigma } \exp (-2\pi \sigma ^2n^2)\), for all \(n \in \mathbb {Z}\). The nth Fourier series coefficient of P is then given by \(p_ng_n\), and we show that the Fourier series \(\sum _{n \in \mathbb {Z}} p_ng_n e^{2\pi int}\) converges to P(t) for all \(t \in \mathbb {T}\) using Dirichlet’s theorem [22, Thm. 2.1], whose applicability conditions we verify next. Since \(\sum _{n \in \mathbb {Z}} \left| p_n\right| < \infty \), \(\sum _{n \in \mathbb {Z}} \left| g_n\right| < \infty \), and \(\left| e^{2\pi int}\right| = 1\), for all \(t \in \mathbb {T}\), by the Weierstrass M-test, the series \(\sum _{n \in \mathbb {Z}} p_ne^{2\pi int}\) and \(\sum _{n \in \mathbb {Z}} g_ne^{2\pi int}\) converge absolutely and uniformly. This implies that the functions p and \(g\) are both continuous on \(\mathbb {T}\). Moreover, \(g\) is continuously differentiable on \(\mathbb {R}\) as \(\sum _{n \in \mathbb {Z}} \left| ng_n\right| < \infty \). As a result, the function P is continuously differentiable on \(\mathbb {R}\), and by application of Dirichlet’s theorem, it follows that
$$\begin{aligned} \forall t \in \mathbb {T}, \quad P(t) = \sum _{n \in \mathbb {Z}} g_n p_n e^{2\pi int}. \end{aligned}$$
For \(n \in \mathbb {Z}\), we have
$$\begin{aligned} g_n p_n&= \sqrt{2\sigma }\exp \left( -2\pi \sigma ^2n^2\right) \sqrt{2\sigma }\exp \left( -2\pi \sigma ^2 n^2\right) \\&\quad \times \int _{-1/2}^{1/2} \exp \left( -4\pi \sigma ^2 u^2\right) \exp \left( -8\pi \sigma ^2 nu\right) \mathrm {d}u \\&= 2\sigma \exp \left( -4\pi \sigma ^2 n^2\right) \int _{-1/2}^{1/2} \exp \left( -4\pi \sigma ^2 u^2\right) \exp \left( -8\pi \sigma ^2 nu\right) \mathrm {d}u \\&= 2\sigma \int _{-1/2}^{1/2} \exp \left( -4\pi \sigma ^2(u + n)^2\right) \mathrm {d}u. \end{aligned}$$
Now fix \(t \in [0, 1)\). If \(t = 0\), we have
$$\begin{aligned} P(t) = P(0)&= \sum _{n \in \mathbb {Z}} g_n p_n = 2\sigma \sum _{n \in \mathbb {Z}} \int _{-1/2}^{1/2} \exp \left( -4\pi \sigma ^2(u+n)^2\right) \mathrm {d}u \\&= 2\sigma \sum _{n \in \mathbb {Z}} \int _{n-1/2}^{n+1/2} \exp (-4\pi \sigma ^2v^2)\mathrm {d}v = 2\sigma \int _{-\infty }^\infty \exp (-4\pi \sigma ^2v^2)\mathrm {d}v = 1, \end{aligned}$$
and if \(t \ne 0\), we get
$$\begin{aligned} P(t)&= \sum _{n \in \mathbb {Z}} \left( 2\sigma \int _{-1/2}^{1/2} \exp \left( -4\pi \sigma ^2(u+n)^2\right) \mathrm {d}u\right) e^{2\pi int} \nonumber \\&= \int _{-1/2}^{1/2} 2\sigma \sum _{n \in \mathbb {Z}} \exp \left( -4\pi \sigma ^2(u+n)^2\right) e^{2\pi int} \mathrm {d}u \end{aligned}$$
(129)
$$\begin{aligned}&= \int _{-1/2}^{1/2} 2\sigma \sum _{n \in \mathbb {Z}} \exp \left( -4\pi \sigma ^2(u+n)^2\right) e^{2\pi i(u+n)t} e^{-2\pi i tu}\mathrm {d}u \end{aligned}$$
(130)
$$\begin{aligned}&= \int _{-1/2}^{1/2} \psi (u)\varphi (-u)\mathrm {d}u = \xi (0). \end{aligned}$$
(131)
Here, \(\varphi \) is the 1-periodic function defined by \(\varphi (u) :=e^{2\pi it u}\), \(u \in [-1/2, 1/2)\), and \(\psi \) and \(\xi \) are given by
$$\begin{aligned} \begin{array}{lll} \forall u \in \mathbb {T}, \qquad &{}\psi (u) &{}:=\displaystyle 2\sigma \sum _{n \in \mathbb {Z}} \exp \left( -4\pi \sigma ^2(u+n)^2\right) e^{2\pi it(u+n)} \\ \forall x \in \mathbb {T}, \qquad &{}\xi (x) &{}:=\displaystyle \int _{-1/2}^{1/2} \varphi (u)\psi (x-u)\mathrm {d}u. \end{array} \end{aligned}$$
The order of summation and integration in (129) is interchangeable thanks to
$$\begin{aligned} \sum _{n \in \mathbb {Z}} \int _{-1/2}^{1/2} \exp \left( -4\pi \sigma ^2(u+n)^2\right) \mathrm {d}u&= \sum _{n \in \mathbb {Z}} \int _{n-1/2}^{n+1/2} \exp \left( -4\pi \sigma ^2 v^2\right) \mathrm {d}v \\&= \int _{-\infty }^\infty \exp \left( -4\pi \sigma ^2 v^2\right) \mathrm {d}v < \infty . \end{aligned}$$
The function \(\xi \) can be expanded into a Fourier series. Specifically, it holds that
$$\begin{aligned} \forall x \in \mathbb {T}, \quad \xi (x) = \sum _{n \in \mathbb {Z}} \varphi _n\psi _n e^{2\pi inx}, \end{aligned}$$
where \(\varphi _n\) and \(\psi _n\) denote the nth Fourier series coefficients of \(\varphi \) and \(\psi \), respectively. We have
$$\begin{aligned} \varphi _n = \int _{-1/2}^{1/2} \varphi (x)e^{-2\pi i nx}\mathrm {d}x = \int _{-1/2}^{1/2} e^{2\pi i(t-n)x}\mathrm {d}x = \frac{(-1)^n\sin (\pi t)}{\pi (t-n)} \end{aligned}$$
and
$$\begin{aligned} \psi _n&= \int _{-1/2}^{1/2} \psi (x)e^{-2\pi inx} \mathrm {d}x \\&= \int _{-1/2}^{1/2} 2\sigma \sum _{m \in \mathbb {Z}} \exp \left( -4\pi \sigma ^2(x+m)^2\right) e^{2\pi it(x+m)} e^{-2\pi inx}\mathrm {d}x \\&= 2\sigma \sum _{m \in \mathbb {Z}} \int _{m-1/2}^{m+1/2} \exp \left( -4\pi \sigma ^2v^2\right) e^{2\pi itv} e^{-2\pi inv}\mathrm {d}v \\&=2\sigma \int _{-\infty }^\infty \exp (-4\pi \sigma ^2v^2)e^{2\pi i(t-n)v}\mathrm {d}v \\&= \exp \left( -\frac{\pi (t-n)^2}{4\sigma ^2}\right) , \end{aligned}$$
for \(n \in \mathbb {Z}\). It follows that
$$\begin{aligned} \forall t \in \mathbb {T}, \quad P(t) = \xi (0) = \sum _{n \in \mathbb {Z}} \varphi _n\psi _n = \sum _{n \in \mathbb {Z}} \frac{(-1)^n\sin (\pi t)}{\pi (t-n)}\exp \left( -\frac{\pi (t-n)^2}{4\sigma ^2}\right) . \end{aligned}$$
We then get
$$\begin{aligned} \forall t \in \mathbb {T}, \quad P(t)D_K(t)&= \sum _{n \in \mathbb {Z}} \frac{(-1)^n\sin ((2K+1)\pi t)}{\pi (t - n)}\exp \left( -\frac{\pi (t-n)^2}{4\sigma ^2}\right) \\&= (2K+1)\sum _{n \in \mathbb {Z}} {{\mathrm{sinc}}}((2K+1)\pi (t-n))R(t-n), \end{aligned}$$
where \(R\) was defined in (15). Similarly, we can show that
$$\begin{aligned} \forall t \in \mathbb {T}, \quad Q(t)D_K(t) = (2K+1)\sum _{n \in \mathbb {Z}} {{\mathrm{sinc}}}((2K+1)\pi (t-n))R'(n-t). \end{aligned}$$
This finally yields
$$\begin{aligned} \forall t \in \mathbb {T}, \quad (\mathcal {A}_g^*c_0)(t) = \sum _{\ell = 1}^L\sum _{n \in \mathbb {Z}}&\Big (\alpha _\ell {{\mathrm{sinc}}}((2K+1)\pi (t - t_\ell -n))R(t - t_\ell - n) \\&+ \beta _\ell {{\mathrm{sinc}}}((2K+1)\pi (t - t_\ell -n))R'(n - t + t_\ell )\Big ) \\ =\sum _{\ell = 1}^L\sum _{n \in \mathbb {Z}}&\Big (\alpha _\ell u(t - t_\ell -n) + \beta _\ell v(t - t_\ell -n)\Big ), \end{aligned}$$
where we set
$$\begin{aligned} \forall t \in \mathbb {R}, \quad u(t)&:=R(t){{\mathrm{sinc}}}(2\pi f_c' t) \end{aligned}$$
(132)
$$\begin{aligned} \forall t \in \mathbb {R}, \quad v(t)&:=R'(-t){{\mathrm{sinc}}}(2\pi f_c' t) = \frac{\pi t}{2\sigma ^2}R(t){{\mathrm{sinc}}}(2\pi f_c' t) \end{aligned}$$
(133)
as in (86) and (87) with \(f_c' :=K+1/2\). Analogously to the proof of Theorem 10 we can define the operators
$$\begin{aligned} \begin{array}{llll} \mathcal {U}_p :&{} \mathbb {C}^L &{} \longrightarrow &{} \mathbb {C}^L \\ &{} \alpha = \{\alpha _\ell \}_{\ell =1}^L &{} \longmapsto &{} \left\{ \sum \limits _{m= 1}^L\sum \limits _{n \in \mathbb {Z}} \alpha _mu^{(p)}(t_\ell - t_m- n)\right\} _{\ell = 1}^L \end{array} \end{aligned}$$
and
$$\begin{aligned} \begin{array}{llll} \mathcal {V}_p :&{} \mathbb {C}^L &{} \longrightarrow &{} \mathbb {C}^L \\ &{} \beta = \{\beta _\ell \}_{\ell \in \Omega } &{} \longmapsto &{} \left\{ \sum \limits _{m= 1}^L\sum \limits _{n \in \mathbb {Z}} \beta _mv^{(p)}(t_\ell - t_m- n)\right\} _{\ell = 1}^L, \end{array} \end{aligned}$$
where \(p \in \{0, 1\}\). Then, given \(\varepsilon = \{\varepsilon _\ell \}_{\ell = 1}^L\) with \(\left| \varepsilon _\ell \right| = 1\), \(\ell \in \{1, 2, \ldots , L\}\), we can solve the equation system
$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {U}_0\alpha + \mathcal {V}_0\beta = \varepsilon \\ \mathcal {U}_1\alpha + \mathcal {V}_1\beta = 0 \end{array}\right. \end{aligned}$$
(134)
to determine \(\alpha \in \mathbb {C}^L\) and \(\beta \in \mathbb {C}^L\) such that the interpolation conditions \((\mathcal {A}_g^*c_0)(t_\ell ) = \varepsilon _\ell \), for all \(\ell \in \{1, 2, \ldots , L\}\), are satisfied and \(\mathcal {A}_g^*c_0\) has a local extremum at every \(t_\ell \), \(\ell \in \{1, 2, \ldots , L\}\). As in the proof of Theorem 10, if the operators \(\mathcal {V}_1\) and \(\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1\) are invertible, then one can choose \(\alpha = (\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1)^{-1}\varepsilon \) and \(\beta = -\mathcal {V}_1^{-1}\mathcal {U}_1\alpha \) to satisfy (134). Proving the invertibility of \(\mathcal {V}_1\) and \(\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1\) is essentially identical to the corresponding part in the proof of Theorem 10 with \(f_c\) replaced by \(f_c'\). Verifying that \(\left| (\mathcal {A}_g^*c_0)(t)\right| < 1\) for all \(t \in \mathbb {T}\setminus T\), where \(T= \{t_\ell \}_{\ell = 1}^L\), is also done in a fashion similar to the proof of Theorem 10 (see Appendix Section “\(|{(\mathcal {A}_g^*c_0)(t)|} < 1\) for all \(t \in \mathbb {R}{\setminus } T\)”).
