1 Introduction

The trigonometric series considered here are of the type

$$\begin{aligned} S(t):=\sum _{\nu } a(\nu )e(\nu t), \end{aligned}$$

where \(e(x):= e^{2\pi ix}\) and \(a(\nu )\in {{\mathbb {C}}}\) for a strictly increasing sequence \(\nu \in {{\mathbb {R}}}\). For any given \(T\in (0,+\infty )\) and \(p\in [1,+\infty )\), let us denote the \(L^p\)-norm of \(f:{{\mathbb {R}}}\rightarrow {{\mathbb {C}}}\) by \(\Vert f\Vert _{p}:=\big (\int _{{{\mathbb {R}}}}|f(t)|^p\,dt\big )^{1/p}\) and set \(\Vert f\Vert _{p,T}:=\big (\int _{-T}^T|f(t)|^p\,dt\big )^{1/p}\). The space of the complex-valued functions defined on \({{\mathbb {R}}}\) with a finite \(L^p\)-norm is denoted by \(L^p({{\mathbb {R}}},{{\mathbb {C}}})\). By \(f\in L^1_{loc}({{\mathbb {R}}},B)\), with \(B\subseteq {{\mathbb {C}}}\), we mean that \(f:{{\mathbb {R}}}\rightarrow B\) is summable on each compact subset of \({{\mathbb {R}}}\). The Fourier transform of an integrable \(f:{{\mathbb {R}}}\rightarrow {{\mathbb {C}}}\) is denoted by

$$\begin{aligned} {\widehat{f}}(y):=\int _{{{\mathbb {R}}}}f(t)e(-ty)\,dt. \end{aligned}$$

The convolution of \(f,g\in L^1({{\mathbb {R}}},{{\mathbb {C}}})\) is

$$\begin{aligned} (f *g)(x)=\int _{{{\mathbb {R}}}}f(t)g(x-t)\,dt. \end{aligned}$$

For any weight \(w:{{\mathbb {R}}} \rightarrow {{\mathbb {C}}}\) and any \(\delta >0\), we set \(w_\delta (x)=w(x)\) or 0 according as \(|x|\le \delta\) or not. Further, we denote the discrete convolution of \(a(\nu )\) and \(w_\delta (x)\) as

$$\begin{aligned} (a\star w_\delta )(y):=\sum _{\nu }a(\nu )w_\delta (y-\nu ). \end{aligned}$$

The function that is identically 1 we denote by \(\mathbf{1}\), so that \(\mathbf{1}_{\delta }\) is the characteristic function of the interval \((-\delta ,\delta )\). Recall that \(\widehat{\mathbf{1}_{\delta }}(y)=2\delta \hbox {sinc}(2\delta y)\), where

$$\begin{aligned} \hbox {sinc}\, x:= \left\{ \begin{array}{ll} \displaystyle { {{\sin (\pi x)}\over {\pi x}} }\ &{}\text{ if } x\not =0,\\ 1\ &{}\text{ if } x=0.\ \end{array} \right. \end{aligned}$$

Definition 1.1

Let \(1<p\le q<\infty\) be fixed and \(p'=p/(p-1)\). If \(U,V\in L^1_{loc}({{\mathbb {R}}},{{\mathbb {R}}}_+)\) are even, U is non-decreasing on \((0,+\infty )\), V is non-increasing on \((0,+\infty )\) and

$$\begin{aligned} \sup _{s>0}\biggl (\int _0^{s}U(t)^{1-p'}\,dt\biggr )^{1/p'}\!\biggl (\int _0^{1/s} V(t)\,dt\biggr )^{1/q}<+\infty , \end{aligned}$$

then we say that (U, V) is a (p, q)-pair of Pitt weights.

In Sect. 2 the following result is proved.

Theorem 1.1

Let \(\delta ,T\in (0,+\infty )\) and \(1<p\le q<\infty\) be fixed. If \(w\in L^1_{loc}({{\mathbb {R}}},{{\mathbb {C}}})\) is even, (U, V) is a (p, q)-pair of Pitt weights and \(S(t)=\sum _{\nu } a(\nu )e(\nu t)\) is absolutely convergent, then there exists \(C=C(p,q,U.V)>0\) such that

$$\begin{aligned} \Vert V^{1/q}S\Vert _{q,T}\le \frac{C}{\min _{|t|\le T}|\widehat{w_\delta }(t)|}\Vert U^{1/p}(a\star w_\delta )\Vert _p. \end{aligned}$$
(1.1)

More generally, the same inequality holds if \(w_\delta\) is replaced by \(w\in L^1({{\mathbb {R}}},{{\mathbb {C}}})\).

Remark 1.1

Unless otherwise stated, hereafter the constant C is not necessarily the same at each occurrence. Further, let us assume henceforth that \(\displaystyle {\min _{|t|\le T}|\widehat{w_\delta }(t)|}\not =0\), for otherwise (1.1) is trivial.

In order to prove Theorem 1.1, we apply a weighted Fourier transform norm inequality, called Pitt’s inequality, which is an old result in Fourier analysis ([15]). Such an inequality is quoted in Sect. 2, before the proof of Theorem 1.1, by referring to the version essentially given by Heinig et al. ([8]), [10, 14] in 1983–1984 (see also [2, 6] for advanced and more recent versions). It is well-known that Pitt’s inequality yields the Hardy-Littlewood inequality, the Hausdorff-Young inequality and the Plancherel identity. In the latter case, i.e., \(p=q=2\), \(U=V=\mathbf{1}\), \(C=1\), the inequality (1.1) becomes

$$\begin{aligned} \int _{-T}^T|S(y)|^2\,dy \le \frac{1}{\min _{|t|\le T}|\widehat{w_\delta }(t)|^2}\int _{{{\mathbb {R}}}}|(a\star w_\delta )(y)|^2\,dy. \end{aligned}$$
(1.2)

In the present paper we continue our study of (1.2), that was started in [5]. We point out that most of the next considerations can be easily adapted to the general case (1.1). Let us recall that a first consequence of (1.2) is the following majorant properties established in [5]. Let \(\delta , T\in (0,+\infty )\) and let \(w\in L^1_{loc}({{\mathbb {R}}},{{\mathbb {C}}})\) be even such that

$$\begin{aligned} (w_\delta *w_\delta )(x):= \int _{{{\mathbb {R}}}}w_\delta (t)w_\delta (x-t)\,dt \ge 0,\quad \forall x\in {{\mathbb {R}}}. \end{aligned}$$

If \(S(t)=\sum _{\nu } a(\nu )e(\nu t)\) and \(B(y)=\sum _\nu b(\nu )e(\nu y)\) are absolutely convergent, and the first sum is a majorant for the second one, i.e., \(|a(\nu )|\le b(\nu )\) for all \(\nu\), then

$$\begin{aligned} \int _{-T}^T|S(y)|^2\,dy\le \frac{1}{\min _{|t|\le T}|\widehat{w_\delta }(t)|^2} \int _{{{\mathbb {R}}}}|(b\star w_\delta )(x)|^2\,dx. \end{aligned}$$
(1.3)

Now, it is worthwhile to recall that by a straightforward application of the Hardy-Littlewood majorant principle (see [13, Ch.7] or [11]) one gets

$$\begin{aligned} \int _{-T}^T|S(y)|^2dy\le 3\int _{-T}^T|B(y)|^2dy. \end{aligned}$$

The combination of such an inequality with (1.2) for B(y) gives a weaker inequality than (1.3) because of the factor 3 in the right-hand side. Moreover, we recall that such a factor is best possible in the Hardy-Littlewood majorant principle (see [12]).

Example 1.1

A well-known instance of (1.2) is given by taking the weight \(u_{\delta }\) associated to the unit step function

$$\begin{aligned} \displaystyle { u(t):= \left\{ \begin{array}{ll} 1\ &{}\hbox { if}\ t>0\\ 0\ &{}\text{ otherwise. }\ \end{array} \right. } \end{aligned}$$

It is easily seen that \(\widehat{u_{\delta }}(y)=e(-\delta y/2)\, \delta \, \hbox {sinc}(\delta y)\). Therefore, for \(0<\delta T<1\), from (1.2) we find the Gallagher inequality [7]

$$\begin{aligned} \Vert S\Vert ^2_{2,T}\le {1\over {\delta ^2\hbox {sinc}^2(\delta T)}} \int _{{{\mathbb {R}}}}\big |\sum _{x<\nu \le x+\delta }a(\nu )\big |^2\,dx. \end{aligned}$$

We refer the reader to [9] for a first introduction to the applications of such an inequality within the analytic number theory.

Example 1.2

Another remarkable instance of (1.2) is the special case of the Cesàro weight

$$\begin{aligned} C_\delta (y):=\max (1-\delta ^{-1}|y|,0), \end{aligned}$$
(1.4)

whose Fourier transform is \(\widehat{C_{\delta }}(y)=\delta \hbox { sinc}^2(\delta y)\). In the literature, this is known as Fejer kernel [4]. For \(\delta T\in (0,1)\), the inequality (1.2) gives

$$\begin{aligned} \Vert S\Vert ^2_{2,T}\le {1\over {\delta ^2\hbox {sinc}^4(\delta T)}} \int _{{{\mathbb {R}}}}\big |\sum _{|\nu -x|\le \delta }\big (1-{|\nu -x|\over \delta }\big )a(\nu )\big |^2\,dx. \end{aligned}$$

In [5] we have applied the latter inequality to the Dirichlet polynomials and to the distribution of certain arithmetical functions in short intervals.

In the present paper we focus on some further aspects of (1.2), mostly concerning the optimal choice of the weight w. To this aim and for the convenience of the reader, we need to quote some definitions and properties already introduced in [5]. First, for every \(w\in L^1_{loc}({{\mathbb {R}}},{{\mathbb {C}}})\) let us define the normalized self-convolution of \(w_\delta\), with \(\delta >0\), as

$$\begin{aligned} \widetilde{w_\delta }(x):= {1\over {2\delta }}(w_{\delta } *w_{\delta })(x). \end{aligned}$$

For example, the Cesàro weight (1.4) is the normalized self-convolution of the restriction to \([-\delta /2,\delta /2]\) of \(\mathbf{1}\):

$$\begin{aligned} C_{\delta }(x) ={1\over \delta }\!\!\!\mathop {\int }_{\begin{array}{c} {|t|\le \delta /2}\\ {|x-t|\le \delta /2} \end{array}}dt ={1\over {\delta }}(\mathbf{1}_{\delta /2}*\mathbf{1}_{\delta /2})(x) =\!\widetilde{\mathbf{1}_{\delta /2}}(x). \end{aligned}$$

It is well-known that the iteration of the self-convolution gives rise to a process of smoothing (see [4]). Moreover, the support of \(\widetilde{w_\delta }\) is doubled with respect to the support of \(w_\delta\) in the sense that it is a subset of \([-2\delta ,2\delta ]\). Because of the normalizing factor \((2\delta )^{-1}\), that takes into account the length of the integration interval, the magnitude of \(w_\delta\) is not altered much by the normalized self-convolution. More precisely, if one has \(w_{\delta }\asymp 1\), i.e., \(1\ll w_{\delta }\ll 1\), in an interval of length \(\gg \delta\), then there exists an interval of length \(\gg \delta\) (not necessarily the same) where \(\widetilde{w_\delta }\asymp 1\). Recall that \(A\ll _{\theta } B\) stands for \(|A|\le cB\), where \(c>0\) is an unspecified constant depending on \(\theta\).

From another well-known property of the convolution it follows that the Fourier transform of \(\widetilde{w_\delta }\) is \(\widehat{\widetilde{w_\delta }}(y)= (2\delta )^{-1}\widehat{w_{\delta }}(y)^2\). In particular, from \(\widehat{\mathbf{1}_{\delta }}(y)=2\delta \hbox {sinc}(2\delta y)\) we rediscover that \(\widehat{C_{\delta }}(y)=\widehat{\widetilde{\mathbf{1}_{\delta /2}}}(y)= \delta ^{-1}\widehat{\mathbf{1}_{\delta /2}}(y)^2= \delta \hbox {sinc}^2(\delta y)\).

The normalized self-convolution generates recursively the family of weights:

$$\begin{aligned} C^{\langle j\rangle }_{\delta }(x):= \widetilde{C^{\langle j-1\rangle }_{\delta /2}}(x), \ j\ge 1, \end{aligned}$$

with the base steps \(C^{\langle 0\rangle }_{\delta }(x):= \mathbf{1}_{\delta }(x)\) and \(C^{\langle 1\rangle }_{\delta }(x):= C_{\delta }(x)\).

In [5] we proved the following inductive formula for the Fourier transform of such weights:

$$\begin{aligned} \widehat{C^{\langle j\rangle }_\delta }(y)= {{4\delta }\over {2^{2^j}}}\hbox {sinc}^{2^j}\big ({{\delta y}\over {2^{j-1}}}\big ),\quad \forall j\ge 0. \end{aligned}$$
(1.5)

Consequently, \(\widehat{C^{\langle j\rangle }_\delta }(y) \asymp _j \delta\) at least for \(0\le |y|\le 2^{j-2}\delta ^{-1}\).

Such a process of continuous smoothing through the self-convolution of a weight \(w_\delta\) has a discrete counterpart given by the autocorrelation of \(w_\delta\) (since no confusion can arise in the following, we will use the simpler term correlation):

$$\begin{aligned} {\mathscr {C}}_{w_\delta }(a):= \mathop {\sum \sum }_{\begin{array}{c} {n \quad m}\\ {n-m=a} \end{array}}w_\delta (n)\overline{w_\delta (m)}. \end{aligned}$$

For example, since it turns out that

$$\begin{aligned} C_\delta (t) ={1\over \delta } \sum _{a\le \delta -|t|}1 ={1\over \delta } \mathop {\sum \sum }_{\begin{array}{c} {a,b\le \delta }\\ {b-a=t} \end{array}}1 ={{\mathscr {C}}_{u_\delta }(t)\over \delta }, \end{aligned}$$

the Cesàro weight (1.4) is the normalized correlation of the unit step weight \(u_\delta\). Note that the Cesàro weight (1.4) is generated by both types of smoothing from the function \(\mathbf{1}\). Moreover, through an iteration of the normalized correlation one might parallel the self-convolution process to generate the whole family of weights \(C^{\langle j\rangle }_\delta\), \(j\ge 1\).

An important aspect is that if the coefficients of a trigonometric series are correlations of w, then such a series is non-negative. More precisely,

$$\begin{aligned} \sum _h {\mathscr {C}}_{w_\delta }(h)e(h\nu )=\sum _h \mathop {\sum \sum }_{n-m=h}w_\delta (n)\overline{w_\delta (m)}e(h\nu ) = \big |\sum _n w_\delta (n)e(n\nu )\big |^2. \end{aligned}$$

A particularly well-known case is the Fejér kernel (compare the \(\widehat{C_\delta }\) formula)

$$\begin{aligned} \delta \sum _h C_\delta (h)e(h\nu )=\sum _h {\mathscr {C}}_{u_\delta }(h)e(h\nu )= \big |\sum _{1\le n\le \delta }e(n\nu )\big |^2. \end{aligned}$$

Such a positivity property is the complete analogous of the aforementioned fact that the Fourier transform of a self-convolution is a square.

Now, let us introduce our comparison argument for the weight w in view of possible refinements of (1.2). First, recall that

$$\begin{aligned} \displaystyle {m_{\delta ,T}:= \min _{|t|\le T}|\widehat{w_\delta }(t)|^2}\not =0 \end{aligned}$$

by assumption in order to avoid triviality. Thus, we define

$$\begin{aligned} {{\mathcal {E}}}_{w}={{\mathcal {E}}}_{w}(T,\delta ):=\{y\in {{\mathbb {R}}}: |\widehat{w_\delta }(y)|^2< m_{\delta ,T}\} \end{aligned}$$

and write

$$\begin{aligned} \Vert S\Vert ^2_{2,T}&\le m_{\delta ,T}^{-1} \int _{{{\mathbb {R}}}\setminus {{\mathcal {E}}}_{w}}|S(y)\widehat{w_\delta }(y)|^2\,dy + \int _{{{\mathcal {E}}}_{w}}|S(y)|^2\,dy \nonumber \\&\le m_{\delta ,T}^{-1} \int _{{{\mathbb {R}}}\setminus {{\mathcal {E}}}_{w}}|S(y)\widehat{w_\delta }(y)|^2\,dy + |{{\mathcal {E}}}_{w}|\sup _{y\in {{\mathcal {E}}}_{w}}|S(y)|^2. \end{aligned}$$
(1.6)

In order to take advantage of such an inequality one would require \({{\mathcal {E}}}_{w}\) either with a small measure or free of peaks of S(y), if it does not support a sufficiently small \(L^2\)-norm. For example, let us examine the possible scenario for the weights (1.5). To this aim, recall that

$$\begin{aligned} \widehat{C^{\langle 0\rangle }_{\delta }}(y)=\widehat{\mathbf{1}_{\delta }}(y)= \left\{ \begin{array}{ll} \displaystyle { {{\sin (2\pi \delta y)}\over {\pi y}}}&{}\text{ if } y\not =0,\\ 2\delta &{} \text{ if } y=0. \end{array} \right. \end{aligned}$$

The aforementioned triviality for (1.2) does not occur if we assume \(\delta T\in (0,1/2)\), for otherwise it would be \(\delta t_0=1/2\) for some \(t_0\in (0,T]\) and consequently

$$\begin{aligned} \min _{|t|\le T}\big |\widehat{\mathbf{1}_\delta }(t)\big |^2= \min _{|t|\le T}{\sin ^2(2\pi \delta t)\over (\pi t)^2}= {\sin ^2(2\pi \delta t_0)\over (\pi t_0)^2}=0. \end{aligned}$$

On the other hand, it is plain that if \(\delta T\in (0,1/2)\), then one has

$$\begin{aligned} \{y\in {{\mathbb {R}}}: |y|>T_0\}\subseteq {{\mathcal {E}}}_{\mathbf{1}}(T,\delta ) =\biggl \{y\in {{\mathbb {R}}}: \big |\widehat{\mathbf{1}}_\delta (y)\big |^2< {\sin ^2(2\pi \delta T)\over (\pi T)^2}\biggr \} \end{aligned}$$

for some \(T_0\ge T\). It is easily seen that analogous considerations hold for the whole family \(\{C^{\langle j\rangle }_{\delta }\}_{j\ge 0}\). However, in several applications of (1.2), like in the case of Dirichlet polynomials (see [5]), the contribution from the tail \(|y|>T\) might amount to a remainder term. On the other hand, note that the inequality

$$\begin{aligned} \Vert S\Vert ^2_{2,T}\le m_{\delta ,T}^{-1}\Vert \widehat{w_\delta } S\Vert ^2_{2,T} \end{aligned}$$

trivially holds for any choice of \(w_\delta\) such that \(m_{\delta ,T}\not =0\). Even for this reason, it is worthwhile to compare weights in view of possible refinements of the first term on the right-hand side of (1.6). To this end, we give the following definition.

Definition 1.2

Let us suppose that \(w_\delta , v_\delta\) are such that \(\widehat{w_\delta }(t)\not =0\), \(\widehat{v_\delta }(t)\not =0\), \(\forall t\in [-T,T]\). We say that \(v_\delta\) is T-better than \(w_\delta\) if

$$\begin{aligned} \frac{\displaystyle {\min _{|t|\le T}|\widehat{v_\delta }(t)|^2}}{\displaystyle {\min _{|t|\le T}|\widehat{w_\delta }(t)|^2}} \ge {{|\widehat{v_\delta }(y)|^2} \over {|\widehat{w_\delta }(y)|^2}} \quad \forall y\in [-T,T]. \end{aligned}$$
(1.7)

If so, we write \(v_\delta \prec _{_T} w_\delta\).

If \(v_\delta \prec _{_T} w_\delta\), it is plain that \(v_\delta\) yields a refinement of the upper bound of \(\Vert S\Vert ^2_{2,T}\) with respect to \(w_\delta\), at least in \([-T,T]\). In §3 we prove the next result on the family of weights (1.5) and discuss the comparison involving also some weights as \(C_{\delta }*\mathbf{1}_\delta\) and the Lanczos weight.

Theorem 1.2

Let \(\delta , T\in (0,+\infty )\). For every \(j\ge 0\), if \(\delta T<2^{j-1}\), then \(C^{\langle j+1\rangle }_\delta \prec _{_T} C^{\langle j\rangle }_\delta\). Moreover, the inequality

$$\begin{aligned} \frac{\displaystyle {\min _{|t|\le T}\big |\widehat{C^{\langle j+1\rangle }_\delta }(t)\big |^2}}{\displaystyle {\min _{|t|\le T}\big |\widehat{C^{\langle j\rangle }_\delta }(t)\big |^2}} \ge {\big |\widehat{C^{\langle j+1\rangle }_\delta }(y)\big |^2\over \big |\widehat{C^{\langle j\rangle }_\delta }(y)\big |^2} \end{aligned}$$
(1.8)

still holds for all \(\displaystyle { |y|\in \bigcup _{k\in {{\mathbb {N}}}} \big [2^{j-1}(2k+1)/\delta +\varepsilon '_k,2^{j-1}(2k+3)/\delta -\varepsilon ''_k\big ]}\), where

$$\begin{aligned} \varepsilon '_k=\varepsilon '_k(j,\delta ,T),\quad \varepsilon ''_k=\varepsilon ''_k(j,\delta ,T)\in (0,2^{j-1}/\delta ) \end{aligned}$$

are such that \(\varepsilon '_k, \varepsilon ''_k\sim \delta T^2/(2^jk)-T/(2k)\) as \(\delta T\rightarrow 2^{j-1}\), \(\forall j,k\in {{\mathbb {N}}}\).

Remark 1.2

For any fixed \(j,k\in {{\mathbb {N}}}\), the interval \(\big [2^{j-1}(2k+1)\delta ^{-1}+\varepsilon '_k,2^{j-1}(2k+3)\delta ^{-1}-\varepsilon ''_k\big ]\) tends to cover the whole interval \([(2k+1)T,(2k+3)T]\) as \(\delta T\rightarrow 2^{j-1}\). Somehow this means that (1.8) holds almost everywhere in \({{\mathbb {R}}}\setminus [-T,T]\) as long as \(\delta T\rightarrow 2^{j-1}\) and it gives a further chance to get even a refinement of the second term on the left-hand side of (1.6) by replacing \(C^{\langle j\rangle }_\delta\) with \(C^{\langle j+1\rangle }_\delta\). Furthermore, such a refinement might rely also on the value of the series \(S((2k+1)T)\), \(\forall k\in {{\mathbb {Z}}}\).

Remark 1.3

An effective use of (1.2) with \(C^{\langle j\rangle }_{\delta }\) requires finding explicit expressions of such weights. For example, the normalized self-convolution of \(C_{\delta /2}\) is the so-called Jackson-de La Vallé Poussin weight \(C^{\langle 2\rangle }_{\delta }\), that is the following cubic spline [4, Problem 5.1.2 (v)]:

$$\begin{aligned} \delta ^{-1}(C_{\delta /2}*C_{\delta /2})(t)= \left\{ \begin{array}{lll} \displaystyle { {6|t|^3-6\delta t^2+\delta ^3}\over {3\delta ^3}}&{}\text{ if } |t|\le \delta /2,\\ \displaystyle {{2(\delta -|t|)^3\over 3\delta ^3}}&{} \text{ if } \delta /2<|t|\le \delta ,\\ 0&{} \text{ if } |t|>\delta . \end{array} \right. \end{aligned}$$

Note that the support of \(C^{\langle 2\rangle }_{\delta }\) is \([-\delta ,\delta ]\), as expected.

2 Proof of Theorem 1.1

Here we recall Pitt’s inequality:

Let \(1<p\le q<\infty\) be fixed and (U, V) be a (p, q)-pair of Pitt weights. There exists \(C=C(p,q,U.V)>0\) such that

$$\begin{aligned} \Vert V^{1/q}{\widehat{f}}\Vert _{q}\le C\Vert U^{1/p}f\Vert _p. \end{aligned}$$
(2.1)

for all f such that \(U^{1/p}f\in L^p({{\mathbb {R}}},{{\mathbb {C}}})\).

Proof

First, note that from the hypotheses on S and w it follows that

$$\begin{aligned} \sum _{\nu }\int _{{{\mathbb {R}}}} |a(\nu )w_\delta (x-\nu )|\,dx&= \sum _{\nu }|a(\nu )|\int _{|x-\nu |\le \delta }|w(x-\nu )|\,dx \\&=\Vert w\Vert _{1,\delta }\sum _{\nu } |a(\nu )|<\infty . \end{aligned}$$

Therefore, by applying Lebesgue’s dominated convergence theorem [16, Th.1.38] it turns out that \(a\star w_\delta \in L^1({{\mathbb {R}}},{{\mathbb {C}}})\) and

$$\begin{aligned} \widehat{(a\star w_\delta )}(y):=&\nonumber \int _{{{\mathbb {R}}}}(a\star w_\delta )(x)e(-xy)\,dx\\ =&\sum _{\nu } a(\nu )\int _{{{\mathbb {R}}}}w_\delta (x-\nu )e(-xy)\,dx\\ =&\sum _{\nu } a(\nu )e(-\nu y)\int _{{{\mathbb {R}}}}w_\delta (t)e(-ty)\,dt=S(-y)\widehat{w_\delta }(y). \end{aligned}$$

Then, observe that we can assume \(U^{1/p} (a\star w_\delta )\in L^p({{\mathbb {R}}},{{\mathbb {C}}})\) for \(1<p<\infty\), for otherwise the inequality

$$\begin{aligned} \Vert V^{1/q}S\, \widehat{w_\delta }\Vert _{q}\le C\, \Vert U^{1/p}(a\star w_\delta )\Vert _p. \end{aligned}$$
(2.2)

would be trivial. Hence, (2.2) follows by applying Pitt’s inequality (2.1), being plain that (recall that w, V are even functions)

$$\begin{aligned} \int _{{\mathbb {R}}}|S(y)\widehat{w_\delta }(y)|^qV(y)\,dy=\int _{{{\mathbb {R}}}}|S(-y)\widehat{w_\delta }(y)|^qV(y)\,dy. \end{aligned}$$

By taking any fixed \(T\in (0,+\infty )\), clearly (2.2) implies (1.1).

Now, let us prove that

$$\begin{aligned} \Vert V^{1/q}S\, {\widehat{w}}\Vert _{q}\le C\Vert U^{1/p}(a\star w)\Vert _p. \end{aligned}$$
(2.3)

As before, we can clearly assume that \(a\star w\in L^p({{\mathbb {R}}},{{\mathbb {C}}})\) and apply (2.2) with \(\delta =n\in {{\mathbb {N}}}\) to write

$$\begin{aligned} \Vert V^{1/q}S\, \widehat{w_n}\Vert _{q}\le C\Vert U^{1/p}(a\star w_n)\Vert _p. \end{aligned}$$
(2.4)

Since \(w_n(t)e(-ty)\) converges to \(w(t) e(-ty)\), with \(|w_n(t)e(-ty)|\le |w(t)|\) and \(w\in L^1({{\mathbb {R}}},{{\mathbb {C}}})\), the dominated convergence theorem yields

$$\begin{aligned} \lim _n \Vert V^{1/q}S\, \widehat{w_n}\Vert _{q}=\Vert V^{1/q}S\, {\widehat{w}}\Vert _{q}. \end{aligned}$$

On the other hand, the same theorem implies that

$$\begin{aligned} \lim _n\int _{{{\mathbb {R}}}}|(a\star w_n)(x)|^pU(x)\,dx= \int _{{{\mathbb {R}}}}|(a\star w)(x)|^pU(x)\,dx. \end{aligned}$$

Hence, (2.3) follows from (2.4) after passage to the limit as \(n\rightarrow \infty\). Again, by taking any fixed \(T\in (0,+\infty )\), it is plain that (1.1) holds if \(w_\delta\) is replaced by \(w\in L^1({{\mathbb {R}}},{{\mathbb {C}}})\). \(\square\)

Remark 2.1

It is well konwn that, by taking \(V(x)=|x|^{-\alpha q}\), \(U(x)=|x|^{\beta p}\) with

$$\begin{aligned} \max \{0,1/p+1/q-1\}\le \alpha <1/q\ \hbox {and}\ \beta :=\alpha +1-1/p-1/q, \end{aligned}$$
(2.5)

Pitt’s inequality (2.1) yields three classical inequalities in Fourier analysis:

  • the Hardy-Littlewood inequality if \(p=q\ge 2\), \(\alpha =0\) or \(1<p=q\le 2\), \(\beta =0\);

  • the Hausdorff-Young inequality if \(q=1-1/p\ge 2\), \(\alpha =\beta =0\);

  • the Plancherel identity if \(p=q=2\), \(\alpha =\beta =0\).

Accordingly, for \(V(x)=|x|^{-\alpha q}\) and \(U(x)=|x|^{\beta p}\) the inequality (2.2) turns into

$$\begin{aligned} \Vert \mathrm{v}^{-\alpha }\widehat{w_\delta }\, S\Vert _{q}\le C\Vert \mathrm{v}^{\beta }(a\star w_\delta )\Vert _p, \end{aligned}$$

where we have set \(\mathrm{v}(x)=|x|\). Thus, (1.1) specializes to the following instances paralleling the aforementioned three properties:

$$\begin{aligned}&\Vert S\Vert _{p,T}\le \frac{C}{\min _{|t|\le T}|\widehat{w_\delta }(t)|}\Vert \mathrm{v}^{1-2/p}(a\star w_\delta )\Vert _p,\quad \hbox {with}\ p=q\ge 2; \\&\Vert \mathrm{v}^{1-2/p}S\Vert _{p,T}\le \frac{C}{\min _{|t|\le T}|\widehat{w_\delta }(t)|}\Vert a\star w_\delta \Vert _p,\quad \hbox {with}\ 1<p=q\le 2; \\&\Vert S\Vert _{q,T}\le \frac{C}{\min _{|t|\le T}|\widehat{w_\delta }(t)|}\Vert a\star w_\delta \Vert _p,\quad \hbox {with}\ q=1-1/p\ge 2, \end{aligned}$$

(in the latter \(C=p^{1/(2p)}/q^{1/(2q)}\) is the so-called Beckner’s constant);

$$\begin{aligned} \Vert S\Vert _{2,T}\le \frac{C}{\min _{|t|\le T}|\widehat{w_\delta }(t)|}\Vert a\star w_\delta \Vert _2. \end{aligned}$$

Finally, we conclude this remark by recalling that the original version of Pitt’s inequality was tailored for the Fourier series and the power series [15]. In particular, by assuming that \(\nu \in {{\mathbb {Z}}}\) for our Fourier series S, Theorem 2 of [15] yields

$$\begin{aligned} \Big (\sum _{\nu \in {{\mathbb {Z}}}} |a(\nu )\nu ^{-\alpha }|^q\Big )^{1/q} \le C \Big (\int _0^1 |t^{\beta }S(t)|^p\,dt\Big )^{1/p}, \\ \Big (\int _0^1 |t^{-\alpha }S(t)|^q\,dt\Big )^{1/q} \le C \Big (\sum _{\nu \in {{\mathbb {Z}}}} |a(\nu )\nu ^{\beta }|^p\Big )^{1/p}, \end{aligned}$$

where \(p,q,\alpha ,\beta\) are as in (2.5).

3 Proof of Theorem 1.2: comparison of weights

Proof

Let us start with \(j=0\) and show that \(C_\delta\) is T-better than \(\mathbf{1}_\delta\) by assuming that \(\delta T<1/2\), namely we prove that

$$\begin{aligned} {\displaystyle {\min _{|t|\le T}\big |\widehat{C_\delta }(t)\big |^2}\over \displaystyle {\min _{|t|\le T}\big |\widehat{\mathbf{1}_\delta }(t)\big |^2}} \ge {\big |\widehat{C_\delta }(y)\big |^2\over \big |\widehat{\mathbf{1}_\delta }(y)\big |^2} \end{aligned}$$
(3.1)

holds for all \(y\in [-T,T]\). First, recall that \(\delta T<1/2\) yields

$$\begin{aligned} \min _{|t|\le T}\big |\widehat{\mathbf{1}_\delta }(t)\big |^2= {\sin ^2(2\pi \delta T)\over (\pi T)^2}\not =0,\quad \min _{|t|\le T}\big |\widehat{C_\delta }(t)\big |^2= {\sin ^4(\pi \delta T)\over (\pi T)^4\delta ^2}\not =0. \end{aligned}$$

Since

$$\begin{aligned} {\displaystyle {\min _{|t|\le T}\big |\widehat{C_\delta }(t)\big |^2}\over \displaystyle {\min _{|t|\le T}\big |\widehat{\mathbf{1}_\delta }(t)\big |^2}}= {\tan ^2(\pi \delta T)\over 4(\pi \delta T)^2}\ge \frac{1}{4}={\big |\widehat{C_\delta }(0)\big |^2\over \big |\widehat{\mathbf{1}_\delta }(0)\big |^2}, \end{aligned}$$

we can assume that \(y\not =0\) and set \(x=y/T\), \(\theta =\delta T\). Thus, (3.1) becomes

$$\begin{aligned} G_\theta (x):= {\big |\widehat{C_\delta }(xT)\big |^2\over \big |\widehat{\mathbf{1}_\delta }(xT)\big |^2}= {\tan ^2(\pi \theta x)\over (2\pi \theta x)^2} \le G_\theta (1). \end{aligned}$$

It is easy to see that \(G_\theta (x)\) satisfies the following properties:

  1. (i)

    \(G_\theta (x)\) is even with respect to both x and \(\theta\)

  2. (ii)

    \(G_\theta (x)\) is strictly increasing with respect to \(x\in (0,1]\)

  3. (iii)

    \(\displaystyle {\lim _{x\rightarrow 0} G_\theta (x)=1/4}\) \(\forall \theta \in (0,1/2)\)

  4. (iv)

    \(G_\theta (1)\) is strictly increasing with respect to \(\theta \in (0,1/2)\)

  5. (v)

    \(\displaystyle {\lim _{\theta \rightarrow 0}G_\theta (1)=1/4}\), \(\displaystyle {\lim _{\theta \rightarrow 1/2}G_\theta (1)=+\infty }\)

  6. (vi)

    \(G_\theta (x)=0\Longleftrightarrow x=k/\theta \ \forall k\in {{\mathbb {Z}}}{\setminus }\{0\}\) (note that \(\forall k\in {{\mathbb {Z}}}\setminus \{0\}\) and \(\forall \theta \in (0,1/2)\) one has \(|k|/\theta > 2|k|\ge 2\))

  7. (vii)

    \(G_\theta (x)\rightarrow +\infty\) as \(x\rightarrow (2k+1)(2\theta )^{-1}\ \forall k\in {{\mathbb {Z}}}\).

Note that \(|2k+1|(2\theta )^{-1}> |2k+1|\ge 1\), \(\forall k\in {{\mathbb {Z}}}\) and \(\forall \theta \in (0,1/2)\).

From properties (i)-(v) it follows that \(G_\theta (x) \le G_\theta (1)\) for all \(x\in [-1,1]{\setminus }\{0\}\), that is to say that (3.1) holds for all \(|y|\le T\), i.e., \(C_\delta \prec _{_T} \mathbf{1}_\delta\).

In order to prove the second part of the theorem for \(j=0\), observe that properties (i), (vi) and (vii) imply that the above inequality for \(G_\theta\) is true for all

$$\begin{aligned} |x|\in \bigcup _{k\in {{\mathbb {N}}}} {{\mathcal {I}}}_{\theta ,k}:= \bigcup _{k\in {{\mathbb {N}}}} [(2k+1)(2\theta )^{-1}+\Delta '_{k},(2k+3)(2\theta )^{-1}-\Delta ''_{k}], \end{aligned}$$

where \(0<\Delta '_{k}=\Delta '_{k}(\theta ), \Delta ''_{k}=\Delta ''_{k}(\theta )<(2\theta )^{-1}\) are such that the endpoints of the intervals \({{\mathcal {I}}}_{\theta ,k}\) are the solutions of the equation \(G_\theta (x)=G_\theta (1)\). In particular, this yields

$$\begin{aligned} \tan ^2(\pi /2+\pi \theta \Delta '_{k})=G_\theta (1) \big ((2k+1)\pi +2\pi \theta \Delta '_{k}\big )^2, \end{aligned}$$

with \(0<\theta \Delta '_{k}<1/2\). From the latter equation and from (v) we deduce that \(\Delta '_{k}\sim (\theta -1/2)k^{-1}\) as \(\theta =\delta T\rightarrow 1/2\), for all \(k\in {{\mathbb {N}}}\). An analogous property holds for \(\Delta ''_k\). Hence, we set \(\varepsilon '_k(0,\delta ,T)=T\Delta '_{k}, \varepsilon ''_k(0,\delta ,T)=T\Delta ''_{k}\) to get the theorem proved in the case \(j=0\). For \(j>0\) it sufficies to note that (see (1.5))

$$\begin{aligned} {\big |\widehat{C^{\langle j+1\rangle }_\delta }(y)\big |^2\over \big |\widehat{C^{\langle j\rangle }_\delta }(y)\big |^2}= \big ({\tan (\pi \delta y/2^j)\over \delta \pi y/2^{j-1}}\big )^{2^{j+1}} =\biggl ({\tan (\pi \theta x)\over 2\pi \theta x}\biggr )^{2^{j+1}}=G_\theta (x)^{2^{j}}, \end{aligned}$$

where we have set \(\theta =\delta T/2^j\), \(x=y/T\) and \(G_\theta (x)=(2\pi \theta x)^{-2}\tan ^2(\pi \theta x)\) as before. Since \(0<\delta T< 2^{j-1}\) if and only if \(0<\theta < 1/2\), the conclusion follows from what we have showed in the case \(j=0\). \(\square\)

Remark 3.1

Theorem 1.2 seems to indicate that one could generate more and more T-better weights by increasing the iteretad convolutions of \(\widehat{\mathbf{1}_\delta }\). However, the following examples suggest that one must exercise some caution about the choice of the weights to be compared.

Example 3.1

Let us compare \(C_{2\delta }\) and \(D_{2\delta }:= C_{\delta }*\mathbf{1}_\delta\). First, recall that

$$\begin{aligned} \widehat{C_{2\delta }}(y)=2\delta \hbox {sinc}^2(2\delta y),\ \widehat{D_{2\delta }}(y)=\widehat{C_\delta }(y)\widehat{\mathbf{1}_\delta }(y)= 2\delta ^2\hbox {sinc}^2(\delta y)\hbox {sinc}(2\delta y), \end{aligned}$$

so that

$$\begin{aligned} {\big |\widehat{D_{2\delta }}(y)\big |^2\over \big |\widehat{C_{2\delta }}(y)\big |^2} =\frac{\delta ^2\hbox {sinc}^4(\delta y)}{\hbox { sinc}^2(2\delta y)}= \left\{ \begin{array}{ll} \displaystyle { {{\tan ^2(\pi \delta y)}\over {(\pi y)^2}}}&{}\text{ if } y\not =0,\\ \delta ^2&{} \text{ if } y=0. \end{array} \right. \end{aligned}$$

Further, for \(\delta T<1/2\), one has

$$\begin{aligned} \min _{|t|\le T}\big |\widehat{D_{2\delta }}(t)\big |^2&= 4\delta ^4\min _{|t|\le T}\big (\hbox {sinc}^4(\delta t)\hbox {sinc}^2(2\delta t)\big )\\ {}&= 4\delta ^4\hbox {sinc}^6(\delta T)\cos ^2(\pi \delta T)\not =0, \\ \min _{|t|\le T}\big |\widehat{C_{2\delta }}(t)\big |^2&=4\delta ^2 \hbox {sinc}^4(2\delta T)\not =0. \end{aligned}$$

Since it turns out that

$$\begin{aligned} {\displaystyle {\min _{|t|\le T}\big |\widehat{D_{2\delta }}(t)\big |^2}\over \displaystyle {\min _{|t|\le T}\big |\widehat{C_{2\delta }}(t)\big |^2}}= {{\tan ^2(\pi \delta T)}\over {(\pi T)^2}}={\big |\widehat{D_{2\delta }}(T)\big |^2\over \big |\widehat{C_{2\delta }}(T)\big |^2}\ge {\big |\widehat{D_{2\delta }}(y)\big |^2\over \big |\widehat{C_{2\delta }}(y)\big |^2},\ \forall y\in [-T,T], \end{aligned}$$

then we conclude that \(D_{2\delta }\prec _{_T}C_{2\delta }\) once \(\delta T<1/2\).

Note that both \(C_{2\delta }\) and \(D_{2\delta }\) have support in \([-2\delta ,2\delta ]\). On the other hand, by comparing \(C_{\delta }\) and \(D_{2\delta }\) we see that

$$\begin{aligned} {\big |\widehat{D_{2\delta }}(y)\big |^2\over \big |\widehat{C_{\delta }}(y)\big |^2} ={|\widehat{C_\delta }(y)\widehat{\mathbf{1}_\delta }(y)|^2\over \big |\widehat{C_{\delta }}(y)\big |^2} =\big |\widehat{\mathbf{1}_\delta }(y)\big |^2= \left\{ \begin{array}{ll} \displaystyle { {{\sin ^2(2\pi \delta y)}\over {(\pi y)^2}}}&{}\text{ if } y\not =0,\\ 4\delta ^2&{} \text{ if } y=0, \end{array} \right. \end{aligned}$$

and, for \(\delta T<1/2\), one has

$$\begin{aligned} {\displaystyle {\min _{|t|\le T}\big |\widehat{D_{2\delta }}(t)\big |^2}\over \displaystyle {\min _{|t|\le T}\big |\widehat{C_{\delta }}(t)\big |^2}}= {{\sin ^2(2\pi \delta T)}\over {(\pi T)^2}}\le \big |\widehat{\mathbf{1}_\delta }(y)\big |^2= {\big |\widehat{D_{2\delta }}(y)\big |^2\over \big |\widehat{C_{2\delta }}(y)\big |^2},\quad \forall y\in [-T,T], \end{aligned}$$

that is to say, if \(\delta T<1/2\), then \(C_{\delta }\prec _{_T}D_{2\delta }\). Analogously, it is easy to see that \(C^{\langle j\rangle }_\delta \prec _{_T}C^{\langle j\rangle }_{2\delta }\) for all \(j\ge 0\) (and consequently \(C^{\langle j\rangle }_\delta \prec _{_T}C^{\langle j-1\rangle }_{2\delta }\) for all \(j\ge 1\), because of Theorem 1.2), as long as \(\delta T<2^{j-1}\).

Example 3.2

Given real numbers \(\delta \ge \gamma >0\), the Lanczos weight is defined as [4, Problem 5.1.2 (v)]

$$\begin{aligned} {{\mathcal {L}}}_{\delta ,\gamma }(x):={1\over {\gamma }}(\mathbf{1}_{{\delta -\gamma /2}}*\mathbf{1}_{\gamma /2})(x) = \left\{ \begin{array}{lll} 1&{}\text{ if } |x|\le \delta -\gamma ,\\ \displaystyle {{\delta -|x|}\over {\gamma }}&{} \text{ if } \delta -\gamma <|x|\le \delta ,\\ 0&{} \text{ if } |x|>\delta , \end{array} \right. \end{aligned}$$

whose Fourier transform is

$$\begin{aligned} \widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(y)&= (2\delta -\gamma )\hbox {sinc}(\gamma y)\hbox {sinc}((2\delta -\gamma ) y)\\ {}&= \left\{ \begin{array}{lll} \displaystyle { {{\sin (\pi \gamma y)\sin (2\pi \delta y-\pi \gamma y)}\over {(\pi y)^2\gamma }}}&{}\text{ if } y\not =0,\\ 2\delta -\gamma &{} \text{ if } y=0. \end{array} \right. \end{aligned}$$

The diagram of \({{\mathcal {L}}}_{\delta ,\gamma }\) is an isosceles trapezium. Note that \(\mathbf{1}_\delta \ge {{\mathcal {L}}}_{\delta ,\gamma }\ge C_\delta ={{\mathcal {L}}}_{\delta ,\delta }\) for any \(\delta \ge \gamma >0\). Since \({{\mathcal {L}}}_{\delta ,\delta }=C_\delta\), then we can assume that \(\delta>\gamma >0\). Further, by taking \(\gamma T<\delta T<(2\delta -\gamma )T<1/2\) one has \(\widehat{{\mathbf{1}}_{\delta }}(y)\not =0\) and \(\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(y)\not =0\) for all \(y\in [-T,T]\). Therefore, under such assumptions we can compare \({{\mathcal {L}}}_{\delta ,\gamma }\) and \(\mathbf{1}_\delta\) by verifying the inequality

$$\begin{aligned} {\big |\widehat{\mathbf{1}_\delta }(y)\big |^2\over \big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(y)\big |^2}&={{(2\delta )^2\hbox {sinc}^2(2\delta y)}\over { (2\delta -\gamma )^2\hbox {sinc}^2(\gamma y)\hbox {sinc}^2((2\delta -\gamma )y)}} \\&\ge {{(2\delta )^2\hbox {sinc}^2(2\delta T)}\over { (2\delta -\gamma )^2\hbox {sinc}^2(\gamma T)\hbox {sinc}^2((2\delta -\gamma )T)}} ={\displaystyle {\min _{|t|\le T}\big |\widehat{\mathbf{1}_\delta }(t)\big |^2}\over \displaystyle {\min _{|t|\le T}\big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(t)\big |^2}}. \end{aligned}$$

To this aim, let us write

$$\begin{aligned} {\displaystyle {\min _{|t|\le T}\big |\widehat{\mathbf{1}_\delta }(t)\big |^2}\over \displaystyle {\min _{|t|\le T}\big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(t)\big |^2}} =&\biggl ({{\pi \gamma T\sin (2\pi \delta T)}\over { \sin (\pi \gamma T)\sin (\pi (2\delta -\gamma ) T)}}\biggr )^2 \\ =&(\pi \gamma T)^2\biggl ({{\sin (\pi (2\delta -\gamma ) T+\pi \gamma T)}\over { \sin (\pi \gamma T)\sin (\pi (2\delta -\gamma ) T)}}\biggr )^2 \\ =&(\pi \gamma T)^2\biggl (\cot (\pi \gamma T)+\cot \big (\pi (2\delta -\gamma ) T\big )\biggr )^2 \\ =&F(\gamma T)^2+\frac{\gamma ^2}{(2\delta -\gamma )^2}F((2\delta -\gamma )T)^2+\\&\frac{2\gamma }{2\delta -\gamma }F(\gamma T) F((2\delta -\gamma )T), \end{aligned}$$

where \(F(x):= (\pi x)\cot (\pi x)\). Since \(F(x)\le 1\) for \(|x|\le 1/2\), we get

$$\begin{aligned} {\displaystyle {\min _{|t|\le T}\big |\widehat{\mathbf{1}_\delta }(t)\big |^2}\over \displaystyle {\min _{|t|\le T}\big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(t)\big |^2}} \le 1+\frac{\gamma ^2}{(2\delta -\gamma )^2} +\frac{2\gamma }{2\delta -\gamma } = {{(2\delta )^2}\over { (2\delta -\gamma )^2}}={\big |\widehat{\mathbf{1}_\delta }(0)\big |^2\over \big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(0)\big |^2}. \end{aligned}$$

Therefore, we can assume that \(y\not =0\) and write

$$\begin{aligned} {\big |\widehat{\mathbf{1}_\delta }(y)\big |^2\over \big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(y)\big |^2}= F(\gamma y)^2+\frac{\gamma ^2}{(2\delta -\gamma )^2}F((2\delta -\gamma )y)^2 +\frac{2\gamma }{2\delta -\gamma }F(\gamma y) F((2\delta -\gamma )y). \end{aligned}$$

Since it is readily seen that \(F(x)\ge F(x_0)\) for \(|x|\le x_0<1/2\), we conclude that \({{\mathcal {L}}}_{\delta ,\gamma }\prec _{_T}\mathbf{1}_\delta\) provided \(\gamma T<\delta T<(2\delta -\gamma )T<1/2\).

Finally, we show that \(C_{\delta }\prec _{_T}{{\mathcal {L}}}_{\delta ,\gamma }\), under the same conditions on \(\gamma\), \(\delta\) and T. Indeed, we prove that the inequality

$$\begin{aligned} {\big |\widehat{C_{\delta }}(y)\big |^2\over \big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(y)\big |^2}&={{\delta ^2\hbox {sinc}^4(\delta y) }\over {(2\delta -\gamma )^2\hbox {sinc}^2(\gamma y)\hbox { sinc}^2((2\delta -\gamma )y) }} \\&\le {{\delta ^2\hbox {sinc}^4(\delta T) }\over {(2\delta -\gamma )^2\hbox {sinc}^2(\gamma T)\hbox { sinc}^2((2\delta -\gamma )T) }}={\displaystyle {\min _{|t|\le T}\big |\widehat{C_{\delta }}(t)\big |^2}\over \displaystyle {\min _{|t|\le T}\big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(t)\big |^2}} \end{aligned}$$

holds for all \(y\in [-T,T]\). To this aim, let us recall that, being concave in \((-\pi ,\pi )\), the function \(\hbox {sinc}\) is also logarithmically concave in the same interval (see [3, Ch.3]), i.e., for all \(\lambda \in [0,1]\) and all \(x_1,x_2\in (-1,1)\) one has

$$\begin{aligned} \hbox {sinc}(\lambda x_1+(1-\lambda )x_2)\ge \hbox {sinc}^\lambda (x_1)\hbox {sinc}^{1-\lambda }(x_2). \end{aligned}$$

In particular, by taking \(\lambda =1/2\), \(x_1=\gamma T\) and \(x_2=(2\delta -\gamma )T\), with \(\gamma T<\delta T<(2\delta -\gamma )T<1/2\), this yields

$$\begin{aligned} {\displaystyle {\min _{|t|\le T}\big |\widehat{C_{\delta }}(t)\big |^2}\over \displaystyle {\min _{|t|\le T}\big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(t)\big |^2}} \ge {{\delta ^2 }\over {(2\delta -\gamma )^2 }}={\big |\widehat{C_{\delta }}(0)\big |^2\over \big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(0)\big |^2}. \end{aligned}$$

Therefore, let assume that \(y\not =0\) and write

$$\begin{aligned} {\big |\widehat{C_{\delta }}(y)\big |^2\over \big |\widehat{{{\mathcal {L}}}_{\delta ,\gamma }}(y)\big |^2}&= {{\gamma ^2\sin ^4(\pi \delta y) }\over {\delta ^2\sin ^2(\pi \gamma y)\sin ^2(\pi (2\delta -\gamma )y) }} \\&= {{\gamma ^2 }\over {(2\delta )^2 }} \biggl (1+{{1-\cos (\pi \gamma y)\cos (\pi (2\delta -\gamma )y) }\over {\sin (\pi \gamma y)\sin (\pi (2\delta -\gamma )y) }}\biggr )^2. \end{aligned}$$

Thus, it is plain that we get the desired conclusion once we prove that, for any fixed real numbers a, b such that \(0<a<b<\pi /(2T)\), the function

$$\begin{aligned} G_{a,b}(y):={{1-\cos (ay)\cos (by) }\over {\sin (ay )\sin (by) }}=\frac{1}{\sin (ay )\sin (by)}- \frac{1}{\tan (ay)\tan (by)} \end{aligned}$$

is monotone increasing in (0, T]. Indeed, it turns out that

$$\begin{aligned} \frac{d}{dy}G_{a,b}(y)= \frac{\cos (ay)-\cos (by) }{\sin ^2(ay )\sin ^2(by)}\big (b\sin (ay)-a\sin (by)\big )>0, \end{aligned}$$

because \(0<a<b<\pi /(2T)\) and \(y\in (0,T]\) yield \(\cos (ay)-\cos (by)>0\) and \((ay)^{-1}\sin (ay)>(by)^{-1}\sin (by)\).