1 Introduction

It is known that exponential Riesz bases \(\{e^{i \lambda _n t}\}\) (with \(\lambda _n\in \mathbb R\)) are stable in the sense that a small perturbation of a Riesz basis produces a Riesz basis; it is proved by Paley and Wiener ([5, 9]). The proof of the Paley–Wiener theorem does not provide an explicit stability bound. The celebrated theorem by M. I. Kadec shows that \(1/4\) is the stability bound for the exponential basis on \(L^2[-\pi ,\pi ]\).

The proof of theorem, as reported in the Young’s textbook [9], applies for sequences of real numbers. Even earlier, however, Duffin and Eachus [2] shows that the Paley–Wiener criterion is satisfied whenever the sequences are complex and \(\frac{\log 2}{\pi }\) is a stability bound. For Young (page 38): “Whether the constant \(\frac{\log 2}{\pi }\) can be replaced by \(1/4\) (for complex \(\lambda _n\)) remains an unsolved problem.” With Theorem C and Theorem D on  [2] they consider sets which are on the borderline of being near a given orthonormal set, while the last part of their paper gives a simple formula for constructing sets near a given orthonormal set. Afterward, Duffin and Eachus apply this result (Theorem D) to the sequence of functions \(\{e^{i\lambda _n x}\}\), where \(\{\lambda _n\}\), \(n=0,\pm 1, \pm 2,\ldots \) is a sequence of complex constants satisfying \(|\lambda _n-n|\le L\) for some constant L. The Duffin and Eachus’s approach is deeper and more general than one of Young; in fact their work speaks of orthonormal sets and not of basis. In their paper can be read the following: “The above results on the non-harmonic Fourier series are an extension of previous knowledge in two respects. In the first place, Paley and Wiener were forced to assume that \(\{\lambda _n\}\) was a real sequence. Secondly, they obtained the value \(1/\pi ^2\) where we have \(\ln 2/\pi \). The best value for L is not known; however a theorem of Levinson gives an upper limit of \(1/4\)”.

Theorem 1 seeks to overcome the limitations exhibited in the paper of Duffin and Eachus and in the book of Young for the Riesz basis, introducing a limitation on the imaginary part of \(\lambda _n\). A consequence of Theorem 1 and its corollary, is that the constant \(\frac{\log 2}{\pi }\) can be replaced by \(1/4\) (for complex \(\lambda _n\)).

Lastly, an example that shows \(1/4\) cannot be replaced by a larger constant for complex case, are given in the appendix. For the latest results on generalizations and extensions of Kadec’s theorem see: [1, 3, 6].

2 A class of sequences that improves the estimation of Duffin and Eachus

Theorem 1

If \(\{\bar{\lambda }_n\}=\{\lambda _n+i\mu _n\}\) is a sequence of complex numbers for which

$$\begin{aligned} |\lambda _n-n|\leqq L<\frac{1}{4}, \ \ n=0, \pm 1, \pm 2,\ldots \end{aligned}$$
(1)

and

$$\begin{aligned} |\mu _n|\leqq \tau (L)<\frac{1}{\pi } \ln \left( \frac{2}{2-\cos \pi L+ \sin \pi L}\right) , \ \ n=0, \pm 1, \pm 2,\ldots \end{aligned}$$
(2)

then \(\{e^{i \bar{\lambda }_n t}\}\) satisfies the Paley–Wiener criterion and so forms a Riesz basis for \(L^2[-\pi ,\pi ]\).

Proof

It is to be shown that \(\left\| \sum _{n}^{+\infty } c_n \left( e^{i n t}-e^{i \bar{\lambda }_n t}\right) \right\| <1\) whenever \(\sum _n |c_n|^2\leqq 1\). Write

$$\begin{aligned} e^{i n t}-e^{i \bar{\lambda }_n t}&= e^{i n t}\left( 1-e^{i \delta _n t}\ e^{-\mu _n t}\right) \nonumber \\&= e^{i n t}\left[ 1-e^{-\mu _n t}+e^{-\mu _n t}\left( 1-e^{i \delta _n t}\right) \right] \end{aligned}$$
(3)

where \(\delta _n=\lambda _n-n\). This time again, the trick is to expand the function \(1-e^{i \delta t}\) (\(-\pi \le t\le \pi \)) in a Fourier series relative to the complete orthonormal system \(\left\{ 1,\cos nt, \sin \big (n-\frac{1}{2}\big )t\right\} _{n=1}^{\infty }\) and then exploit the fact that \(|\lambda _n - n|\) is not too large. Then the expansion of \(1-e^{i \delta t}\) is the same as the previous theorem. Let \(\{c_n\}\) be an arbitrary finite sequence of scalars such that \(\sum |c_n|^2\le 1\). By interchanging the order of summation, using triangle inequality and the notation introduced in the Kadec’s theorem on [9], it shows

$$\begin{aligned}&\left\| \sum _{n}^{+\infty } c_n e^{i n t}\left[ 1-e^{-\mu _n t}+e^{-\mu _n t}\left( 1-e^{i \delta _n t}\right) \right] \right\| \end{aligned}$$
(4)
$$\begin{aligned}&\le \sup _n\left| 1-e^{-\mu _n t}\right| \left\| \sum _{n}^{+\infty } c_n e^{i n t}\right\| + \sup _n\left( e^{-\mu _n t}\right) \left( A+B+C\right) \end{aligned}$$
(5)

From the assumptions of the theorem it is easily seen that \(\sup _n \left( e^{-\mu _n t}\right) \le e^{\tau \pi }\) and \(\sup _n \Big |1-e^{-\mu _n t}\Big |\le e^{\tau \pi }-1\) where \(\tau =\tau (L)\). Now by some estimates on fraction expansions proved in  [3], it has that

$$\begin{aligned} \left\| \sum _{n}^{+\infty } c_n \left( e^{i n t}-e^{i \bar{\lambda }_n t}\right) \right\| \le e^{|M|}-1+ e^{|M|}\left( 1- \cos \pi L+ \sin \pi L\right) =:\lambda \end{aligned}$$
(6)

It is observed that with arbitrary \(L<1/4\) and

$$\begin{aligned} \tau (L)<\frac{1}{\pi } \ln \left( \frac{2}{2-\cos \pi L+ \sin \pi L}\right) \end{aligned}$$
(7)

is obtained \(\lambda <1\). \(\square \)

The following result shows that, in the hypotheses of the Theorem 1, it has \(\{e^{i \bar{\lambda }_n t}\}\) satisfies the Paley–Wiener criterion for \(|\bar{\lambda }_n-n|< 1/4\) even when \(\{\bar{\lambda }_n\}\) is a complex sequence.

Corollary 1

For each \(L<\frac{1}{4}\), one has

$$\begin{aligned} (i)\ \ |\mu _n|\le \frac{\ln 2}{\pi }; \ \ \ \ \ \ (ii) \ \ |\bar{\lambda }_n-n|\le \frac{1}{4} \end{aligned}$$
(8)

Proof

The proof of first relation (i) is trivial and is left to the reader. Noting that

$$\begin{aligned} |\bar{\lambda }_n-n|\le |\lambda _n - n|+|\mu _n|\le L+\frac{1}{\pi } \ln \left( \frac{2}{2-\cos \pi L+ \sin \pi L}\right) \end{aligned}$$
(9)

relation (ii) is verified if \(\bar{x}-\ln \left( 1+\frac{\sin \bar{x}-\cos \bar{x}}{2}\right) \le \frac{\pi }{4}\) with \(\bar{x}=\pi L\). Let us consider the function \(f(\bar{x})\), defined as follow:

$$\begin{aligned} f(\bar{x})=\bar{x}-\ln \left( 1+\frac{\sin \bar{x}-\cos \bar{x}}{2}\right) \end{aligned}$$
(10)

It comes to prove that the function \(f(\bar{x})-\bar{x}:=g(\bar{x})\) is convex. Rewrite the function \(g(\bar{x})\) using the relationship \((\sin \bar{x}-\cos \bar{x})/2=\frac{\sqrt{2}}{2}\sin \left( \bar{x}-\frac{\pi }{4}\right) \) and so \(g(x)=-\ln \left( 1+\frac{\sqrt{2}}{2}\sin x\right) \) for \(x=\bar{x}-\pi /4\). Bearing in mind that a function is convex if and only if it is midpoint convex, it must be demonstrated that \(2g\left( \frac{x+y}{2}\right) \le g(x)+g(y)\), and hence

$$\begin{aligned} -2\ln \left( 1+\frac{\sqrt{2}}{2}\sin \frac{x+y}{2}\right) \le -\ln \left( 1+\frac{\sqrt{2}}{2}\sin x\right) -\ln \left( 1+\frac{\sqrt{2}}{2}\sin y\right) \nonumber \\ \end{aligned}$$
(11)

where \(y=\bar{y}-\pi /4\). From properties of logarithms and by applying Prosthaphaeresis formulas, Werner formulas, and half-angle formulae, it has

$$\begin{aligned} \sqrt{2}\sin \left( -\frac{x+y}{2}\right) \le \cos ^2 \frac{x-y}{4} \end{aligned}$$
(12)

Rewriting \(-\frac{x+y}{2}=\frac{\pi }{4}-\frac{\bar{x}-\bar{y}}{2}-\bar{y}\) \(\le \frac{\pi }{4}-t\) with \(t=\frac{\bar{x}-\bar{y}}{2}\in [0,\pi /4]\), it becomes \(\sqrt{2}\sin \left( \frac{\pi }{4}-t\right) \le \cos ^2 \frac{t}{2}\) , that is verified over \([0,\pi /4]\). Then \(f(x)\) is convex. Denoting with \(P_1(0,\ln 2)\), \(P_2(\pi /4,\pi /4)\) two points belonging to graphic of \(f(x)\) and from an obvious properties of convex functions: \(f(x)\le \frac{\pi -\ln 16}{\pi }x+\ln 2\) (the straight line for \(P_1, P_2\)), by the right side term that is less than \(\frac{\pi }{4}\) if \(x\le \frac{\pi }{4}\), it is concluded the claim.

3 Conclusions

Theorem 1 and Corollary 1 responding to the outstanding questions of Duffin, Eachus and Young, essentially because this paper shows that the constant \(\frac{\log 2}{\pi }\) can be replaced by \(1/4\), also for the complex case. Moreover, from Corollary 1, it has \(\{e^{i \bar{\lambda }_n t}\}\) satisfies the Paley–Wiener criterion for \(|\bar{\lambda }_n-n|< 1/4\) even when \(\{\bar{\lambda }_n\}\) is a complex sequence. Two lemmas present in appendix (an extension to complex case of result present on [9]) prove that Kadec’s \(1/4\)-theorem is “best possible”: the system \(\{e^{i \bar{\lambda }_n t}\}\) constitutes a basis for \(L^2[-\pi ,\pi ]\) whenever every \(\bar{\lambda }_n\) is complex and \(|\lambda _n-n|\leqq L\), \(|\mu _n|\leqq \tau (L)\) but not constitute a basis when \(L=1/4\). Equally interesting is the fact that \(\tau (L)\) is not specified in the proofs of Lemmas 1 and 2 and, into this proofs, it is not necessary that it assumes the logarithmic expression (2).

In Duffin and Eachus [2] one reads: “It is a curious parallelism that \(\log 2/\pi \) and \(1/4\) are in the same ratio as the limits of Takenaka and Schoenberg in a somewhat similar unsolved problem”. See: [7, 8]. In  [8] is reported a particular case of one of Takenaka’s theorems [7]: “If every derivative of an integral function \(f(z)\) has a zero inside or on the unit circle and if \(\limsup _{r\rightarrow \infty }\frac{\log M(r)}{r}<\log 2\) then \(f(z)\) is a costant”. [\(M(r)\) is the maximum modulus in \(|z|\le r\) of function]. The author write that this condition is probably not “best possible”: \(\sin \frac{\pi }{4}z-\cos \frac{\pi }{4}z\) shows that \(\log 2\) cannot be replaced by any number larger than \(\pi /4\), and this may well be the true value. A possible development of the work would be compare proof of Kadec’s-\(1/4\) theorem (complex case) with question in [8].