Continuity of some operators arising in the theory of superoscillations

Abstract

The study of superoscillations naturally leads to the analysis of a large class of convolution operators acting on spaces of entire functions. In particular, the key point is often the proof of the continuity of these operators on appropriate spaces. Most papers in the current literature utilize abstract methods from functional analysis to establish such continuity. In this paper, on the other hand, we rely on some recent advances in the study of entire functions, to offer explicit proofs of the continuity of such operators. To demonstrate the applicability and the flexibility of these explicit methods, we will use them to study the important case of superoscillations associated with quadratic Hamiltonians. The paper also contains a list of interesting open problems, and we have collected as well, for the convenience of the reader, some well-known results, and their proofs, on Gamma and Mittag–Leffler functions that are often used in our computations.

Appendix

We state in this section some well-known results on the gamma function and the Mittag–Leffler functions that we have used in the proofs.

Lemma 5.1

Let j, $k\in \mathbb {N}$, then we have

$$\begin{aligned} (j+k)!\le 2^{j+k}j!k!. \end{aligned}$$

Proof

Let $p\atopwithdelims ()j$ be the binomial coefficients, then it is well known that from the Newton binomial formula, we have

$$\begin{aligned} 2^p=\sum _{j=0}^p{p\atopwithdelims ()j}=\sum _{j=0}^p\frac{p!}{j!(p-j)!}, \end{aligned}$$

so

$$\begin{aligned} \frac{p!}{j!(p-j)!}\le 2^p \end{aligned}$$

and setting $p-j=k$ we get the statement. $\square$

Lemma 5.2

Let n, $k\in \mathbb {N}$, then we have

$$\begin{aligned} \Gamma (n+1)\Gamma (k+1)\le \Gamma (n+k+2). \end{aligned}$$

Proof

Let

$$\begin{aligned} B(p,q):=\int _0^1t^{p-1}(1-t)^{q-1}\mathrm{d}t \end{aligned}$$

be the beta function B. Its relation with the gamma function $\Gamma$ is given by

$$\begin{aligned} B(p,q)=\frac{\Gamma (p)\Gamma (q)}{\Gamma (p+q)}. \end{aligned}$$

This can be shown in two steps. First, with the change of variable $t=\cos ^2(\vartheta )$ the beta function can be written as

$$\begin{aligned} B(p,q)=2\int _0^{\pi /2}\cos ^{2p-1}(\vartheta )\sin ^{2q-1}(\vartheta ) \, \mathrm{d}\vartheta \end{aligned}$$

Second, we observe that

$$\begin{aligned} \Gamma (p)\Gamma (q)=\int _0^\infty \mathrm{e}^{-t}t^{p-1}\, \mathrm{d}t\int _0^\infty \mathrm{e}^{-s}s^{q-1}\, \mathrm{d}s \end{aligned}$$

and with the change of variables

$$\begin{aligned} t=r^2\cos ^2(\vartheta ),\quad s=r^2\sin ^2(\vartheta ) \end{aligned}$$

the previous formula becomes

$$\begin{aligned} \Gamma (p)\Gamma (q)=4\int _0^{\pi /2}\int _0^\infty r^{2p+2q-1} \mathrm{e}^{-r^2} \cos ^{2p-1}(\vartheta )\sin ^{2q-1}(\vartheta ) \, \mathrm{d}\vartheta \,\mathrm{d}r . \end{aligned}$$

By setting $r^2=u$ in the above relation we obtain

$$\begin{aligned} \Gamma (p)\Gamma (q)=\Gamma (p+q)B(p,q). \end{aligned}$$

Finally, we observe that

$$\begin{aligned} \frac{\Gamma (n+1)\Gamma (k+1)}{\Gamma (n+k+2)}=B(n+1,k+1)\le \int _0^1t^{n}(1-t)^{k}\mathrm{d}t\le \int _0^1 \mathrm{d}t= 1; \end{aligned}$$

since, for $t\in [0,1]$ it is $t^{n}(1-t)^{k}\le 1$, and this ends the proof. $\square$

We conclude with a useful estimate that we have not used in this paper, but it enters into several problems in convolution operators associated with superoscillations.

Lemma 5.3

Let $q\in [1,\infty )$. Then we have

$$\begin{aligned} \Gamma \Big (\frac{n}{q}+1\Big )\le (n!)^{1/q}. \end{aligned}$$

Proof

It is a direct consequence of Hölder inequality. Consider p and q such that $1/p+1/q=1$, observe that

$$\begin{aligned} \Gamma \Big (\frac{n}{q}+1\Big )= & {} \int _0^\infty \mathrm{e}^{-t}t^{n/q}\, \mathrm{d}t\\= & {} \int _0^\infty \mathrm{e}^{-t(1/p+1/q)}t^{n/q}\, \mathrm{d}t, \end{aligned}$$

so we obtain

$$\begin{aligned} \Gamma \left( \frac{n}{q}+1\right)= & {} \int _0^\infty \mathrm{e}^{-t/q}t^{n/q}\ \mathrm{e}^{-t/p}\, \mathrm{d}t\\\le & {} \left( \int _0^\infty \mathrm{e}^{-t}t^{n}\ \, \mathrm{d}t\right) ^{1/q} \left( \int _0^\infty \mathrm{e}^{-t}\, \mathrm{d}t\right) ^{1/p}\\= & {} \left( \int _0^\infty \mathrm{e}^{-t}t^{n}\ \, \mathrm{d}t\right) ^{1/q}\\= & {} (n!)^{1/q}. \end{aligned}$$

$\square$

1.1 On the Mittag–Leffler function

The Mittag–Leffler function is defined by its power series

$$\begin{aligned} E_\alpha (z)=\sum _{k=0}^\infty \frac{z^k}{\Gamma (\alpha k+1)},\quad \alpha \in \mathbb {C},\quad \mathrm{Re}(\alpha )>0. \end{aligned}$$

The series converges in the whole complex plane for all $\alpha \in \mathbb {C},\ \mathrm{Re}(\alpha )>0$. For all $\mathrm{Re}(\alpha )<0$ it diverges everywhere on $\mathbb {C}\setminus \{0\}$. For $\mathrm{Re}(\alpha )=0$ the radius of convergence is $R=\mathrm{e}^{\pi |Im(\alpha )|/2}$. The most interesting fact is that for $\mathrm{Re}(\alpha )>0$ the Mittag–Leffler function is an entire function of finite order. Indeed using Stirling’s asymptotic formula

$$\begin{aligned} \Gamma (\alpha k+1)=\sqrt{2\pi } (\alpha k)^{\alpha k+1/2}\mathrm{e}^{-\alpha k}(1+o(1)),\quad \text {for} \ k\rightarrow \infty , \end{aligned}$$

so that for

$$\begin{aligned} c_k=\frac{1}{\Gamma (\alpha k+1)} \end{aligned}$$

for $\alpha >0$ we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }\frac{k\ln k}{\ln \frac{1}{|c_k|}}=\limsup _{k\rightarrow \infty }\frac{k\ln k}{\ln |\Gamma (\alpha k+1)|}=\frac{1}{\alpha } \end{aligned}$$

and

$$\begin{aligned} \limsup _{k\rightarrow \infty }\left( k^{1/\rho } \root k \of {|c_k|}\right) =\limsup _{k\rightarrow \infty }\left( k^{1/\rho } \root k \of {\frac{1}{|\Gamma (\alpha k+1)|}}\right) =(e/\alpha )^{\alpha }. \end{aligned}$$

This means that:

for each $\alpha \in \mathbb {C}$ such that $Re(\alpha )>0$ the Mittag–Leffler function is an entire function of order $\rho =1/Re(\alpha )$ and of type $\sigma =1$.

This function provides a generalization of the exponential function because we replace $k!=\Gamma (k+1)$ by $(\alpha k)!=\Gamma (\alpha k+1)$ in the denominator of the power terms of the exponential series. A useful generalization that we have used in the computations of this paper is the two-parametric Mittag–Leffler function

$$\begin{aligned} E_{\alpha ,\beta }(z)=\sum _{k=0}^\infty \frac{z^k}{\Gamma (\alpha k+\beta )},\quad \alpha ,\ \beta \in \mathbb {C},\quad \mathrm{Re}(\alpha )>0. \end{aligned}$$

The function $E_{\alpha ,\beta }(z)$ for $\alpha ,\ \beta \in \mathbb {C}$ and $Re(\alpha )>0$ is an entire function of $\rho =1/\mathrm{Re}(\alpha )$ and of type $\sigma =1$ for every $\beta \in \mathbb {C}$.