# Analyticity and supershift with regular sampling

F. Colombo  (FC) Politecnico di Milano
Dipartimento di Matematica
Via E. Bonardi, 9
20133 Milano
Italy<EMAIL_ADDRESS>,  I. Sabadini  (IS) Politecnico di Milano
Dipartimento di Matematica
Via E. Bonardi, 9
20133 Milano
Italy<EMAIL_ADDRESS>,  D. C. Struppa  (DCS) The Donald Bren
Presidential Chair in Mathematics
Chapman University, Orange, CA 92866
USA<EMAIL_ADDRESS>and  A. Yger  (AY) IMB, Université de Bordeaux,
33405, Talence, France<EMAIL_ADDRESS>
###### Abstract.

The notion of supershift (in itself a generalization of the notion of
superoscillation arising in quantum mechanics) expresses the fact that the
sampling of a function in an interval allows to compute the values of the
function far from the interval. In this paper we study the relation between
supershift and real analyticity. We use a classical result due to Serge
Bernstein to show that real analyticity for a complex valued function implies
a strong form of supershift. On the other hand, we use a parametric version of
a result by Leonid Kantorovitch to show that the converse is not true. We
additionally study the stability of functions that satisfy suitable supershift
requirements under multiplication by complex numbers or primitivization.

The first and second author are supported by MUR grant Dipartimento di
Eccellenza 2023-2027.

Keywords. Real analyticity; supershift; superoscillation; sampling.

AMS classification 42C10, 26E05

##  1\. Introduction

In their seminal paper [1], Aharonov and his collaborators introduced a new
concept in quantum mechanics, the so-called weak value, and as a consequence
they obtained some apparently paradoxical results (for a complete review of
the notion of weak value see [17]). One of the reasons why such results were
possible is the discovery of a family of functions from ${\mathbb{R}}$ to
${\mathbb{C}}$ with the surprising property that they are completely
determined, on the real axis, by their values on a countable subset of a
finite interval in $\mathbb{R}$. The first example of such a function is the
exponential function $f(x,a):a\longrightarrow e^{iax}$ regarded as a function
of the real variable $a$, with $x$ as a parameter. Specifically, it is
possible to write the function $a\longmapsto e^{iax}$, for arbitrary large
values of $a$, in terms of the values $e^{i\lambda_{N,\nu}x}$ with
$\lambda_{N,\nu}=1-2\nu/N$ for $N\in\mathbb{N}^{*}$ and $0\leq\nu\leq N$, and
therefore $|\lambda_{N,\nu}|\leq 1.$ To show that this is the case, one
considers the function

$F_{N}(x,a):=\big{(}\cos\frac{x}{N}+ia\sin\frac{x}{N}\big{)}^{N},$ (1.1)

which (by simply using the Euler formulas for sine and cosine, and the Newton
binomial formula) can be rewritten as

$F_{N}(x,a)=\sum_{\nu=0}^{N}C_{\nu}(N,a)e^{i(1-2\nu/N)x},$

where

$C_{\nu}(N,a):={\binom{N}{\nu}\Big{(}\frac{1+a}{2}\Big{)}^{N-\nu}\Big{(}\frac{1-a}{2}\Big{)}^{\nu}}.$
(1.2)

Since (1.1) immediately implies that
$\lim_{N\rightarrow+\infty}\,F_{N}(x,a)=e^{iax}$, we obtain that the value of
the function $a\longmapsto f(x,a)=e^{iax}$ for $a$ arbitrarily large is
retrievable if one knows the values of the function on the countable set of
values $\lambda_{N,\nu}\in[-1,1]$.

Since in this particular example the variable $a$ is the frequency of the
oscillating exponential function, now to be considered as a function of $x$,
this phenomenon is usually referred to as superoscillation and we call a
sequence such as $\\{x\longmapsto F_{N}(x,a)\\}_{N\geq 1}$ a superoscillating
sequence, and the limit function a superoscillating function.

The fact that large oscillations can be obtained by superposition of much
smaller oscillations, so that a highly oscillating function could be seen as a
limit of band-limited functions, led to a series of surprising physical
phenomena, some of which are described in [9, 10] and were recently the
subject of a more popular discussion in [21].

One of the first questions that Aharonov posed was whether this interesting
phenomenon would be conserved if one evolves a superoscillating sequence
according to the Schrödinger equation for the free particle. As we have shown
in [5, 6], the answer is positive, indeed one can prove that the solution to
the Cauchy problem

$i\frac{\partial\psi(x,t)}{\partial t}=-\frac{\partial^{2}\psi(x,t)}{\partial
x^{2}}\psi(x,t),\ \ \ \ \psi(x,0)=F_{N}(x,a),$

is given by

$\psi_{N}(x,t)=\sum_{k=0}^{N}C_{k}(N,a)e^{i(1-2k/N)x}e^{-i(1-2k/N)^{2}t},$
(1.3)

and that $\psi_{N}(x,t)$ converges to $e^{iax-ia^{2}t}$ uniformly on every
compact set in $\mathbb{R}^{2}$.

This result is of independent interest, but it also shows the existence of a
simple case of superoscillations in two variables. This realization led to the
development of an independent theory of superoscillations in several
variables, see [4, 7]. Once the case of the free particle is settled, the next
natural question is to evolve a superoscillatory sequence with different
potentials. The authors have obtained a large number of results in this
direction but even the next simplest case, namely the harmonic oscillator
studied in [13], highlights a completely new phenomenon. Indeed we proved that
the Cauchy problem for the quantum harmonic oscillator

$i\frac{\partial\psi(t,x)}{\partial
t}=\frac{1}{2}\Big{(}-\frac{\partial^{2}}{\partial
x^{2}}+x^{2}\Big{)}\psi(t,x),\ \ \ \ \psi(0,x)=F_{N}(x,a)$

has the solution

$\psi_{N}(t,x)=(\cos t)^{-1/2}e^{-(i/2)x^{2}\tan
t}\sum_{k=0}^{N}C_{k}(N,a)e^{ix(1-2k/N)/\cos t-(i/2)(1-2k/N)^{2}\tan t}$ (1.4)

and when we take the limit for $N\to\infty$ we get

$\lim_{N\to\infty}\psi_{N}(t,x)=(\cos t)^{-1/2}e^{-(i/2)x^{2}\tan
t}e^{-(i/2)a^{2}\tan t+iax/\cos t}.$

As one can see, the function in (1.4) is not superoscillating as the function
in (1.3). The phenomenon that arises is now the fact that a function can be
described in points away from the origin if one knows its value in enough
points near the origin. This phenomenon (of which superoscillations is the
simplest example) is what we called supershift (a precise definition was given
in [14] and will be repeated later on in Section 2). As it turned out, we were
able to demonstrate this phenomenon in a very large number of cases, and we
noticed that the possibility of doing so seemed to be a direct consequence of
the analyticity of the functions for which supershift was to be demonstrated.
This raises the very natural question of whether the two notions are indeed
connected. In this paper we will show that in fact analyticity implies
supershift (this result is already implied in some of our previous works) but
that the converse is not true. This result goes full circle because one of
Aharonov’s early remarks was that, in some naive sense, the property of
superoscillations can be considered akin to analyticity. Indeed, an analytic
function is completely determined by countably many data at the origin (the
value, and all of its derivatives) and a superoscillating function is
completely determined by countably many data near the origin (its values in
the points $\lambda_{N,\nu}$).

The plan of the paper is the following. Section 2 contains the various
definitions and refinements of the notions of superoscillation and supershift
that we shall use later. In Section 3 we show that the notion of analyticity
is stronger than a suitably formalized notion of supershift. Specifically, in
Theorem 3.4, using a parametric reformulation of a classical result due to
Serge Bernstein we show that real analyticity for a complex valued function
implies a strong form of supershift. On the other hand, we use a parametric
version of a result by Leonid Kantorovitch to show, see Theorem 3.8, that the
converse is not true. In Section 4 we additionally study the stability of
functions that satisfy suitable supershift requirements under essential
operations such as complex multiplication or primitivization (Theorems 4.1 and
4.2).

##  2\. Preliminary results

In this section we will offer some preliminary definitions and results that
will be useful in the sequel. To begin with, let

$H=\begin{bmatrix}&h_{1,0},...,h_{1,\nu(1)}\\\
&h_{2,0},h_{2,1},...,h_{2,\nu(2)}\\\ &\vdots\\\
&h_{N,0},h_{N,1},h_{N,2},...,h_{N,\nu(N)}\\\ &\vdots\end{bmatrix}$ (2.1)

be a collection of real numbers (interpreted as frequencies) which all belong
to the closed interval $[-1,1]$ with $h_{N,\nu}\geq h_{N,\nu+1}$ for
$N\in\mathbb{N}^{*}$ and $0\leq\nu\leq\nu(N)-1$. The simplest example of such
double index sequence can be obtained by setting $\nu(N)=N$ and
$h_{N,\nu}=1-2(\nu+1)/N$, $0\leq\nu\leq N-1$.

######  Definition 2.1.

A sequence of generalized trigonometric polynomials $(T_{H,N})_{N\geq 1}$ of
the form

${T_{H,N}}(x)=\sum\limits_{\nu=0}^{\nu(N)}C_{H,\nu}(N)\,\exp(ixh_{N,\nu}),\quad({\rm
with}\ C_{H,\nu}(N)\in\mathbb{C}\quad{\rm for}\ N\in\mathbb{N}^{*},\
0\leq\nu\leq\nu(N))$

is called an $H$-sequence of generalized trigonometric polynomials.

Given an open subset $A\subset\mathbb{R}$, an $(A,H)$-sequence of generalized
trigonometric polynomials $\\{T_{H,N}[a]:a\in A\\}_{N\geq 1}$ is by definition
an $A$-parametrized $H$-sequence of generalized trigonometric polynomials such
that each complex amplitude $C_{H,\nu}(N,a)$ depends continuously on $a\in A$,
for any $N\in\mathbb{N}^{*}$ .

######  Definition 2.2 (Superoscillating sequence).

An $(A,H)$-sequence of generalized trigonometric polynomials is said to be
superoscillating if there are two continuous functions
$g_{H}\leavevmode\nobreak\ :A\rightarrow\mathbb{C}$ and
$C_{H}:A\rightarrow\mathbb{R}$ such that both $V=\\{a\in
A\,:\,C_{H}(a)\not=0,\ |g_{H}(a)|>1\\}$ and the open subset $U$ of points
$x\in\mathbb{R}$ about which

$\lim_{N\rightarrow+\infty}T_{H,N}[a]\,(x)=C_{H}(a)\,\exp(i\,g_{H}(a)\,x),$
(2.2)

locally uniformly with respect to $(a,x)$, are non-empty open subsets
respectively of $A$ and $\mathbb{R}$. The set $U\cap V$ is then called the
superoscillating subset of the $(A,H)$-superoscillating sequence
$\\{T_{N}[a]\,:\,a\in A\\}_{N\geq 1}$.

######  Remark 2.3.

What motivates the terminology here is the fact that, for any $a\in A$, each
$T_{H,N}[a]$ is a low-band signal, with frequencies in $[-1,1]$, while the
signal $x\mapsto C_{H}(a_{0})\,e^{ig_{H}(a_{0})x}$ is a true elementary
oscillating signal with frequency $g_{H}(a_{0})$ outside $[-1,1]$ as soon as

$a_{0}\in\\{a\in A\,:\,C_{H}(a)\not=0,\ |g_{H}(a)|>1\\}\not=\emptyset.$

For a further insight to superoscillations see also [11]. Throughout this
paper, we shall consider $(\mathbb{R},H)$-superoscillating sequences
subordinate to the choice of $g={\rm Id}_{\mathbb{R}}$ and $C\equiv 1$.

Generalizing the original description of Aharonov, it is easy to realize
prototypical examples of superoscillating $(\mathbb{R},H)$-sequences of
generalized trigonometric polynomials as we describe below.

######  Definition 2.4 (Regular sampling).

We will call regular sampling of the frequency interval $[-1,1]$ the sampling
defined by $H^{\boldsymbol{\epsilon}}=[h_{N,\nu}^{\boldsymbol{\epsilon}}]$
where $\boldsymbol{\epsilon}=(\epsilon_{N})_{N\geq 1}$ is a sequence of
elements in $[0,1[$ which tends to $0$ when $N$ tends to infinity and

$h_{N,\nu}^{\boldsymbol{\epsilon}}=1-2\,\Big{(}\frac{\nu+\epsilon_{N}(N-\nu)}{N}\Big{)},\
0\leq\nu\leq N\leavevmode\nobreak\ .$ (2.3)

######  Remark 2.5.

The $(\mathbb{R},H^{\boldsymbol{\epsilon}})$-sequence
$\\{T_{N}^{\boldsymbol{\epsilon}}[a]\,:\,a\in\mathbb{R}\\}_{N\geq 1}$, where

$\begin{split}T_{N}^{\boldsymbol{\epsilon}}[a]\,(x)&=\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\,\Big{(}\frac{1+a}{2}\Big{)}^{N-\nu}\Big{(}\frac{1-a}{2}\Big{)}^{\nu}\exp\big{(}ih_{N,\nu}^{\boldsymbol{\epsilon}}\,x\big{)}\\\
&=\exp(i\,\epsilon_{N}x)\,\Big{(}\frac{1+a}{2}\,\exp\Big{(}ix\,\frac{1-\epsilon_{N}}{N}\Big{)}+\frac{1-a}{2}\,\exp\Big{(}-ix\,\frac{1-\epsilon_{N}}{N}\Big{)}\Big{)}^{N}\\\
&=\exp(i\,\epsilon_{N}x)\,\Big{(}\cos\Big{(}x\,\frac{1-\epsilon_{N}}{N}\Big{)}+i\,a\,\sin\Big{(}x\,\frac{1-\epsilon_{N}}{N}\Big{)}\Big{)}^{N}\end{split}$
(2.4)

is superoscillating on $U=\mathbb{R}$.

######  Remark 2.6.

When the array $H$ in (2.1) is such that $\forall\,N\in\mathbb{N}^{*},\
\nu(N)=N$ and so ${\rm card}\,\\{h_{N,\nu}\,:\,\nu=0,...,\nu(N)\\}=N+1$, the
sequence of Lagrange-Hermite interpolators (with respect to the parameter $a$)

$\Big{\\{}x\longmapsto T_{H,N}^{\rm
Lag}[a](x):=\sum\limits_{\nu=0}^{N}\Big{(}\prod_{\nu^{\prime}\not=\nu}\frac{a-h_{N,\nu^{\prime}}}{h_{N,\nu}-h_{N,\nu^{\prime}}}\Big{)}\,e^{ih_{N,\nu}x}\,:\,a\in\mathbb{R}\Big{\\}}$
(2.5)

is also superoscillating since

$\begin{split}\Big{|}e^{iax}-T_{H,N}^{\rm
Lag}[a](x)\Big{|}=\frac{\prod_{\nu=0}^{N}|a-h_{N,\nu}|}{(N+1)!}\sup_{|\xi|\leq\max(1,|a|)}\Big{|}\Big{(}\frac{\partial}{\partial
a}\Big{)}^{N+1}[e^{iax}](\xi)\Big{|}\leq\frac{\big{(}(|a|+1)|x|\big{)}^{N+1}}{(N+1)!}\end{split}$
(2.6)

for any $(a,x)\in\mathbb{R}^{2}$ according to the expression of the reminder
term in the Lagrange interpolation formula, see [3, Theorem 2.2]. Such a
superoscillating sequence $\\{T_{H,N}^{\rm
Lag}[a]\,:\,a\in\mathbb{R}\\}_{N\geq 1}$ could be interpreted as corresponding
to an irregular sampling of the low-frequency domain $[-1,1]$, which occurs as
soon as an infinitely many rows $H_{N_{\iota}}$ of the array $H$ are such that
the function $\nu\in\\{0,...,N_{\iota}-1\\}\mapsto
h_{N_{\iota},\nu}-h_{N_{\iota},\nu+1}>0$ is not constant.

Let us introduce here the important concept of Supershift Property $({\rm
SP})_{\mathscr{F}}$.

######  Definition 2.7.

(Supershift Property $({\rm SP})_{\mathscr{F}}$) Let $A$ be an open interval
of $\mathbb{R}$, possibly $\mathbb{R}$ itself, which contains $[-1,1]$ and let
$\psi:a\in A\longmapsto\psi_{a}\in\mathscr{F}$ be a continuous map from $A$ to
a topological $\mathbb{C}$-vector space $\mathscr{F}$. Let

$\left\\{{T_{H,N}}[\lambda](x)=\sum\limits_{\nu=0}^{\nu(N)}C_{H,\nu}(N,\lambda)\,\exp(ixh_{N,\nu})\,:\,\lambda\in\mathbb{R}\right\\}_{N\geq
1}$

be an $(\mathbb{R},H)$-superoscillating sequence. The map $\psi$ is said to
satisfy the Supershift Property $({\rm SP})_{\mathscr{F}}$ on $A$ with respect
to $\\{T_{H,N}[\lambda]\,:\,\lambda\in\mathbb{R}\\}_{N\geq 1}$ if the sequence
of functions

$a\in
A\longmapsto\sum\limits_{\nu=0}^{\nu(N)}C_{H,\nu}(N,a)\,\psi_{h_{N,\nu}}\in\mathscr{F},\quad
N=1,2,...$ (2.7)

converges to $\psi_{a}$ in the space of continuous functions
$\mathcal{C}(A,\mathscr{F})$, with respect to the topology of uniform
convergence on any compact subset, i.e.

$\lim_{N\to\infty}\sum\limits_{\nu=0}^{\nu(N)}C_{H,\nu}(N,a)\,\psi_{h_{N,\nu}}=\psi_{a}.$

######  Remark 2.8.

The reader will notice that this definition reduces, for $C_{\nu}(N,a)$ as in
(1.2), to the original example of Aharonov, which can in fact be obtained by
taking $A={\mathbb{R}}$, and
$H=H^{\boldsymbol{0}}:=\\{1-2\nu/N\,:\,N\in\mathbb{N}^{*},\ 0\leq\nu\leq
N\\}$.

Given a topological $\mathbb{C}$-vector space $\mathscr{F}$, we also introduce
for continuous maps from an open interval $A\subset\mathbb{R}$ with length
strictly larger than $2(={\rm length}\,([-1,1])$) to $\mathscr{F}$, the so-
called Translation-Commuting Supershift Property $({\rm TCSP})_{\mathscr{F}}$
with respect to an $(\mathbb{R},H)$-superoscillating sequence.

######  Definition 2.9.

(Translation-Commuting Supershift Property $({\rm TCSP})_{\mathscr{F}}$) Let
$A$ be as above and for $a\in A$, let $\psi_{a}\in\mathscr{F}$ be a continuous
map from $A$ to $\mathscr{F}$. Let
$\\{T_{H,N}[\lambda]\,:\,\lambda\in\mathbb{R}\\}_{N\geq 1}$ be a
$(\mathbb{R},H)$-superoscillating sequence. The continuous map $\psi$ is said
to satisfy the $\mathscr{F}$-Translation-Commuting Supershift Property $({\rm
TCSP})_{\mathscr{F}}$ on $A$ with respect to the superoscillating sequence
$\\{T_{H,N}[\lambda]\,:\,\lambda\in\mathbb{R}\\}_{N\geq 1}$ if the sequence of
functions

$\left(\sum\limits_{\nu=0}^{\nu(N)}C_{H,\nu}(N,a)\,\psi_{a^{\prime}+h_{N,\nu}}\right)\subset\mathscr{F},\quad
N=1,2,...$ (2.8)

defined for

$(a,a^{\prime})\in\mathbb{A}:=\big{\\{}(a,a^{\prime})\in\mathbb{R}\times
A\,:\,a^{\prime}+[-1,1]\subset A,\ a+a^{\prime}\in A\big{\\}}$

converges to $\psi_{a+a^{\prime}}$ in $\mathcal{C}(\mathbb{A},\mathscr{F})$
with respect to the topology of uniform convergence on any compact subset of
$\mathbb{A}$.

Observe that $({\rm TCSP})_{\mathscr{F}}$ on an open interval $A$ of
$\mathbb{R}$ with respect to such an $(\mathbb{R},H)$ superoscillating
sequence implies $({\rm SP})_{\mathscr{F}}$ with respect to the same
superoscillating sequence provided that $A$ contains $[-1,1]$.

######  Remark 2.10.

If $A=[-\alpha,\beta]$ with $\alpha,\beta>1$, then the set $\mathbb{A}$ can be
equivalently described as

$\mathbb{A}=\\{(x,y)\in\mathbb{R}^{2}\ :1-\alpha\leq y\leq-1+\beta,\
-\alpha\leq x+y\leq\beta\\}.$

Let $\mathcal{H}(\mathbb{C})$ be the space of entire functions and let us
consider

${\rm Exp}(\mathbb{C})=\\{\Phi\in\mathcal{H}(\mathbb{C})\ :\
\exists\,R>0\quad{\rm such\
that}\quad\sup_{z\in\mathbb{C}}e^{-R|z|}|\Phi(z)|=0\\}.$ (2.9)

For any $R>0$, let also consider the subspace of ${\rm Exp}(\mathbb{C})$
defined by:

${\rm Exp}_{<R}(\mathbb{C})=\\{\Phi\in\mathcal{H}(\mathbb{C})\ :\
\exists\,\varepsilon\in\,]0,R]\quad{\rm such\
that}\quad\sup_{z\in\mathbb{C}}e^{-(R-\epsilon)|z|}|\Phi(z)|<+\infty\\}.$
(2.10)

Both spaces are topological spaces with their AU-structures, see for example
[8, Definition 4.1.10]. We recall that the dual space of ${\rm
Exp}_{<R}(\mathbb{C})$ is isomorphic to
$\mathcal{H}(\mathbb{D}_{\mathbb{C}}(0,R))$, see [25, §6, Example 8] or also
[8, Proposition 4.1.16].

Let $A$ be an open interval of $\mathbb{R}$ which contains $[-1,1]$. Among the
$(\mathbb{R},H)$-superoscillating sequences, those for which the sequence

$\Big{(}z\in\mathbb{C}\longmapsto\sum\limits_{\nu=0}^{\nu(N)}C_{N,\nu}(N,a)e^{ih_{N,\nu}z}\Big{)}_{N\geq
1}$ (2.11)

converges locally uniformly with respect to $a\in A$ towards $z\mapsto
e^{iaz}$ in ${\rm Exp}(\mathbb{C})$ (or at least in ${\rm
Exp}_{<R}(\mathbb{C})$ for some $R>\sup_{A}|a|>1$ when $A$ is bounded) will be
of particular interest for us.

Let $\mathscr{F}$ be a topological $\mathbb{C}$-vector space and let
${\boldsymbol{K}}:{\rm Exp}(\mathbb{C})\rightarrow\mathscr{F}$ be a continuous
operator. Then the continuous map

$\psi_{{\boldsymbol{K}}}\leavevmode\nobreak\ :a\in
A\longmapsto{\boldsymbol{K}}(z\mapsto e^{iaz}),$ (2.12)

(or, more briefly, ${\boldsymbol{K}}(e^{iaz})$) satisfies $({\rm
SP})_{\mathscr{F}}$ on $A$ with respect to the
$(\mathbb{R},H)$-superoscillating sequence which is considered here. The same
argument applies to the continuous operator ${\boldsymbol{K}}_{R}:{\rm
Exp}_{<R}(\mathbb{C})\rightarrow\mathscr{F}$ and the continuous map

$\psi_{{\boldsymbol{K}}_{R}}\leavevmode\nobreak\ :a\in
A\longmapsto{\boldsymbol{K}}_{R}(e^{iaz})\ {\rm when}\ A\subset]-R,R[.$

Moreover, since the Taylor sequence

$\Big{(}\sum\limits_{\kappa=0}^{M}(ia)^{\kappa}\,\frac{z^{\kappa}}{\kappa!}\Big{)}_{M\geq
0}$ (2.13)

converges in ${\rm Exp}(\mathbb{C})$ towards $z\mapsto e^{iaz}$ locally
uniformly with respect to $a\in A$, one has

$\begin{split}\psi_{{\boldsymbol{K}}}(a)&=\lim\limits_{M\rightarrow+\infty}\sum\limits_{\kappa=0}^{M}{\boldsymbol{K}}\Big{(}\frac{(iz)^{\kappa}}{\kappa!}\Big{)}\,a^{\kappa}\end{split}$
(2.14)

locally uniformly with respect to $a\in A$, which shows that
$\psi_{{\boldsymbol{K}}}$ inherits analyticity. Similarly, when (2.13)
converges in ${\rm Exp}_{<R}(\mathbb{C})$, in the case $A\subset]-R,R[$, then
the limit function

$\begin{split}\psi_{{\boldsymbol{K}}_{R}}(a)&=\lim\limits_{M\rightarrow+\infty}\sum\limits_{\kappa=0}^{M}{\boldsymbol{K}}_{R}\Big{(}\frac{(iz)^{\kappa}}{\kappa!}\Big{)}\,a^{\kappa}\end{split}$
(2.15)

also inherits analyticity. More precisely, given any element $T$ in the
topological $\mathbb{C}$-dual of $\mathscr{F}$,
$T\circ\psi_{{\boldsymbol{K}}}$ is the restriction to $A$ of an entire
function, while $T\circ\psi_{{\boldsymbol{K}}_{\mathbb{R}}}$ is the
restriction to $A$ of a holomorphic function in $D_{\mathbb{C}}(0,R)$. It is
easy to see that such setting can be easily modified when $A$ is any open
interval of $\mathbb{R}$ with length strictly larger than $2$. If the
$(\mathbb{R},H)$-superoscillating sequence is such that the sequence of entire
functions

$\Big{(}z\in\mathbb{C}\longmapsto\sum\limits_{\nu=0}^{\nu(N)}C_{N,\nu}(N,a)e^{i(a^{\prime}+h_{N,\nu})z}\Big{)}_{N\geq
1}$

parametrized now by $(a,a^{\prime})\in\mathbb{A}$ as in (2.8) converges in
${\rm Exp}(\mathbb{C})$ (or at least in ${\rm Exp}_{<R}(\mathbb{C})$ in case
$A\subset]-R,R[$) to $z\longmapsto\exp(i(a+a^{\prime})z)$, this time locally
uniformly with respect to both parameters $(a,a^{\prime})\in\mathbb{A}$, the
continuous maps (2.12) satisfy the $({\rm TCSP})_{\mathscr{F}}$ property on
$A$. Such continuous functions inherit on $A$ the analyticity property (2.14)
in view of the continuity of ${\boldsymbol{K}}$ (or ${\boldsymbol{K}}_{R}$).

Let us now consider a specific Cauchy problem for the Schrödinger equation.
Let $U$ be an open interval in ${\mathbb{R}}$, and consider for
$a\in{\mathbb{R}}$

$\Big{(}i\,\frac{\partial}{\partial t}+\frac{\partial^{2}}{\partial
x^{2}}-V(x)\Big{)}\psi(t,x)=0,\quad\psi(0,x)=e^{iax},$ (2.16)

where $x\in U\mapsto V(x)$ is a suitable real continuous potential. The most
classical examples of open sets $U\subset\mathbb{R}$ which we shall consider
are $U=\mathbb{R}$ or $U=\mathbb{R}_{>0}$. When a Green function
$K_{V}\leavevmode\nobreak\ :(x,y,t)\in U\times
U\times\mathbb{R}_{>0}\longmapsto K_{V}(x,y,t)$ is explicitly known for such
Cauchy problem, the evolution of the initial datum
$x\mapsto\psi(0,x)=e^{iax}$, when $a\in\mathbb{R}$, is described (at least
formally), by

${\psi_{a}(t,x)}=\int_{U}e^{iay}\,K_{V}(x,y,t)\,dy.$

The classical potentials $V$ we have in mind are quadratic. For example
$U=\mathbb{R}$, $V(x)=x^{2}/2$ (quantum harmonic oscillator), see [8, §6.4] or
also [16, §5.4 & 6]; $U=\mathbb{R}_{>0}$, $V(x)=\varpi/x^{2}$ with $\varpi>0$
an absolute constant (centrifugal potential), see [16, §5.3], or possibly the
more general case where $x\mapsto V(t,x)$ is quadratic in $x$ as a polynomial,
see [27]. Under some requirements imposed to the explicit expression of the
Green kernel $K_{V}$, among them the essential fact that it extends as a
function of $y$ to an holomorphic function of $z$ in a single (or double)
sector with vertex $0$ about the real axis, see [2, Assumption 3.1], the
formal expression

$\int_{U}e^{iay}\,K_{V}(x,y,t)\,dy$

can be realized concretely as a regularized integral in the Fresnel sense, see
[16, §4 & §5] and [2, §2 & §3], which implies that

$\begin{split}&e^{iax}\longmapsto\int_{U}e^{iay}K_{V}(x,y,t)\,dy,\quad(t,x)\in\mathbb{R}_{>0}\times
U\ {\rm or}\ \mathbb{R}_{\geq 0}\times U\\\
&e^{iax}\longmapsto\Big{\langle}\varphi(t,x)\,,\,\int_{U}\,e^{iay}\,K_{V}(x,y,t)\,dy\Big{\rangle},\quad\varphi\in\mathcal{D}(\mathbb{R}_{>0}\times
U,\mathbb{C})\ {\rm or}\ \mathcal{D}(\mathbb{R}_{\geq 0}\times
U,\mathbb{C})\end{split}$ (2.17)

are realized in terms of the action of continuous operators from ${\rm
Exp}(\mathbb{C})$ (or ${\rm Exp}_{<R}(\mathbb{C})$ for some $R>0$) to
$\mathbb{C}$. Therefore the maps which are obtained by composition on the
right with the map (parametrized as in (2.17) by $(t,x)$ or $\varphi$,
considered as elements in $\mathscr{F}^{\prime}$) with the map which
associates to $a\in A\subset\mathbb{R}$ the initial datum $x\mapsto e^{iax}$
inherit automatically real analyticity besides the properties $({\rm
SP})_{\mathbb{C}}$ or $({\rm TCSP})_{\mathbb{C}}$ with respect to particular
$(\mathbb{R},H)$-superoscillating sequences. Such is the case, in particular,
when the superoscillating sequence which is involved is the
$(\mathbb{R},H^{\boldsymbol{\epsilon}})$-superoscillating sequence defined by
(2.3) and (2.4).

We conclude this section by discussing an example in which the evolution of
$e^{iax}$ can be calculated explicitly in the sense of distributions and yet
real analyticity does not follow.

######  Example 2.11.

We consider the case when $U=\mathbb{R}$ and
$V(x)=\sum_{\ell\in\mathbb{Z}}\widehat{V}(\ell)\,e^{i\ell x}$ is a smooth non-
constant $2\pi$-periodic real potential independent of $t$, the simplest
example being $V(x)=\cos(x-\phi)$ with $\phi\in\mathbb{R}$. For each $t>0$ and
$\ell\in\mathbb{Z}$, let
${\boldsymbol{k}}_{t,\ell}:\ell^{\infty}_{\mathbb{Z}}\longrightarrow\ell^{\infty}_{\mathbb{Z}}$
be the isometric operator defined by

${\boldsymbol{k}}_{t,\ell}:\
(u_{\kappa})_{\kappa\in\mathbb{Z}}\in\ell^{\infty}_{\mathbb{Z}}\longmapsto\big{(}-i\,e^{i(\kappa^{2}-(\kappa-\ell)^{2})t}\,u_{\kappa-\ell}\big{)}_{\kappa\in\mathbb{Z}}\in\ell^{\infty}_{\mathbb{Z}}.$

Inspired by the formal computations performed in [18], one can prove, see [15,
§4], that, in the sense of distributions on $\mathbb{R}_{>0}\times\mathbb{R}$,

$\psi_{a}(t,x)=e^{-ia^{2}t}e^{iax}\,\sum\limits_{\kappa\in\mathbb{Z}}(e^{iat})^{-2\kappa}\,v_{t,\kappa}(e^{ia(\cdot)})\,e^{-i\kappa^{2}t}\,e^{i\kappa
x},$ (2.18)

where

$\big{(}v_{t,\kappa}(e^{ia(\cdot)})\big{)}_{\kappa\in\mathbb{Z}}=\exp\Big{(}\int_{0}^{t}\Big{(}\sum\limits_{\ell\in\mathbb{Z}}\widehat{V}(\ell)\,(e^{ia\tau})^{2\ell}\,{\boldsymbol{k}}_{\tau,\ell}\Big{)}\,d\tau\Big{)}(\delta_{0}).$
(2.19)

One can interpret (2.18), when combined with (2.19), as a substitute for the
Green representation formula

$\psi_{a}(t,x)=\int_{U}e^{iay}\,K_{V}(x,y,t)dy,\ \ \ x\in U,\ \ t>0,$ (2.20)

for $K_{V}$ the Green function associated with the periodic potential $V$. If
$\mathscr{F}$ denotes the topological space of complex-valued distributions
$\mathcal{D}^{\prime}(\mathbb{R}_{>0}\times\mathbb{R},\mathbb{C})$ equipped
with its strong dual topology, a continuous map $a\mapsto\psi_{a}$ from
$\mathbb{R}$ to $\mathscr{F}$ is said to be real analytic (according for
example to [22, Definition 1.5.1 & Proposition 1.5.2]) if and only if it
extends to a neighborhood ${\boldsymbol{U}}$ of $\mathbb{R}$ in the complex
plane as a continuous function
$z\in{\boldsymbol{U}}\rightarrow\psi_{z}\in\mathscr{F}$ such that
$z\in{\boldsymbol{U}}\longmapsto\langle\psi_{z},\varphi\rangle$ is holomorphic
for any test-function
$\varphi\in\mathcal{D}(\mathbb{R}_{>0}\times\mathbb{R},\mathbb{C})$. Such is
not the case for the function

$a\in\mathbb{R}\longmapsto\sum\limits_{\kappa\in\mathbb{Z}}(e^{iat})^{-2\kappa}\,v_{t,\kappa}(e^{ia(\cdot)})\,e^{-i\kappa^{2}t}\,e^{i\kappa
x}\in\mathscr{F},$

hence for the function $a\in\mathbb{R}\longrightarrow\psi_{a}$ in view of
(2.18). It may then happen, as it is the case for such class of non-constant
smooth $2\pi$-periodic real potentials, that the evolution
$(t,x)\mapsto\psi_{a}(t,x)$ of the initial datum $x\mapsto e^{iax}$ in (2.16)
fails to inherit real analyticity in $a$, when analyticity of distribution-
valued continuous functions is understood as above.

##  3\. Regular $\mathbb{C}$-supershifts, Bernstein polynomials and
Kantorovitch examples

It is convenient in this section to re-scale the real line $\mathbb{R}$
through the affine bijective map $\Upsilon:a\mapsto 2a-1$ from $\mathbb{R}$ to
itself that maps $[0,1]$ onto $[-1,1]$. The $(\mathbb{R},H)$ superoscillating
sequences that we consider are those which correspond to regular sampling on
$[-1,1]$, namely

$H=H^{\boldsymbol{\epsilon}}:=\big{\\{}h_{N,\nu}^{\boldsymbol{\epsilon}}=1-2\Big{(}\frac{\nu+\epsilon_{N}(N-\nu)}{N}\Big{)}\,:\,N\in\mathbb{N}^{*},\
0\leq\nu\leq N\big{\\}}$ (3.1)

as in (2.3) and

$C_{H^{\boldsymbol{\epsilon}},\nu}(N,a):=C_{H^{\boldsymbol{0}},\nu}(N,a)=\binom{N}{\nu}\Big{(}\frac{1+a}{2}\Big{)}^{N-\nu}\Big{(}\frac{1-a}{2}\Big{)}^{\nu},\
0\leq\nu\leq N.$ (3.2)

Let us first recall why such sequences are indeed superoscillating.

######  Proposition 3.1.

Let $\boldsymbol{\epsilon}=(\epsilon_{N})_{N\geq 1}$, $\epsilon_{N}\in[0,1[$,
such that $\lim_{N\rightarrow+\infty}\epsilon_{N}=0$ and $a\in\mathbb{R}$.
Then, the sequence of entire functions with exponential growth whose $N$-th
element is

$T_{N}^{\boldsymbol{\epsilon}}[a](z)=\exp(i\,\epsilon_{N}z)\,\Big{(}\cos\Big{(}z\,\frac{1-\epsilon_{N}}{N}\Big{)}+i\,a\,\sin\Big{(}z\,\frac{1-\epsilon_{N}}{N}\Big{)}\Big{)}^{N}$
(3.3)

converges in ${\rm Exp}(\mathbb{C})$ towards the entire function $z\longmapsto
e^{iaz}$. Moreover the convergence is uniform with respect to the parameters
$a$ and $\boldsymbol{\epsilon}$ provided $a$ remains in a compact
$K_{\mathbb{R}}$ of $\mathbb{R}$ and $\boldsymbol{\epsilon}$ belongs to a
family
$\\{\boldsymbol{\epsilon}_{\iota^{\prime}}=(\epsilon_{\iota^{\prime},N})_{N\geq
1}\,:\,\iota^{\prime}\in I^{\prime}\\}$ of sequences
$\boldsymbol{\epsilon}_{\iota^{\prime}}\in([0,1])^{\mathbb{N}^{*}}$ which all
converge towards $0$, such convergence being uniform with respect to
$\iota^{\prime}$. Then the $(\mathbb{R},H^{\boldsymbol{\epsilon}})$-sequence
$\\{T_{N}^{\boldsymbol{\epsilon}}[a]\,:\,a\in\mathbb{R}\\}_{N\geq 1}$ is
superoscillating.

###### Proof.

Such result is known. We refer for example to [16, Lemma 4.1] with $z$
replaced by $z(1-\epsilon_{N})$. The uniformity properties (with respect to
$a\in K_{\mathbb{R}}$) and to $\boldsymbol{\epsilon}$ in a family
$\\{\boldsymbol{\epsilon}_{\iota^{\prime}}=(\epsilon_{\iota^{\prime},N})_{N\geq
1}\,:\,\iota^{\prime}\in I^{\prime}\\}$ such as described in the statement are
also immediate to check. ∎

######  Remark 3.2.

Let $A$ be an open interval of $\mathbb{R}$ which contains the segment
$[-1,1]$. To say that a continuous map $\psi:A\rightarrow\mathbb{C}$ satisfies
the $({\rm SP})_{\mathbb{C}}$ property on $A$ with respect to the
superoscillating sequence
$\\{T_{N}^{\boldsymbol{\epsilon}}[a]\,:\,a\in\mathbb{R}\\}_{N\geq 1}$, see
Definition 2.7 is equivalent to say that the sequence of complex valued
continuous functions

$\Big{(}b\in
B\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}\,\Psi\Big{(}\nu\,\frac{1-\epsilon_{N}}{N}\Big{)}\Big{)}_{N\geq
1},$ (3.4)

where

$B:=\Upsilon^{-1}(A),\ \Psi=\psi\circ\Upsilon,$ (3.5)

converges to $\Psi$ in $\mathcal{C}(B,\mathbb{C})$. When $A$ is an interval of
$\mathbb{R}$ with length strictly larger than $2$, to say that a continuous
map $\psi:A\rightarrow\mathbb{C}$ satisfies the $({\rm TCSP})_{\mathbb{C}}$
property on $A$ with respect to the superoscillating sequence
$\\{T_{N}^{\boldsymbol{\epsilon}}[a]\,:\,a\in\mathbb{R}\\}_{N\geq 1}$, see
Definition 2.9, is equivalent to say that the sequence of complex valued
continuous functions

$\Big{(}(b,b^{\prime})\in\mathbb{B}\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}\,\Psi\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{N}}{N}\Big{)}\Big{)}_{N\geq
1},$ (3.6)

where

$B=\Upsilon^{-1}(A),\
\mathbb{B}:=\Upsilon^{-1}(\mathbb{A})=\\{(b,b^{\prime})\in\mathbb{R}\times
B\,:\,b^{\prime}+[0,1]\subset B,\ b+b^{\prime}\in B\\},\
\Psi=\psi\circ\Upsilon,$ (3.7)

converges to $(b,b^{\prime})\in\mathbb{B}\longmapsto\Psi(b+b^{\prime})$ in
$\mathcal{C}(\mathbb{B},\mathbb{C})$.

Let us first recall in our context a well known result relating Serge
Bernstein’s approximation operators with iterated forward discrete
differentiation on the real line.

######  Proposition 3.3.

Let $A$ be an interval of $\mathbb{R}$ with length strictly larger than $2$,
$\psi\in\mathcal{C}(A,\mathbb{C})$, and let $\mathbb{B}$, $\Psi$ be defined as
in (3.7). For any sequence $\boldsymbol{\epsilon}=(\epsilon_{N})_{N\geq 1}$ of
elements in $[0,1[$ which tends to $0$ when $N$ tends to infinity one has, for
any $(b,b^{\prime})\in\mathbb{B}$ and $N\in\mathbb{N}^{*}$, that

$\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}\,\Psi\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{N}}{N}\Big{)}=\sum\limits_{\kappa=0}^{N}\frac{N!}{(N-\kappa)!}\frac{(\Delta_{(1-\epsilon_{N})/N}^{+})^{\kappa}[\Psi]\,(b^{\prime})}{\kappa!}\,b^{\kappa},$
(3.8)

where

$\begin{split}(\Delta_{(1-\epsilon_{N})/N}^{+})^{0}[\Psi]\,(b^{\prime})&=\Psi(b^{\prime})\\\
(\Delta_{(1-\epsilon_{N})/N}^{+})^{1}[\Psi]\,(b^{\prime})&=\Psi(b^{\prime}+(1-\epsilon_{N})/N)-\Psi(b^{\prime})\\\
(\Delta_{(1-\epsilon_{N})/N}^{+})^{2}[\Psi]\,(b^{\prime})&=(\Delta_{1-\epsilon_{N}/N}^{+})^{1}[\Psi](b^{\prime}+(1-\epsilon_{N})/N)-(\Delta_{(1-\epsilon_{N})/N}^{+})^{1}[\Psi]\,(b^{\prime})\\\
&=\Psi(b^{\prime}+2(1-\epsilon_{N})/N)-2\,\Psi(b^{\prime}+(1-\epsilon_{N})/N)+\Psi(b^{\prime})\\\
(\Delta_{(1-\epsilon_{N})/N}^{+})^{3}[\Psi]\,(b^{\prime})&=\cdots\end{split}$
(3.9)

are the successive forward (+) differences of $\Psi$ at $b^{\prime}$ (with
rate of sampling $(1-\epsilon_{N})/N$).

###### Proof.

When $\boldsymbol{\epsilon}=\boldsymbol{0}$, the result follows from the
classical combinatorial identity

$\sum\limits_{\nu=0}^{N}\binom{N}{\nu}X^{\nu}(1-X)^{N-\nu}\,c(\nu)=\sum\limits_{\nu=0}^{N}\binom{N}{\kappa}\,\Delta_{1/N}^{\kappa}[\,c\,](0)\,X^{\kappa},$

see for example [23, Theorem 7.1.1]. Replacing $\boldsymbol{0}$ by
$\boldsymbol{\epsilon}$ amounts just to change the rate $1/N$ used to compute
right-divided differences to $(1-\epsilon_{N})/N$. For each $b^{\prime}$ such
that $b^{\prime}+[0,1]\subset B$, it remains to take

$c\,(\nu\,(1-\epsilon_{N})/N)=c_{b^{\prime}}(\nu\,(1-\epsilon_{N})/N))=\Psi(b^{\prime}+\nu\,(1-\epsilon_{N})/N).$

Formula (3.8) then follows immediately if one evaluates $X$ at $b$. ∎

We will also need in this section the following parametric version of another
important result due to Serge Bernstein [12, Theorem A].

######  Theorem 3.4.

Let $c\in[0,1]$, $\rho>\max(c,1-c)$, $\\{G_{\iota}\,:\,\iota\in I\\}$ be a
bounded family in the space $\mathcal{H}(D(c,\rho))$ of holomorphic functions
in $D_{\mathbb{C}}(c,\rho)$ and let
$\\{\boldsymbol{\epsilon}_{\iota^{\prime}}=(\epsilon_{\iota^{\prime},N})_{N\geq
1}\,:\,\iota^{\prime}\in I^{\prime}\\}$ be a family of sequences
$\boldsymbol{\epsilon}_{\iota^{\prime}}\in([0,1])^{\mathbb{N}^{*}}$ which all
converge to $0$ uniformly with respect to $\iota^{\prime}$. Then, for any
$\iota\in I$, $\iota^{\prime}\in I^{\prime}$, the sequence of polynomial maps

$\Big{(}z\in
D_{\mathbb{C}}(c,\rho)\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}\,G_{\iota}\Big{(}\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}$ (3.10)

converges to $G_{\iota}(z)$ in $\mathcal{H}(D_{\mathbb{C}}(c,\rho))$. Moreover
the convergence on every compact subset of $D_{\mathbb{C}}(c,\rho)$ is uniform
with respect to the parameters $\iota$ and $\iota^{\prime}$.

###### Proof.

We follow the proof presented in [20, Theorem 4.1.1], slightly adapting it in
order to incorporate the parametric setting. Let $\rho_{0}<\rho$ and, for each
$\iota\in I$, let
$G_{\iota}(z)=\sum_{\kappa=0}^{\infty}\gamma_{\iota,\kappa}(z-c)^{\kappa}$,
with $\gamma_{\iota,\kappa}\in\mathbb{C}$ the Taylor coefficients of
$G_{\iota}$ about $c$. It follows from the fact that the family
$G=\\{G_{\iota}\,:\,\iota\in I\\}$ is bounded in
$\mathcal{H}(D_{\mathbb{C}}(c,\rho))$ that there is a positive constant
$M=M_{G}(\rho_{0})$ such that

$\forall\,\iota\in I,\
\sum_{\kappa=0}^{\infty}|\gamma_{\iota,\kappa}|\,\rho_{0}^{\kappa}\leq M.$
(3.11)

For each $\boldsymbol{\epsilon}\in[0,1[^{\mathbb{N}^{*}}$ which tends to $0$
when $N$ tends to infinity, for each $N\in\mathbb{N}^{*}$ and
$\kappa\in\mathbb{N}$, let

$B_{N,\kappa}^{\boldsymbol{\epsilon}}(X)=\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\Big{(}\frac{1-\epsilon_{N}}{N}\nu-c\Big{)}^{\kappa}\,X^{\nu}(1-X)^{N-\nu}\in\mathbb{R}[X].$

Let us introduce the (for the time being formal) generating function of the
sequence of polynomial maps
$(B_{N,\kappa}^{\boldsymbol{\epsilon}}(z))_{\kappa\geq 0}$, namely

$(u,z)\longmapsto\sum\limits_{\kappa=0}^{\infty}B_{N,\kappa}^{\boldsymbol{\epsilon}}(z)\,\frac{u^{\kappa}}{\kappa!}=\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}\Big{(}\sum\limits_{\kappa=0}^{\infty}\Big{(}\frac{1-\epsilon_{N}}{N}\,\nu-c\Big{)}^{\kappa}\frac{u^{\kappa}}{\kappa!}\Big{)}\\\
=\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}\Big{(}\exp\Big{(}(1-c)\,\frac{1-\epsilon_{N}}{N}\Big{)}\Big{)}^{\nu}\Big{(}\exp\Big{(}-c\,\frac{1-\epsilon_{N}}{N}\,u\Big{)}\Big{)}^{N-\nu}\\\
=\Big{(}z\,\exp\Big{(}(1-c)\,\frac{1-\epsilon_{N}}{N}\,u\Big{)}+(1-z)\,\exp\Big{(}-c\,\frac{1-\epsilon_{N}}{N}\,u\Big{)}\Big{)}^{N}\\\
=\Big{(}\sum_{\kappa=0}^{\infty}\big{(}z(1-c)^{\kappa}+(1-z)(-c)^{\kappa}\big{)}\,\Big{(}\frac{1-\epsilon_{N}}{N}\Big{)}^{\kappa}\,\frac{u^{\kappa}}{\kappa!}\Big{)}^{N}.$
(3.12)

As proved in [20, (6), page 89], we have that

$\big{|}z(1-c)^{\kappa}+(1-z)(-c)^{\kappa}\big{|}\leq|z-c|^{\kappa},$ (3.13)

for any $z$ such that $|z-c|\geq\min(c,1-c)$ and for any
$\kappa\in\mathbb{N}$. Therefore, for $z$ such that $|z-c|\geq\min(c,1-c)$,
the coefficients of the (formal) power series in $u$ (raised to the power $N$
in the last line of (3.12)) are bounded from above in absolute value by those
of

$u\longmapsto\exp\Big{(}\rho_{0}\,\frac{1-\epsilon_{N}}{N}\,u\Big{)}.$

The moduli of the coefficients in the formal expansion in the powers
$u^{\kappa}$ of

$(z,u)\longmapsto\sum\limits_{\kappa=0}^{\infty}B_{N,\kappa}^{\boldsymbol{\epsilon}}(z)\,\frac{u^{\kappa}}{\kappa!}$

do not then exceed those of the corresponding coefficients in the expansion in
the powers $u^{\kappa}$ of

$u\longmapsto\exp\big{(}\rho_{0}\,(1-\epsilon_{N})\,u\big{)}.$

For any $\iota\in I$, $\iota^{\prime}\in I^{\prime}$, $N\in\mathbb{N}^{*}$ and
$z\in\overline{D_{\mathbb{C}}(c,\rho_{0})}$, one has

$\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}G_{\iota}\Big{(}\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}=\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}\Big{(}\sum\limits_{\kappa=0}^{\infty}\gamma_{\iota,\kappa}\,\Big{(}\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}-c\Big{)}^{\kappa}\Big{)}\\\
=\sum\limits_{\kappa=0}^{\infty}\gamma_{\iota,\kappa}\,\Big{(}\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\,\Big{(}\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}-c\Big{)}^{\kappa}z^{\nu}(1-z)^{N-\nu}\Big{)}.$

Since

$\Big{|}\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\,\Big{(}\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}-c\Big{)}^{\kappa}z^{\nu}(1-z)^{N-\nu}\Big{|}=|B_{N,k}^{\boldsymbol{\epsilon}_{\iota}}(z)|\leq(\rho_{0}(1-\epsilon_{N}))^{k},$

according to the reasoning above, the sequence of polynomial functions

$\Big{(}\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}G_{\iota}\Big{(}\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}$

is uniformly bounded in $D_{\mathbb{C}}(c,\rho)$ with respect both to
$\iota\in I$ and $\iota^{\prime}\in I^{\prime}$. Since the family
$\\{G_{\iota}\,;\,\iota\in I\\}$ is bounded in $\mathcal{H}(V)$, where $V$ is
any open neighborhood of $[0,1]$, the family
$\\{(G_{\iota})_{|[0,1]}\,:\,\iota\in I\\}$ admits a uniform modulus of
continuity $[0,1]$ Classical facts about Bernstein’s approximation of
continuous functions on $[0,1]$, see for example [20, §1.6], ensure that for
any $\iota\in I$ and $\iota^{\prime}\in I^{\prime}$, the sequence of functions

$\Big{(}\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}G_{\iota}\Big{(}\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}$

converges in $\mathcal{C}([0,1],\mathbb{C})$ to $z\in[0,1]\longmapsto
G_{\iota}(z)$, such convergence being uniform with respect to the choice of
$\iota\in I$ or $\iota^{\prime}\in I^{\prime}$ as soon as the convergence of
$(\epsilon_{\iota^{\prime},N})_{N\geq 1}$ towards $0$ is uniform with respect
to the index $\iota^{\prime}$. Since all points in $[0,1]$ as a subset of
$D_{\mathbb{C}}(c,\rho)$ are accumulation points, the assertion of the
proposition follows then from Vitali-Porter theorem, see [26, p. 68] or also
[24, §2.4]. ∎

Theorem 3.4 implies the following result.

######  Theorem 3.5.

Let $A$ be an open interval of $\mathbb{R}$ with length $R$ strictly larger
than $2$ and let $F$ be either an entire function when $A$ is unbounded, or a
holomorphic function on

$W_{A}=\bigcup_{\\{a^{\prime}\in A\,:\,a^{\prime}+[-1,1]\subset
A\\}}\overline{D_{\mathbb{C}}\big{(}a^{\prime}+C(a^{\prime}),{\rm
dist}\big{(}a^{\prime}+C(a^{\prime}),\mathbb{R}\setminus A\big{)}\big{)}},$
(3.14)

where

$C(a^{\prime})=2\,\Big{(}\frac{a^{\prime}-1-\inf A}{R-2}-1\Big{)}\in]-1,1[$
(3.15)

when $A$ is bounded. Then $\psi=F_{|A}$ satisfies on $A$ the $({\rm
TCSP})_{\mathbb{C}}$ property with respect to the superoscillating sequence
$\\{T_{N}^{\boldsymbol{\epsilon}}[a]\,:\,a\in\mathbb{R}\\}_{N\geq 1}$, where
$\boldsymbol{\epsilon}$ is any sequence of elements in $[0,1[$ which tends to
$0$. Moreover, given a family of sequences
$\boldsymbol{\epsilon}_{\iota^{\prime}}\in([0,1])^{\mathbb{N}^{*}}$,
$\iota^{\prime}\in I^{\prime}$, which all converge to $0$ uniformly with
respect to $\iota^{\prime}$ on any compact subset of the all sequences of
functions

$\Big{(}(a,a^{\prime})\in\mathbb{A}\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\Big{(}\frac{1+a}{2}\Big{)}^{N-\nu}\Big{(}\frac{1-a}{2}\Big{)}^{\nu}\,\psi(a^{\prime}+h_{N,\nu}^{\boldsymbol{\epsilon}_{\iota^{\prime}}})\Big{)}_{N\geq
1}$

to $(a,a^{\prime})\in\mathbb{A}\longmapsto\psi(a+a^{\prime})$ is also uniform
with respect to the index $\iota^{\prime}$.

###### Proof.

Consider first the case where $A$ is a bounded open interval with length
$R>2$. Then $B=\Upsilon^{-1}(A)=]b_{\rm min},b_{\rm max}[$ with $b_{\rm
max}-b_{\rm min}=R/2>1$. Let $V$ be an open neighborhood of

$W_{B}=\bigcup_{b_{\rm min}<b^{\prime}<b_{\rm
max}-1}\overline{D_{\mathbb{C}}\big{(}b^{\prime}+c(b^{\prime}),\rho_{0}(b^{\prime})}\big{)}$

where

$c(b^{\prime})=1-\frac{b^{\prime}-b_{\rm
min}}{R/2-1}\in]0,1[,\quad\rho_{0}(b^{\prime})={\rm
dist}\big{(}b^{\prime}+c(b^{\prime}),\mathbb{R}\setminus\Upsilon^{-1}(A)\big{)},$
(3.16)

so that

$\lim_{b^{\prime}\searrow b_{\rm
min}}c(b^{\prime})=1,\quad\lim_{b^{\prime}\nearrow b_{\rm
max}}c(b^{\prime})=0,\ \lim_{b^{\prime}\searrow b_{\rm
min}}\rho_{0}(b^{\prime})=\lim_{b^{\prime}\nearrow b_{\rm
min}}\rho_{0}(b^{\prime})=\max(1,R/2-1).$

Suppose that $\Psi=\psi\circ\Upsilon$ is the restriction to the open interval
$\Upsilon^{-1}(A)$ of a holomorphic function in $V$, which amounts to say that
$\psi$ is the restriction to $A$ of a holomorphic function in the open subset
$2V-1$ of the complex plane. This implies that for any $b^{\prime}\in]b_{\rm
min},b_{\rm max}-1[$, there exists a holomorphic function $G_{b^{\prime}}$ in
an open neighborhood $D_{\mathbb{C}}(c(b^{\prime}),\rho(b^{\prime}))$ of
$\overline{D_{\mathbb{C}}(c(b^{\prime}),\rho_{0}(b^{\prime}))}$ such that
$G_{b^{\prime}}=\Psi(b^{\prime}+\cdot)$ in an open neighborhood of $[0,1]$. It
follows then from Theorem 3.4 with

$\\{G_{\iota}\,:\,\iota\in I\\}=\\{z\mapsto
G_{b^{\prime}\pm\eta}(z)\,:\,0\leq\eta\leq\eta(b^{\prime})\\},$

and from the definition of $\rho_{0}(b^{\prime})<\rho(b^{\prime})$ that, given
$\\{\boldsymbol{\epsilon}_{\iota^{\prime}}=(\epsilon_{\iota^{\prime},N})_{N\geq
1}\,:\,\iota^{\prime}\in I^{\prime}\\}$ a family of sequences
$\boldsymbol{\epsilon}_{\iota^{\prime}}\in([0,1])^{\mathbb{N}^{*}}$ which
converge towards $0$ uniformly with respect to $\iota^{\prime}$, the sequence
of polynomial functions

$\Big{(}b\in]b_{\rm min}-b^{\prime},b_{\rm max}-b^{\prime}[\
\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\,b^{\nu}(1-b)^{N-\nu}G_{b^{\prime}\pm\eta}\Big{(}\nu\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}\\\ =\Big{(}b\in]b_{\rm min}-b^{\prime},b_{\rm max}-b^{\prime}[\
\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\,b^{\nu}(1-b)^{N-\nu}\Psi\Big{(}b^{\prime}\pm\eta+\nu\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}$

converges uniformly on any compact subset of in $\Upsilon^{-1}(A)-b^{\prime}$
towards $G_{b^{\prime}\pm\eta}(b)=\Psi(b+b^{\prime})$, the convergence being
uniform with respect both to $\pm\eta$ such that
$0\leq\eta\leq\eta(b^{\prime})$ and to $\iota^{\prime}$. Since

$2W_{B}-1=\bigcup_{\\{a^{\prime}\in A\,;\,a^{\prime}+[-1,1]\subset
A\\}}\overline{D_{\mathbb{C}}\Big{(}(a^{\prime}-1)+2\Big{(}\frac{a^{\prime}-1-\inf
A}{R-2}\Big{)},2\rho_{0}\Big{(}\frac{a^{\prime}}{2}\Big{)}\Big{)}}\\\
=\bigcup_{\\{a^{\prime}\in A\,;\,a^{\prime}+[-1,1]\subset
A\\}}\overline{D_{\mathbb{C}}\big{(}a^{\prime}+C(a^{\prime}),{\rm
dist}\,\big{(}a^{\prime}+C(a^{\prime}),\mathbb{R}\setminus
A\big{)}\big{)}}=W_{A}$

according to the definition of $a^{\prime}\mapsto C(a^{\prime})$, Theorem 3.5
follows in the case where $A$ is bounded. When $A$ is unbounded, one may
exhaust $A$ with an increasing sequence of bounded intervals $(A_{k})_{k\geq
1}$ whose sequence of lengths $(R_{k})_{k\geq 1}$ tends to infinity. Theorem
3.5 in the unbounded case follows then immediately from Theorem 3.5 in the
bounded case since testing $({\rm TCSP})_{\mathbb{C}}$ with respect to
$(\mathbb{R},H^{\boldsymbol{\epsilon}})$ on $A$ amounts clearly to test such
property on each $A_{k}$. ∎

######  Remark 3.6.

When $A$ is bounded with length $R>2$, one has

$W_{A}\subset\\{z\in\mathbb{C}\,:\,{\rm
dist}(z,\overline{A})\leq\max(2,R-2)\\}.$ (3.17)

Any holomorphic function in an open neighborhood of
$\\{z\in\mathbb{C}\,:\,{\rm dist}(z,\overline{A})\leq\max(2,R-2)\\}$ is then
such that its restriction to $A$ inherits the $({\rm TCSP})_{\mathbb{C}}$
property with respect to the superoscillating sequence
$\\{T_{N}^{\boldsymbol{\epsilon}}[a]\,:\,a\in\mathbb{R}\\}_{N\geq 1}$, where
$\boldsymbol{\epsilon}\in[0,1[^{\mathbb{N}^{*}}$ tends to $0$ when $N$ tends
to infinity (with the uniformity clause with respect to the choice of such
sequence $\boldsymbol{\epsilon}$).

Theorem 3.5 suggests the introduction of the following definition.

######  Definition 3.7.

(Regular $\mathbb{C}$-supershift) Let $A$ be an open interval of $\mathbb{R}$
with length $R$ strictly larger than $2$. A continuous map
$\psi:A\rightarrow\mathbb{C}$ is called a regular $\mathbb{C}$-supershift if
the following two conditions are fulfilled.

  1. (1)

The map $\psi$ satisfies the $({\rm TCSP})_{\mathbb{C}}$ property on $A$ with
respect to any superoscillating sequence
$\\{T_{N}^{\boldsymbol{\epsilon}}[a]\,:\,a\in\mathbb{R}\\}_{N\geq 1}$, as in
(2.3) and (2.4), where $\boldsymbol{\epsilon}$ is any sequence in
$[0,1[^{\mathbb{N}^{*}}$ which tends to $0$ when $N$ tends to infinity.

  2. (2)

Given a family $\\{(\epsilon_{\iota^{\prime},N})_{N\geq 1}\,:\,\iota\in
I^{\prime}\\}$ of such sequences, such that the convergence of all sequences
$\boldsymbol{\epsilon}_{\iota^{\prime}}$ towards $0$ is uniform with respect
to the index $\iota^{\prime}$, the convergence for
$(a,a^{\prime})\in\mathbb{A}$ of the polynomial functions

$\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\Big{(}\frac{1+a}{2}\Big{)}^{N-\nu}\Big{(}\frac{1-a}{2}\Big{)}^{\nu}\psi\Big{(}a^{\prime}+\Big{(}1-2\,\Big{(}\frac{\nu+\epsilon_{\iota^{\prime},N}\,(N-\nu)}{N}\Big{)}\Big{)}\Big{)},\quad
N=1,2,...$ (3.18)

to $\psi(a+a^{\prime})$ in $\mathcal{C}_{c}(\mathbb{A},\mathbb{C})$ is uniform
with respect to the index $\iota^{\prime}$.

It follows from from Theorem 3.5 that such theorem that real analytic
functions, provided they can be extended as holomorphic functions in
sufficiently large complex domains containing $A$, provide examples of regular
supershifts in $A$. Such is the case for example for restrictions to $A$ of
entire functions in case $A$ is unbounded. The natural question is to ask
whether regular $\mathbb{C}$-supershifts inherit some analyticity property. We
will deduce from a result of Leonid Kantorovich [19] that the answer to such
question is NO in the case of bounded intervals $A$.

The following theorem follows from a result in [19], see also [20, Theorem
4.1.3 & subsequent remark]. We modified slightly the construction here in
order to formulate a parametric version, as we did with Bernstein’s theorem in
Theorem 3.4.

######  Theorem 3.8.

Let $G^{-}$ and $G^{+}$ two entire functions such that

$G^{-}(1/2)=G^{+}(1/2),\quad\frac{dG^{-}}{dz}(1/2)\not=\frac{dG^{+}}{dz}(1/2)$
(3.19)

and let $g^{\pm}:b\in\mathbb{R}\rightarrow\mathbb{C}$ be the continuous map
defined by

$g^{\pm}(b)=\begin{cases}G^{-}(b)\ &{\rm if}\ b<1/2\\\ G^{+}(b)\ &{\rm if}\
b\geq 1/2.\end{cases}$ (3.20)

Let $0<\eta<1/2$. There exists a neighborhood $V^{-}_{\eta}$ of $0$ in
$\mathbb{C}$ and a neighborhood $V^{+}_{\eta}$ of $1$ in $\mathbb{C}$ such
that, for any $b^{\prime}\in[-\eta,\eta]$, for any
$\boldsymbol{\epsilon}\in[0,1[^{\mathbb{N}^{*}}$ which tends to $0$ at
infinity, the sequence of polynomial maps

$\Big{(}z\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}g^{\pm}\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{N}}{N}\Big{)}\Big{)}_{N\geq
1}$ (3.21)

converges uniformly on any compact subset of $V_{\eta}^{-}$ to $z\longmapsto
G^{-}(z+b^{\prime})$ and uniformly on any compact subset of $V^{+}_{\eta}$ to
$z\longmapsto G^{+}(z+b^{\prime})$. The convergence is uniform with respect to
the parameter $b^{\prime}\in[-\eta,\eta]$. Moreover, if we consider a family
$\\{(\epsilon_{\iota^{\prime},N})_{N\geq 1}\,:\,\iota^{\prime}\in
I^{\prime}\\}$, where all sequences $\boldsymbol{\epsilon}_{\iota^{\prime}}$
tend to $0$ at infinity uniformly with respect to $\iota^{\prime}$, the above
convergences with $\boldsymbol{\epsilon}_{\iota^{\prime}}$ instead of
$\boldsymbol{\epsilon}$, respectively on compact subsets of $V_{\eta}^{-}$ and
$V_{\eta}^{+}$, are also uniform with respect to the index $\iota^{\prime}$.

###### Proof.

For each $c\in[1/2-\eta,1/2+\eta]$, let $\mathscr{L}_{c}$ be the lemniscate

$\mathscr{L}_{c}=\\{z\in\mathbb{C}\,:\,|z|^{c}|1-z|^{1-c}=c^{c}(1-c)^{1-c}\\},$

$V^{-}(c)$ (respectively $V^{+}(c)$) be the interior of its left loop
(respectively its right loop) about $0$ (respectively $1$),

$V^{-}_{\eta}=\bigcap_{c\in[1/2-\eta,1/2+\eta]}V^{-}(c),\qquad{\rm and}\qquad
V^{+}_{\eta}=\bigcap_{c\in[1/2-\eta,1/2+\eta]}V^{+}(c).$

Fix also $\\{(\epsilon_{\iota^{\prime},N})_{N\geq 1}\,:\,\iota^{\prime}\in
I^{\prime}\\}$ such that all sequences
$\boldsymbol{\epsilon}_{\iota^{\prime}}$ tend to $0$ at infinity uniformly
with respect to $\iota^{\prime}$. It follows from Theorem 3.4 that the two
sequences of polynomial maps

$\begin{split}&\Big{(}z\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}G^{-}\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}\\\
&\Big{(}z\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}G^{+}\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}\end{split}$

converge on any compact subset of the complex plane to respectively
$G^{-}(b^{\prime}+z)$ and $G^{+}(b^{\prime}+z)$, uniformly both with respect
to $b^{\prime}\in[-\eta,\eta]$ and the index $\iota^{\prime}$. Fix a compact
$K^{+}\subset V^{+}_{\eta}$. Since $K^{+}$ is contained in each $V^{+}(c)$ for
$c\in[1/2-\eta,1/2+\eta]$, there is a constant $q_{\eta}(K^{+})<1$ such that
for any $c\in[1/2-\eta,1/2+\eta]$,

$\begin{split}z\in
K^{+}&\Longrightarrow\Big{(}\frac{|z|}{c}\Big{)}^{c}\Big{(}\frac{|1-z|}{1-c}\Big{)}^{1-c}\leq
q_{\eta}(K^{+})<1\\\
&\Longrightarrow\frac{c}{|z|}\,\frac{|1-z|}{1-c}\leq\Big{(}q_{\eta}(K^{+})\,\frac{c}{|z|}\Big{)}^{1/(1-c)}\leq
1,\end{split}$ (3.22)

see the proof of [20, Theorem 4.3.1]. Then one has for each $z\in K^{+}$, for
each $b^{\prime}\in[-\eta,\eta]$ and $H=G^{+}$ or $H=G^{-}$ that

$\Big{|}\sum\limits_{\\{\nu\in\mathbb{N}\,:\,\nu\leq(1/2-b^{\prime})N\\}}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}H\Big{(}b^{\prime}+\nu\,\Big{(}\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}\Big{|}\\\
\leq\sup_{[-b^{\prime},1+b^{\prime}]}|H|\,\sum\limits_{\\{\nu\in\mathbb{N}\,:\,\nu\leq(1/2-b^{\prime})\,N\\}}\binom{N}{\nu}\,(1/2-b^{\prime})^{\nu}(1/2+b^{\prime})^{N-\nu}\Big{(}\frac{|z|}{1/2-b^{\prime}}\Big{)}^{\nu}\Big{(}\frac{|1-z|}{1/2+b^{\prime}}\Big{)}^{N-\nu}\\\
\leq\sup_{[-b^{\prime},1+b^{\prime}]}|H|\,\sum\limits_{\\{\nu\in\mathbb{N}\,:\,\nu\leq(1/2-b^{\prime})N\\}}\Big{(}\frac{|z|}{1/2-b^{\prime}}\Big{)}^{\nu}\Big{(}\frac{|1-z|}{1/2+b^{\prime}}\Big{)}^{N-\nu}.$
(3.23)

Since for any $\nu\in\mathbb{N}$ such that $\nu\leq(1/2-b^{\prime})N$, one has
for $z\in K^{+}$ that

$\Big{(}\frac{|z|}{1/2-b^{\prime}}\Big{)}^{\nu}\Big{(}\frac{|1-z|}{1/2+b^{\prime}}\Big{)}^{N-\nu}\\\
=\Big{(}\Big{(}\frac{|z|}{1/2-b^{\prime}}\Big{)}^{1/2-b^{\prime}}\Big{(}\frac{|1-z|}{1/2+b^{\prime}}\Big{)}^{1/2+b^{\prime}}\Big{)}^{N}\,\Big{(}\frac{1/2-b^{\prime}}{|z|}\,\frac{|1-z|}{1/2+b^{\prime}}\Big{)}^{N(1/2-b^{\prime})-\nu}\leq(q_{\eta}(K^{+}))^{N}$

according to (3.22), one concludes that for any $N\in\mathbb{N}^{*}$ and
$b^{\prime}\in[-\eta,\eta]$, one has from (3.23) that

$\sup_{z\in
K^{+}}\Big{|}\sum\limits_{\\{\nu\in\mathbb{N}\,:\,\nu\leq(1/2-b^{\prime})N\\}}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}H\Big{(}b^{\prime}+\nu\,\Big{(}\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}\Big{|}\\\
\leq(\sup_{[-\eta,1+\eta]}|H|)\,N\,(q_{\eta}(K^{+}))^{N},$ (3.24)

which means that the sequence of polynomial maps

$\Big{(}z\longmapsto\sum\limits_{\\{\nu\in\mathbb{N}\,:\,\nu\leq(1/2-b^{\prime})N\\}}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}H\Big{(}b^{\prime}+\nu\,\Big{(}\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}\Big{)}_{N\geq
1}$

converges uniformly towards $0$ on $K^{+}$, the convergence being uniform with
respect to $b^{\prime}\in[-\eta,\eta]$ and to the index $\iota^{\prime}$.
Since $H$ can be either $G^{-}$ or $G^{+}$ and the sequence of polynomial maps

$\Big{(}z\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}z^{\nu}(1-z)^{N-\nu}G^{+}\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}$

converges uniformly on $K^{+}$ towards $z\mapsto G^{+}(b^{\prime}+z)$, the
convergence being uniform both with respect to $b^{\prime}\in[-\eta,\eta]$ and
the index $\iota^{\prime}$, one concludes that the sequence of polynomial maps
(3.21) converges uniformly on $K^{+}$ towards $z\longmapsto
G^{+}(b^{\prime}+z)$, the convergence being uniform both with respect to
$b^{\prime}\in[-\eta,\eta]$ and the index $\iota^{\prime}$. A similar
reasoning shows that if $K^{-}\subset V_{\eta}^{-}$, the sequence (3.21)
converges uniformly on $K^{-}$ to $z\longmapsto G^{-}(b^{\prime}+z)$ uniformly
both with respect to $b^{\prime}\in[-\eta,\eta]$ and the index
$\iota^{\prime}$. ∎

We now prove the following lemma.

######  Lemma 3.9.

Let $A$ be an open interval of $\mathbb{R}$ with length strictly larger than
$2$ and $\psi:A\rightarrow\mathbb{C}$ be a regular supershift in $A$. Let
$\theta$ be a smooth test-function with support in $[0,\epsilon]$, where
$\epsilon<R-2$ and $A_{\epsilon}=\\{a\in A\,;\,a-\epsilon\in A\\}$. Then the
function

$\psi*\theta:a\in
A_{\epsilon}\longmapsto\int_{0}^{\epsilon}\theta(\alpha)\,\psi(a-\alpha)\,d\alpha=\int_{\mathbb{R}}\theta(\alpha)\,\psi(a-\alpha)\,d\alpha$

is a smooth regular $\mathbb{C}$-supershift in $A_{\epsilon}$.

###### Proof.

Let

$\mathbb{A}_{\epsilon}:=\\{(a,a^{\prime})\in\mathbb{R}\times
A_{\epsilon}\,:\,a^{\prime}+[-1,1]\subset A_{\epsilon},\ a+a^{\prime}\in
A_{\epsilon}\\}$

and $\\{(\epsilon_{\iota^{\prime},N})_{N\geq 1}\,:\,\iota^{\prime}\in
I^{\prime}\\}$ be a family of elements in $[0,1[^{\mathbb{N}^{*}}$ which all
tend to $0$ at infinity, uniformly with respect to $\iota^{\prime}$. For any
$(a_{0},a^{\prime}_{0})\in\mathbb{A}_{\epsilon}$ and any $N\in\mathbb{N}^{*}$,
one has for $(a,a^{\prime})$ in a sufficiently small neighborhood
$U_{a_{0},a^{\prime}_{0}}$ of $(a_{0},a^{\prime}_{0})$

$\int_{0}^{\epsilon}\theta(\alpha)\,\Big{(}\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\Big{(}\frac{1+a}{2}\Big{)}^{N-\nu}\Big{(}\frac{1-a}{2}\Big{)}^{\nu}\psi\Big{(}a^{\prime}-\alpha+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}\,d\alpha\\\
=\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\Big{(}\frac{1+a}{2}\Big{)}^{N-\nu}\Big{(}\frac{1-a}{2}\Big{)}^{\nu}(\psi*\theta)\Big{(}a^{\prime}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}.$
(3.25)

Since $\psi$ is a regular $\mathbb{C}$-supershift on $A$, the sequence of
functions

$\Big{(}(a,a^{\prime})\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\Big{(}\frac{1+a}{2}\Big{)}^{N-\nu}\Big{(}\frac{1-a}{2}\Big{)}^{\nu}\psi\Big{(}a^{\prime}-\alpha+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1},$

considered as indexed by the parameter $\alpha\in[0,\epsilon]$, converges
uniformly on any compact subset of $U_{a_{0},a^{\prime}_{0}}$ towards

$(a,a^{\prime})\longmapsto\psi(a+a^{\prime}-\alpha),$

the convergence being also uniform both with respect to
$\alpha\in[0,\epsilon]$ and to the index $\iota^{\prime}$. As a consequence of
(3.25), the sequence of functions

$\Big{(}(a,a^{\prime})\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}\Big{(}\frac{1+a}{2}\Big{)}^{N-\nu}\Big{(}\frac{1-a}{2}\Big{)}^{\nu}(\psi*\theta)\Big{(}a^{\prime}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}$

converges uniformly on any such compact subset of $U_{a_{0},a^{\prime}_{0}}$
towards

$(a,a^{\prime})\longmapsto\int_{0}^{\epsilon}\theta(\epsilon)\psi(a+a^{\prime}-\alpha)\,d\alpha=(\psi*\theta)(a+a^{\prime}).$

This shows that $\psi*\theta$ is a regular $\mathbb{C}$-supershift in
$\mathbb{A}_{\epsilon}$. It is smooth since it inherits the smoothness of the
regularizing test-function $\theta$. ∎

We deduce then from Theorem 3.8 the following result.

######  Theorem 3.10.

Let $g^{\pm}$ be the continuous complex valued function with $1/2$ as single
non-real analyticity point which is defined by (3.19) and (3.20). There exists
an open interval $A_{0}=]-1-\rho_{0},1+\rho_{0}[\subset\mathbb{R}$ containing
$[-1,1]$ such that $a\in A_{0}\longmapsto\psi_{0}(a)=g^{\pm}((1+a)/2)$ is a
regular $\mathbb{C}$-supershift in $A_{0}$. As a consequence, the fact that
$\psi$ is a smooth regular $\mathbb{C}$-supershift on some open interval
$A\subset\mathbb{R}$ with diameter strictly larger than $2$ does not imply in
general that $\psi$ is real analytic on $A$.

###### Proof.

The fact that the first assertion implies the second one is an immediate
consequence of Lemma 3.9. One can choose two entire functions $G^{-}$ and
$G^{+}$ such that (3.19) is fulfilled and a smooth regularization function
$\theta$ such that $\psi=\psi_{0}*\theta$ shares with $\psi_{0}$ the following
property, namely that there exists at least one point of non-real analyticity
in $]-1,1[$. It remains then to prove the first assertion. Let $\eta>0$,
$V_{\eta}^{-}$ and $V_{\eta}^{+}$ as in Theorem 3.8. Since the family of
continuous functions

$\big{\\{}x\in[0,1]\longmapsto
g^{\pm}(b^{\prime}+x)\,:\,b^{\prime}\in[-\eta,\eta]\\}$

is uniformly Lipschitz, hence admits a uniform modulus of continuity $[0,1]$,
classical facts about Bernstein’s approximation of continuous functions on
$[0,1]$, see for example [20, §1.6], imply that for each
$b^{\prime}\in[-\eta,\eta]$, the sequence of polynomial functions

$\Big{(}b\in[0,1]\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}g^{\pm}\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{N}}{N}\Big{)}\Big{)}_{N\geq
1}$

converges uniformly on $[0,1]$ towards $b\longmapsto g^{\pm}(b+b^{\prime})$,
the convergence being uniform both with respect to $b^{\prime}\in[-\eta,\eta]$
and to the index $\iota^{\prime}$. If we choose $\epsilon_{\eta}\in]0,\eta]$
such that $[-\epsilon_{\eta},\epsilon_{\eta}]\subset V_{\eta}^{-}$ and
$[1-\epsilon_{\eta},1+\epsilon_{\eta}]\subset V_{\eta}^{+}$, Theorem 3.8 shows
that the same result holds on any compact subset of
$]-\epsilon_{\eta},1+\epsilon_{\eta}[$. The first assertion in the theorem
follows if one makes the rescaling $b=(a+1)/2$. ∎

From Theorem 3.10 it is immediate the following statement which follows taking
for example $G^{-}(z)=1-2z$ and $G^{+}(z)=2z-1$, so that $g^{\pm}(x)=|x|$.

######  Corollary 3.11.

There exist smooth regular $\mathbb{C}$-supershifts which are not real
analytic.

##  4\. Operations preserving supershift

Since it is not immediate to establish when a function has the regular
$\mathbb{C}$-supershift property, we conclude this paper by analyzing which
operations preserve this property. The following theorems mention two of them.

######  Theorem 4.1.

Given an open interval $A$ of $\mathbb{R}$ with diameter strictly larger than
$2$, the class of regular $\mathbb{C}$-supershifts on $A$ is stable by
multiplication by the map $a\in A\longmapsto a\in\mathbb{C}$, hence by any
complex valued polynomial map $a\in A\longmapsto p(a)\in\mathbb{C}$.

###### Proof.

Let $\\{(\epsilon_{\iota^{\prime},N})_{N\geq 1}\,:\,\iota^{\prime}\in
I^{\prime}\\}$ be a family of elements in $[0,1[^{\mathbb{N}^{*}}$ which all
tend to $0$ at infinity, uniformly with respect to $\iota^{\prime}$. Let
$B=\Upsilon^{-1}(A)$ and $\Psi=\psi\circ\Upsilon$. An easy computation shows
that if $\Phi$ is the function $\Phi:b\in B\longmapsto b\,\Psi(b)$ then, for
any $N\in\mathbb{N}^{*}$ with $N\geq 2$ and any

$(b,b^{\prime})\in\mathbb{B}=\\{(b,b^{\prime})\in\mathbb{R}\times
B\,:\,b^{\prime}+[0,1]\subset B,\ b+b^{\prime}\in B\\},$

one has

$\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}\Phi\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\\\
=(1-\epsilon_{N})\,b\sum\limits_{\nu=0}^{N-1}\binom{N-1}{\nu}b^{\nu}(1-b)^{N-1-\nu}\Psi\Big{(}b^{\prime}+\frac{1-\epsilon_{\iota^{\prime},N}}{N}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N-1}^{[1]}}{N-1}\Big{)}\\\
+b^{\prime}\,\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}\Psi\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)},$
(4.1)

where

$1-\epsilon_{\iota^{\prime},N-1}^{[1]}=\frac{N-1}{N}(1-\epsilon_{\iota^{\prime},N}).$
(4.2)

Let $K$ be a compact subset of $\mathbb{B}$ and $\chi_{K}>0$ such that
$b^{\prime}+b+\tau\in B$ for any $(b,b^{\prime})\in K$ and
$0\leq\tau\leq\chi_{K}$. Since $\psi$ is a regular $\mathbb{C}$-supershift,
there exists, for any $\varepsilon>0$, an integer $N_{K,\varepsilon}$ such
that

$N\geq N_{K,\varepsilon}\\\
\Longrightarrow\sup_{\stackrel{{\scriptstyle((b,b^{\prime}),\tau)\in
K\times[0,\chi_{K}]}}{{\iota^{\prime}\in
I^{\prime}}}}\Big{|}\Psi(b^{\prime}+\tau+b)-\sum\limits_{\nu=0}^{N-1}\binom{N-1}{\nu}b^{\nu}(1-b)^{N-1-\nu}\Psi\Big{(}b^{\prime}+\tau+\nu\,\frac{1-\epsilon_{\iota^{\prime},N-1}^{[1]}}{N-1}\Big{)}\Big{|}\leq\varepsilon.$

One has in particular for $N\geq\max(N_{K,\varepsilon},1/\chi_{K})$ that

$\sup_{\stackrel{{\scriptstyle(b,b^{\prime})\in K}}{{\iota^{\prime}\in
I^{\prime}}}}\Big{|}\Psi\Big{(}b^{\prime}+\frac{1-\epsilon_{\iota^{\prime},N}}{N}+b\Big{)}-\sum\limits_{\nu=0}^{N-1}\binom{N-1}{\nu}b^{\nu}(1-b)^{N-1-\nu}\Psi\Big{(}b^{\prime}+\frac{1-\epsilon_{\iota^{\prime},N}}{N}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N-1}^{[1]}}{N-1}\Big{)}\Big{|}\leq\varepsilon.$

Observe that the sequences of functions

$\begin{split}&\Big{(}(b,b^{\prime})\in\mathbb{B}\longmapsto\Psi\Big{(}b^{\prime}+\frac{1-\epsilon_{\iota^{\prime},N}}{N}+b\Big{)}\Big{)}_{N\geq
1}\\\
&\Big{(}(b,b^{\prime})\in\mathbb{B}\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}\Psi\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}\end{split}$

converge both locally uniformly on $\mathbb{B}$ towards
$(b,b^{\prime})\longmapsto\Psi(b+b^{\prime})$ on $K$, the convergence being
uniform with respect to the index $\iota^{\prime}$. Comparing the limits when
$N$ tends to infinity of both sides of (4.1), one gets that the sequence of
functions

$\Big{(}(b,b^{\prime})\in\mathbb{B}\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}\Phi\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}$

converges locally uniformly on $\mathbb{B}$ towards
$(b,b^{\prime})\longmapsto\Phi(b+b^{\prime})$, the convergence being uniform
with respect to the index $\iota^{\prime}$. After the rescaling $b=(1+a)/2$,
one gets that $a\in A\longmapsto(2a-1)\psi(a)$ is a regular
$\mathbb{C}$-supershift on $a$, so is $a\in A\longmapsto a\,\psi(a)$ since
$\psi$ already is. The proof of Theorem 4.1 is thus completed. ∎

######  Theorem 4.2.

Given an open interval $A$ of $\mathbb{R}$ with diameter strictly larger than
$2$, the class of regular $\mathbb{C}$-supershifts on $A$ is stable by
primitivization, namely, if $\psi:A\rightarrow\mathbb{C}$ is a
$\mathbb{C}$-regular supershift and $a_{0}\in A$, then

$a\in A\longmapsto\int_{a_{0}}^{a}\psi(\alpha)\,d\alpha$

is a $\mathbb{C}$-regular supershift.

###### Proof.

Let $\psi$ be a regular $\mathbb{C}$-supershift in $A$, $B=\Upsilon^{-1}(A)$
and $\Psi=\psi\circ\Upsilon$, where, as usual,
$\Upsilon(a)=2a-1\in\mathbb{R}$. Let $\\{(\epsilon_{\iota^{\prime},N})_{N\geq
1}\,:\,\iota^{\prime}\in I^{\prime}\\}$ be a family of elements in
$[0,1[^{\mathbb{N}^{*}}$ which all tend to $0$ at infinity, uniformly with
respect to $\iota^{\prime}$. If
$\mathbb{B}=\\{(b,b^{\prime})\in\mathbb{R}\times B\,:\,b^{\prime}+[0,1]\subset
B,\ b+b^{\prime}\in B\\}$, the sequence of functions

$\Big{(}(b,b^{\prime})\in\mathbb{B}\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}\Psi\Big{(}b^{\prime}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}$

converges locally uniformly on $\mathbb{B}$ towards
$(b,b^{\prime})\longmapsto\Psi(b+b^{\prime})$, the convergence being uniform
with respect to the index $\iota^{\prime}$. Let us call (P) such property. Let
$b_{0}^{\prime}\in B$ such that $b_{0}^{\prime}+[0,1]\subset B$. Proving that
any primitive of $\Psi$ in $A$ remains a regular $\mathbb{C}$-supershift
amounts to prove that the function

$b\in B\longmapsto\Phi(b)=\int_{b_{0}^{\prime}}^{b}\Psi(\beta)\,d\beta$

inherits from $\Psi$ the property (P). Theorem 4.2 will then follows after the
rescaling $b=\Upsilon^{-1}(a)=(1+a)/2$. We first observe that the sequence of
functions

$\Big{(}(b,b^{\prime})\in\mathbb{B}\longmapsto\int_{b_{0}^{\prime}}^{b^{\prime}}\Big{(}\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}\Psi\Big{(}\beta+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}\,d\beta\Big{)}_{N\geq
1}$

converges locally uniformly on $\mathbb{B}$ towards

$(b,b^{\prime})\longmapsto\int_{b_{0}}^{b^{\prime}}\Psi(\beta+b)\,d\xi=\Phi(b+b^{\prime})-\Phi(b+b^{\prime}_{0}).$

Therefore, if one proves that the sequence of functions

$\Big{(}b\in\\{b\in\mathbb{R}\,:\,b^{\prime}_{0}+b\in
B\\}\longmapsto\sum\limits_{\nu=0}^{N}\binom{N}{\nu}b^{\nu}(1-b)^{N-\nu}\,\Phi\Big{(}b^{\prime}_{0}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}_{N\geq
1}$ (4.3)

converges locally uniformly on
$-b^{\prime}_{0}+B=\\{b\in\mathbb{R}\,:\,b^{\prime}_{0}+b\in B\\}$ towards
$b\longmapsto\Phi(b+b^{\prime}_{0})$, the convergence on each compact subset
being uniform with respect to the index $\iota^{\prime}$, the fact that $\Phi$
inherits from $\Psi$ the property (P) will follow. One has on
$-b^{\prime}_{0}+B$ that for any $N\geq 2$,

$\frac{d}{db}\Big{(}\sum_{\nu=0}^{N}\binom{N}{\nu}\,b^{\nu}(1-b)^{N-\nu}\Phi\Big{(}b^{\prime}_{0}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}\\\
=(1-\epsilon_{\iota^{\prime},N})\,\sum_{\nu=0}^{N-1}\binom{N-1}{\nu}\,b^{\nu}(1-b)^{N-1-\nu}\,\frac{\Phi\Big{(}b^{\prime}_{0}+(\nu+1)\,\displaystyle{\frac{1-\epsilon_{\iota^{\prime},N}}{N}}\Big{)}-\Phi\Big{(}b^{\prime}_{0}+\nu\,\displaystyle{\frac{1-\epsilon_{\iota^{\prime},N}}{N}}\Big{)}}{\displaystyle{\frac{1-\epsilon_{\iota^{\prime},N}}{N}}}\\\
=(1-\epsilon_{\iota^{\prime},N})\int_{0}^{1}\Big{(}\sum\limits_{\nu=0}^{N-1}\binom{N-1}{\nu}b^{\nu}(1-b)^{N-\nu}\Psi\Big{(}b^{\prime}_{0}+\frac{1-\epsilon_{\iota^{\prime},N}}{N}\,\xi+\nu\,\frac{1-\epsilon_{\iota^{\prime},N-1}^{[1]}}{N-1}\Big{)}\Big{)}\,d\xi,$
(4.4)

where $\epsilon_{\iota^{\prime},N-1}^{[1]}$ is defined in terms of
$\epsilon_{\iota^{\prime},N}$ as in (4.2). Let $K^{\circ}$ be a compact subset
of $-b^{\prime}_{0}+B$, $\chi_{K^{\circ}}>0$ such that
$b+b^{\prime}_{0}+\tau\in B$ for any $\tau\in[0,\chi_{K^{\circ}}]$, and
$\varepsilon>0$. It follows from the fact that $\Psi$ has the property (P)
that there exists $N_{K^{\circ},\varepsilon}$ that

$N\geq N_{K^{\circ},\varepsilon}\\\
\Longrightarrow\sup_{\stackrel{{\scriptstyle(b,\tau)\in
K^{\circ}\times[0,\chi_{K^{\circ}}]}}{{\iota\in
I^{\prime}}}}\Big{|}\Psi(b^{\prime}_{0}+\tau+b)-\sum\limits_{\nu=0}^{N-1}\binom{N-1}{\nu}b^{\nu}(1-b)^{N-\nu}\Psi\Big{(}b^{\prime}_{0}+\tau+\nu\frac{1-\epsilon_{\iota^{\prime},N-1}^{[1]}}{N-1}\Big{)}\Big{|}\leq\varepsilon.$

One has in particular for
$N\geq\max(N_{K^{\circ},\varepsilon},1/\chi_{K^{\circ}})$ such that

$\Big{|}\Psi\Big{(}b^{\prime}_{0}+\frac{1-\epsilon_{\iota^{\prime},N}}{N}\,\xi+b\Big{)}-\sum\limits_{\nu=0}^{N-1}\binom{N-1}{\nu}b^{\nu}(1-b)^{N-\nu}\Psi\Big{(}b^{\prime}_{0}+\frac{1-\epsilon_{\iota^{\prime},N}}{N}\,\xi+\nu\,\frac{1-\epsilon_{\iota^{\prime},N-1}^{[1]}}{N-1}\Big{)}\Big{|}\leq\varepsilon$

for any $(b,\xi)\in K^{\circ}\times[0,1]$ and any $\iota\in I^{\prime}$. It
then follows from (4.4) that the sequence of functions

$\Big{(}b\in
K^{\circ}\longmapsto\frac{d}{db}\Big{(}\sum_{\nu=0}^{N}\binom{N}{\nu}\,b^{\nu}(1-b)^{N-\nu}\Phi\Big{(}b^{\prime}_{0}+\nu\,\frac{1-\epsilon_{\iota^{\prime},N}}{N}\Big{)}\Big{)}\Big{)}_{N\geq
1}$

converges uniformly on $K$ towards $b\longmapsto\Psi(b^{\prime}_{0}+b)$, the
convergence being uniform with respect to the index $\iota^{\prime}\in
I^{\prime}$. This implies that the sequence of corresponding primitives
vanishing at $0$, namely the sequence of functions (4.3), converges locally
uniformly on $-b^{\prime}_{0}+B$ towards $\Phi(b+b^{\prime}_{0})$, which
concludes the proof of Theorem 4.2. ∎

## References

  * [1] Y. Aharonov, D. Albert, L. Vaidman, How the Results of a Measurement of a component of a spin $1/2$ particle can turn out to be 100?, Phys. Rev. Lett. 60 (1988), pp. 1351–1354.
  * [2] Y. Aharonov, J. Behrndt, F. Colombo, P. Schlosser, A unified approach to Schrödinger evolution of superoscillations and supershifts, Journal of Evolution Equations 22, article number: 26 (2022).
  * [3] Y. Aharonov, F. Colombo, I. Sabadini, T. Shushi, D.C. Struppa, J. Tollaksen, A new method to generate superoscillating functions and supershifts, Proc. R. Soc. A. 477 (2249), Paper No. 20210020, 12 pp (2021).
  * [4] Y. Aharonov, F. Colombo, A. N. Jordan, I. Sabadini, T. Shushi, D.C. Struppa, J. Tollaksen, On superoscillations and supershifts in several variables, Quantum Stud. : Math. Found., 9 (2022) : pp. 417-433.
  * [5] Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, On the Cauchy problem for the Schrödinger equation with superoscillatory intitial data, J. Math. Pure Appl., 99 (2013), (9), pp. 165–173.
  * [6] Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, Superoscillating sequences as solutions of generalized Schrödinger equations, J. Math. Pure Appl. 103 (2015), pp. 522–534.
  * [7] Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, Superoscillating sequences in several variables, J. Fourier Anal. Appl. 22 (2016), no. 4, 751–767.
  * [8] Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, The mathematics of superoscillations, Memoirs of the American Mathematical Society 257 (1174), American Mathematical Society, 2017.
  * [9] Y. Aharonov, S. Popescu, D. Rohrlich, How a soft photon can emit a hard photon, TAUP 1847-90, Tel Aviv University, Tel Aviv, Israel, 1991.
  * [10] Y. Aharonov, S. Popescu, D. Rohrlich, On conservation laws in quantum mechanics, Proceedings National Academy of Sciences, 118 n.1, e1921529118, (2020).
  * [11] J. Behrndt, F. Colombo, P. Schlosser, D. C. Struppa, Integral representation of superoscillations via complex Borel measures and their convergence, Trans. Amer. Math. Soc., 376 (2023), 6315-6340.
  * [12] S. Bernstein, Sur la convergence de certaines suites de polynômes, J. Math. Pures Appl. 15, no. 9 (1935), pp. 345-358.
  * [13] R. Buniy, F. Colombo, I. Sabadini, D. C. Struppa, Quantum Harmonic Oscillator with superoscillating initial datum, J. Math. Phys. 55, 113511 (2014).
  * [14] F. Colombo, J. Gantner, D.C. Struppa, Evolution by Schrödinger equation of Aharonov-Berry superoscillations in centrifugal potential, Proc. A. 475 (2019), no. 2225, 20180390, 17 pp.
  * [15] F. Colombo, I. Sabadini, D.C. Struppa, A. Yger, Gauss sums, superoscillations and the Talbot carpet, J. Math. Pures Appl. 147 (2021) pp. 163–178.
  * [16] F. Colombo, I. Sabadini, D.C. Struppa, A. Yger, Superoscillating sequences and supershifts for families of generalized functions, Complex Analysis and Operator Theory 16, article number: 34 (2022).
  * [17] J. Dressel, M. Malik, F. Miatto, A. N. Jordan, R. W. Boyd, Colloquium: Understanding Quantum Weak Values: Basics and Applications, Review of Modern Physics, 86, (2014) 307.
  * [18] B. Gaveau, L.S. Schulman, Explicit time-dependent Schrödinger propagators, Journal of Physics A: Mathematical and General, Volume 19 (1986), no. 10.
  * [19] L. Kantorovich, Sur la convergence de la suite des polynômes de S. Bernstein en dehors de l’intervalle fondamental, Bull. Acad. Sci. URSS (1931), pp. 1103-1115.
  * [20] G.G. Lorentz, Bernstein polynomials, Toronto, 1953 (second edition : Chelsea Publishing Company, New York, 1986).
  * [21] K. McCormick, Puzzling quantum scenario appears not to conserve energy, Quantamagazine, May 16, 2022, quantamagazine.org, retrieved on 12.09.2022.
  * [22] N. Ortner, P. Wagner, Distribution-Valued Analytic Functions-Theory and Applications, edition Swk, Hamburg, 2013.
  * [23] G. M. Phillips, Interpolation and Approximation by Polynomials, CMS Books in Mathematics, Springer Science + Business Media New York, 2003.
  * [24] J. L. Schiff, Normal families, Universitext, Springer-Verlag, New York, 1993.
  * [25] B.A. Taylor, Some locally convex spaces of entire functions, p. 431-467 in Entire functions and related parts of analysis, Proceedings of Symposia in Pure Mathematics 11, American Mathematical Society, Providence, 1968.
  * [26] E. C. Titchmarsh, The theory of functions, Oxford, 1939.
  * [27] K.B. Wolf, On Time-Dependent Quadratic Quantum Hamiltonians, SIAM Journal on Applied Mathematics 40 (3) (1981), pp. 419-431.