On superoscillations and supershifts in several variables

The aim of this paper is to study a class of superoscillatory functions in several variables, removing some restrictions on the functions that we introduced in a previous paper. Since the tools that we used with our approach are not common knowledge we will give detailed proof for the case of two variables. The results proved for superoscillatory functions in several variables can be further extended to supershifts in several variables.


Introduction
Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. Physical phenomena associated with superoscillatory functions have been known for a long time and in more recent years there has been a wide interest both from the physical and the mathematical point of view. In 1952, Toraldo di Francia observed the superoscillation phenomenon in antennas theory, see [43], and Y. Aharonov discovered it in the context of weak values in quantum mechanics, see [1]. An introduction to superoscillatory functions in one variable and some applications to Schrödinger evolution of superoscillatory initial data can be found in [9]. Superoscillatory functions in several variables have been rigorously defined and studied in [8]. The aim of this paper is to remove the restrictions that were used in [8] and to generalize the existing theory to the more general phenomenon of supershift.
Our results are directed to a double audience of physicists and mathematicians and since our tools, that consist of infinite order differential operators acting on spaces of entire holomorphic functions, are not widely known we consider first the case of two superoscillatory variables. In this case, we avoid heavy notations so that the reader can better follow the main points of the proofs.
The prototypical superoscillating function is where a > 1 and the coefficients C j (n, a) can be calculated to be If we fix x ∈ R and we let n go to infinity, we obtain that lim n→∞ F n (x, a) = e iax , and the limit is uniform on the compact sets of the real line. The term superoscillations comes from the fact that in the Fourier representation of the function (1) the frequencies 1 − 2 j/n are bounded by 1, but the limit function e iax has a frequency a that can be arbitrarily larger than 1. A fundamental problem is to determine how large the class of superoscillatory functions. Many of the works in the reference list, as we pointed out before, are devoted to the question of permanence of superoscillations when they are taken as initial values for a given Schrödinger equation, but as a byproduct they also offered a very powerful way to extend the class of superoscillatory functions. These extensions, nevertheless, are still very closely connected to the archetypical function defined earlier on. To address this issue, we have recently introduced, [11], a new method to generate superoscillatory functions for different configurations of points in the interval [−1, 1]. More precisely: let h j (n) be a given set of points in [−1, 1], j = 0, 1, ..., n for n ∈ N, and let a ∈ R be such that |a| > 1. If h j (n) = h i (n) for every i = j the function and therefore when the sequence of holomorphic extensions of ( f n ) converges in the space A 1 of functions of exponential type we have that Two explicit examples are the following: 1. (I) Let n ∈ N and set h j (n) = 1 − 2 n j where j = 0, ..., n; in this case we obtain the superoscillatory function .., n, for a fixed p ∈ N, then we have: In both cases the sequences converge to e iax for every x ∈ R, and numerous other examples can be easily constructed explicitly.
In the paper [8] we described and studied superoscillations in several variables. The methods we used, however, required us to accept some constraints on the kind of superoscillations we could include. In this paper, on the other hand, we develop a new approach that allows us to remove those constraints and study a more general class of superoscillations, by showing how they can be constructed starting from superoscillatory functions in one variable. The main idea is to consider a superoscillating function for some coefficients Z j (n, a) (see Definition 2.1) and to assume that its holomorphic extension to the entire function f n (ξ ) converges to e iaξ in A 1 , i.e. that there exists C ≥ 0 such that For p 1 , p 2 , . . . , p d ∈ N, d ∈ N, we then define In some earlier work, we have also shown that superoscillating functions are a particular case of supershifts, and for this reason we now introduce, and study for the first time, the case of supershifts in d ≥ 2 variables. The results that we have obtained and, even more, the techniques that we have used have convinced us of the existence of an intimate relation between global analyticity in C d and superoscillations and the supershift property on the real space R d . We have discussed this in detail this very subtle question with a colleague [44], and we plan to come back to it with a joint paper in the near future.
The paper is organized into four sections. After this introduction, Section 2 contains the preliminary material on superoscillations, the relevant function spaces and their topology, and we study the continuity of some infinite order differential operators on such spaces. Section 3 is the main part of the paper and contains the definition of superoscillating functions in several variables and some results proved in the specific case of two variables. Section 4 discusses the notion of supershift in several variables.

Superoscillations, function spaces and operators
In this section, we summarize the preliminary definitions and results to treat superoscillatory functions in several variables. We begin with the precise definition of superoscillatory functions in one variable. Definition 2. 1 We call generalized Fourier sequence a sequence of the form where a ∈ R, Z j (n, a) and h j (n) are complex and real valued functions of the variables n, a and n, respectively. The sequence (3) is said to be a superoscillating sequence if sup j,n |h j (n)| ≤ 1 and there exists a compact subset of R, which will be called a superoscillation set, on which f n (x) converges uniformly to e ig(a)x , where g is a continuous real-valued function such that |g(a)| > 1.
The classical Fourier expansion is obviously not a superoscillating sequence since its frequencies are not, in general, bounded. A simple, but important, example is In the recent paper [11], we enlarged the class of superoscillating functions for coefficients and frequencies more general than C j (n, a) and 1 − 2 j/n, and we solved the following problem.

Problem 2.2 Let h j (n) be a given set of points in
.., n, for n ∈ N and let a ∈ R be such that |a| > 1. Determine the coefficients X j (n) of the sequence n (0) = (ia) p mean that the functions x → e iax and x → f n (x) have the same derivatives at the origin, for p = 0, 1, ..., n, and therefore the same Taylor polynomial of order n.

Theorem 2.4 (Solution of Problem 2.2) Let h j (n) be a given set of points in
, for every i = j, then the coefficients X j (n, a) are uniquely determined and given by As a consequence, the sequence Our mathematical tools to study superoscillatory functions in one or in several variables make use of infinite order differential operators. Such operators naturally act on holomorphic functions. This is the reason for which we consider the holomorphic extension to entire functions of the sequence f n (x) defined in (2.1) by replacing the real variable x by the complex variable ξ . For the sequences of entire functions we shall consider, a natural notion of convergence is the convergence in the space A 1 as in the following definition.

Definition 2.5
The space A 1 is the complex algebra of entire functions such that there exists B > 0 such that The space A 1 has a rather complicated topology, see e.g. [20], since it is a linear space obtained via an inductive limit. For our purposes, it is enough to consider, for any fixed B > 0, the set A 1,B of functions f satisfying (5), and to observe that defines a norm on A 1,B , called the B-norm. One can prove that A 1,B is a Banach space with respect to this norm.
Moreover, let f and a sequence ( f n ) n belong to for some C ≥ 0. With these notations and definitions we can make the notion of continuity explicit (see [18]): A linear operator U : The following result, see Lemma 2.6 in [17], gives a characterization of the functions in A 1 in terms of the coefficients appearing in their Taylor series expansion.

Lemma 2.6
The entire function .

Remark 2.7
To say that f ∈ A 1 means that f ∈ A 1,B for some B > 0. The computations in the proof of Lemma 2.6 in [17], show that b = 2eB, and that we can choose C f = f B .
Lemma 2.6 has been proved in [17] and is a crucial fact in the proof of the following results. The reader must not be confused by the fact that the variables x, y appearing in the statement below are real, indeed they have the role of parameters (which can also be considered as complex numbers). We now define two infinite order differential operators that will be used to study superoscillatory functions and supershifts in two variables. We would like to stress, one more time, that the key ingredient in the theory that we have developed is the ability to characterize those operators that act continuously on the space A 1 or, more generally, on spaces of entire functions with growth conditions. Proposition 2.8 Let x, y ∈ R and p, q ∈ N. Denote by D ξ the derivation with respect to the complex variable ξ . We define the formal operator: Then U(x, y, D ξ ) : A 1 → A 1 is continuous for all x, y ∈ R and p, q ∈ N.
Proof Let us consider Taking the modulus we get and Lemma 2.6 gives the estimate on the coefficients f pm− pμ+qμ+k .
We now use the Gamma function estimate to separate the series, and we have Now observe that the series in k satisfies the estimate where C is a positive constant, because of the properties of the Mittag-Leffler function, and the series is convergent to a number denoted by C x,y, p,q > 0. In fact, using the Stirling's formula for the Gamma function, we have and then we deduce and so Now observe that the series (9) has positive coefficients and so it converges if and only if the series we have that the series (9) converges for all x, y ∈ R and p, q ∈ N. So we finally have The estimate (11) shows that U(x, y, D ξ ) f ∈ A 1 , in fact moreover, it also shows that its 2b-norm satisfies We define the formal operator Then V(x, y, D ξ ) : A 1 → A 1 is continuous.
Proof We apply the operator V(x, y, D ξ ) to a function f in A 1 and we have We take the modulus and we use the estimate in Lemma 2.6: With the estimates we separate the series Finally, we get Using (10) we have converge to C x , C y respectively, for x, y ∈ R. So we have from which, recalling that C f = f B and b = 2eB, we deduce Thus we have that the conditions in (6) are satisfied and the statement follows.

Superoscillating functions in several variables
We recall some preliminary definitions related to superoscillatory functions in several variables, then, for the sake of simplicity, we limit our study to the case of two variables and then we discuss how our results can be extended to the general case of d > 2 variables, see [28].
where C j (n, a) are given by (2) and q ∈ N, for = 1, . . . , d. Remark 3.4 In the paper [8], we studied the function theory of superoscillatory functions in several variables under the additional hypothesis that there exist r ∈ N, such that In that case, we proved that for p, q ∈ N, = 1, . . . , d the function is superoscillating when |a| > 1.
In this paper, we work in a more general framework and we are able to remove the restriction (15) on the coefficients p, q for = 1, ..., d and to show that general superoscillating functions as in (3) can be used to define superoscillatory functions in several variables.
We start by proving the following: be a superoscillating function as (3) and assume that its holomorphic extension to the entire function f n (ξ ) converges to e iaξ in the space A 1 . For p and q ∈ N we define F n (x, y) = n j=0 Z j (n, a)e i x(h j (n)) p e iy(h j (n)) q .
Then, we have lim n→∞ F n (x, y) = e i xa p e iya q , and, in particular, F n (x, y) is superoscillating when |a| > 1.
Proof We write the chain of equalities Now observe that using the auxiliary complex variable ξ we have where D ξ is the derivative with respect to ξ and | ξ =0 denotes the restriction to ξ = 0. So we can write and defining the infinite order differential operator we get In Proposition 2.8 we have proved that the operator U(x, y, D ξ ) : A 1 → A 1 is continuous; therefore, we can take the limit inside U(x, y, D ξ ) and we have: Since the limit function is continuous (it is in A 1 ), we can take the restriction to ξ = 0 from which we get the statement.  (21), that still acts on the space A 1 .
We now state the case of d ≥ 2 variables, without giving all the details of the proof.
be superoscillating functions as in (6) and assume that their entire extensions to the functions f n (ξ ) converge to e iaξ in A 1 . Let p 1 , p 2 , . . . , p d with p ∈ N, = 1, 2, . . . , d and d ∈ N, for d ≥ 2. Define Then We write the chain of equalities where we have set We define the infinite order differential operator where and with similar computations as in Proposition 2.8 we can prove that the operator and proceeding as in the case of two variables we get the statement.

Supershifts in several variables
The procedure to define superoscillatory functions can be extended to the case of supershifts. Recall that the supershift property of a function extends the notion of superoscillations and that this concept turned out to be a crucial ingredient for the study of the evolution of superoscillatory functions as initial conditions of the Schrödinger equation (or of other field equations). We define the functions where (c j (n)) is a sequence of complex numbers for j = 0, ..., n and n ∈ N 0 . If for some a ∈ I with |a| > 1, we say that the function ψ n (x), for x ∈ R, admits a supershift.

Remark 4.2
The term supershift comes from the fact that the interval I can be arbitrarily large (it can also be R) and that the constant a can be arbitrarily far away from the interval [−1, 1] where the functions ϕ h j,n (·) are computed, see (22).

Remark 4.3
Superoscillations are a particular case of supershift. In fact, for the sequence (F n ) in (1), we have h j (n) = 1 − 2 j/n, ϕ h j (n) (x) = e i(1−2 j/n)x and c j (n) are the coefficients C j (n, a) defined in (2) where h → G(hx) depends on the parameter x ∈ R. Determine the coefficients Y j (n) in such a way that The solution of Problem 4.4, obtained in [11], is summarized in the following theorem.  (23) and it is given by , Remark 4.6 In the following we will consider those functions G and sequences h j (n) for which the holomorphic extension f n (z) of f n (x) converges in A 1 to G(az).
We can now extend the notion of supershift of a function in several variables.
where (c j (n)) j,n , j = 0, . . . , n, for n ∈ N 0 is a complex-valued sequence, admits the supershift property if be a superoscillating function as in Definition 2.1 and assume that its holomorphic extension to the entire functions f n (z) converges to e iaz in the space A 1 . Get G 1 and G 2 be holomorphic entire functions whose series expansion is given by and define F n (x, y) = n k=0 Z k (n, a)G 1 (xh j (n))G 2 (yh j (n)), where Z k (n, a) are given as in (25). Then F n (x, y) admits the supershift property that is lim n→∞ F n (x, y) = G 1 (xa)G 2 (ya).
Proof We consider g m 2 x m 1 y m 2 (h j (n)) m 1 +m 2 .
We now consider the auxiliary complex variable ξ and we note that for λ ∈ C, ∈ N, where D ξ is the derivative with respect to ξ and | ξ =0 denotes the restriction to ξ = 0, we have .
We now use the operator V(x, y, D ξ ) defined in (13) so that we can write .
Here we use Proposition 2.10 in order to compute the limit and this concludes the proof.

Remark 4.9
The notion of supershift and the previous results can be extended to the case of several variables.