Holomorphic functions, relativistic sum, Blaschke products and superoscillations

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The notion of superoscillation is a particular case of that one of supershift. In the recent years, superoscillating functions, that appear for example in weak values in quantum mechanics, have become an interesting and independent field of research in complex analysis and in the theory of infinite order differential operators. The aim of this paper is to study some infinite order differential operators acting on entire functions which naturally arise in the study of superoscillating functions. Such operators are of particular interest because they are associated with the relativistic sum of the velocities and with the Blaschke products. To show that some sequences of functions preserve the superoscillatory behavior it is of crucial importance to prove that their associated infinite order differential operators act continuously on some spaces of entire functions with growth conditions.


Introduction
Superoscillating function, or sequences, appear in weak values in quantum mechanics as shown in [1,15] and in several other fields like optics or signal processing. Y.  1 Aharonov has suggested several problems associated with superoscillations and the two most important ones are: to establish how large is the class of superoscillatory functions and to study the evolution of superoscillations (associated with weak-values) as initial datum of Schrödinger equation or as initial condition of some relativistic quantum field equations.
The aim of this paper is to contribute to the first problem mentioned above. In fact, we show that one can construct superoscillatory functions or, more in general, functions that admit the supershift property by considering various tools such as the relativistic sum of the velocities and/or the Blaschke products in a way that will be specified in the sequel.
In the last decade a systematic study of superoscillating functions has been carried out from the mathematical point of view and extended to the case of several variables [10]. It has been shown that superoscillations can approximate functions in the Schwartz space S, in the space of distributions of type S , see [27], and of hyperfunctions as shown in [30]. Moreover, these functions appear in the Talbot effect, see [29], and supershifts for families of generalized functions are considered in [28]. We point out that the concept of supershift is more general than the one of superoscillation, which is a particular case. Finally, we mention that superoscillations are also useful in Schur analysis as shown in [16].
The most important superoscillatory function, that appears in connection with weak values in quantum mechanics, is If we fix x ∈ R and we let n go to infinity, we obtain that lim n→∞ F n (x, a) = e iax . Inspired by (1) we define a more general class of superoscillatory functions. We call generalized Fourier sequence a sequence of the form where a ∈ R, k j (n) ∈ R and E j (n, a) ∈ C for all j = 0, ..., n and n ∈ N.
A generalized Fourier sequence Y n (x, a) is said to be a superoscillating sequence if it satisfy the two properties: (P1) sup j,n |k j (n)| ≤ 1 for all j = 0, ..., n and n ∈ N.
(P2) there exists a compact subset of R, which will be called a superoscillation set, on which lim n→∞ Y n (x, a) = e ig(a)x where g is a continuous function on an interval that contains the points a, k j (n) and such that |g(a)| > 1.
As we have already mentioned superoscillations occur as a particular case of sequences of functions that satisfy the so called supershift property. This property is of crucial importance in the study of evolution of superoscillations as initial datum of the Schrödinger equation or of any other field equation in quantum mechanics and is defined below.
Supershift property. Let λ → ϕ λ (x) ∈ C be a continuous function in the variables λ ∈ I, where I ⊆ R is an interval, and x ∈ , where is a domain. We consider x ∈ as parameter for the function λ → ϕ λ (x) where λ ∈ I. When [−1, 1] is contained in I and a ∈ I, we define the sequence in which ϕ λ is computed just on the points 1 − 2 j/n belonging to the interval [−1, 1] and the coefficients C j (n, a) are defined for example as in (2), for j = 0, ..., n and n ∈ N. If for |a| > 1 arbitrary large (but belonging to I), we say that the function λ → ϕ λ (x), for x fixed, admits a supershift in λ.
If we set ϕ λ (x) = e iλx , we obtain the superoscillating sequence described above as a particular case of the supershift. In fact, in this case, we have ψ n (x) = F n (x, a), where F n (x, a) is defined in (1). The name supershift is due to the fact that we are able to obtain ϕ a , for |a| > 1 arbitrarily large, by simply calculating the function λ → ϕ λ in infinitely many points in the neighborhood [−1, 1] of the origin.
With the above definitions in mind we can now state our main results. In Sect. 2 we consider the two superoscillating functions (of supershifts): and we define the pointwise product (related to the coefficients j (n) and k j (n)), denoted by • P , as In particular, we can set j (n) := g(1 − 2 j/n) and k j (n) := h(1 − 2 j/n) where g and h are entire holomorphic functions . Moreover, we assume that g and h belong to the space of entire functions for which there exist constants A, B > 0 such that |g(z)| ≤ Ae B|z| p and |h(z)| ≤ Ae B|z| p for some p ∈ [1, ∞). In Theorem 2.8 we show the supershift property as a consequence of the continuity of an infinite order differential operator associated with the function Y n • P Z n (x, a), see Theorem 2.7. When g and h are such that with the assumption that for a ∈ R we have |g(a)h(a)| < 1 for |a| ≤ 1 and |g(a)h(a)| > 1 for |a| > 1, then the sequence F n • P Y n (x, a) is superoscillating. In the other cases we have the supershift property. To obtain superoscillations, a simple example of choice of f and g is given by g(z) = z p and h(z) = z q for p, q ∈ N.
The relativistic sum of the velocities in special relativity suggests how to define a class of functions that admit the supershift property. We recall that in special relativity the law of addition of the velocities v and w of two objects moving along a line is given by where c is the speed of the light in the vacuum. When we normalize the speed of the light setting c = 1 we have that if |u| < 1 and |v| < 1 then |u ⊕ R v| < 1, see for more details [35]. This fact suggests how to define a binary operation between two sequences bounded by 1 that gives a sequence still bounded by 1. Thus, in Sect. 3 we consider two superoscillating functions Y n (x, a), Z n (x, a) and we define their relativistic product as: Here we assume that j (n) and k j (n) are given by j (n) = g(ε(1 − 2 j/n)) and k j (n) = h(ε(1 − 2 j/n)), for ε ∈ (0, 1), where g and h are holomorphic functions defined on the unit ball D in C that satisfy suitable conditions. Then we prove that for any εa < 1 using suitable defined infinite order differential operators that depend on the relativistic sum. The technique used to prove our main results is based on infinite order differential operators U(x, ∂ ξ ) that are applied to the fundamental superoscillating function (1). We then define where | ξ =0 denotes the restriction to ξ = 0 with respect to the auxiliary complex variable ξ . A crucial fact in the strategy of generating superoscillating functions is the study of the continuity of the operator U(x, ∂ ξ ) on the space of entire functions with growth conditions. As we will see in Sect. 3 the infinite order differential operator associated with the relativistic sum is of the form and the coefficients of the (a p ) p∈N 0 and (b p ) p∈N 0 in the formal operators A(∂ ξ ) and B(∂ ξ ) acting on entire functions are sequences of complex numbers that satisfy suitable growth conditions, see Theorem 3.9. In Sect. 4 we study the infinite order differential operators associated with the Blaschke products and we use such operators to study superoscillatory functions. Blaschke products provide another case that we study in order to get sequences that admit the supershift property. The corresponding infinite order differential operators are studied in Sect. 4.

Remark 1.1
To determine the structure of the infinite order differential operators associated with the relativistic sum and the Blaschke product is part of our task. The precise identification of these infinite order differential operators is useful also for further investigations. For example, the study of the supershift property in the space of hyperfunctions has been done for the infinite order differential operator associated with the quantum harmonic oscillator, see [31].
We conclude this introduction, by pointing out that a different class of superoscillating functions with respect to the ones considered in this paper can be found in [23], and a very general technique to obtain superoscillations has been recently introduced in the paper [5] and explained in the last section.

Superoscillations and supershift
We recall some definitions and results on entire functions, see e.g. [20], which will play an important role in the proofs of the main results. Definition 2.1 Let p ≥ 1. We denote by A p the space of entire functions with order lower than p also called the space of entire functions of order p and finite type. It consists of functions f for which there exist constants B, C > 0 such that The following result gives a characterization of functions in A p in terms of their Taylor coefficients.
where the constant b f is given by Furthermore, a sequence f n in A p tends to zero if and only if C f n → 0 and b f n < b for some b > 0.

Remark 2.3
The constant C > 0 in the estimate (6), i.e., | f (z)| ≤ Ce B|z| p , and the , are in general different. From an inspection of the proof of Lemma 2.2 in [18] the relation between the constants B > 0 and b f > 0 is given by (9), i.e., b f = (2 p Bpe) 1/ p , where e is the Neper number.
The holomorphic extension of F n (x, a) as in (1) converges in A 1 as the following lemma, proved in [28], shows.

Lemma 2.4
Let a ∈ C, for any z ∈ C, consider F n (z, a) := cos z n + i a sin z n n with z, a ∈ C. Then, for every n ∈ N and z ∈ C, one has Now we will consider generalized Fourier sequences ( n j=0 C j (n, a)e ik j (n)x ) n in which the exponential sequences (k j (n)) j,n are defined by holomorphic functions in A q for some q ≥ 1.

Definition 2.5 (Generalized Fourier sequences depending on an entire function) Let
a ∈ R and C j (n, a) ∈ C be as in (2) for all j = 0, ..., n and n ∈ N. Let h ∈ A q , for some q ≥ 1, be such that h maps R to itself, we define the generalized Fourier series Definition 2.6 (The product • P of generalized Fourier sequences) Let Y n (x, a) and Z n (x, a) be generalized Fourier sequences as in Definition 2.5, i.e., where C j (n, a) ∈ C be as in (2) for all j = 0, ..., n and n ∈ N. Let h, g ∈ A q , for some q ≥ 1, be such that h and g map R to itself, j (n) In order to compute the limit lim n→∞ Y n • P Z n (x, a) we need to define a suitable infinite order differential operator acting on A 1 and to prove the continuity of such operator on A 1 . Thanks to the assumptions on the functions h, g ∈ A q , for some q ≥ 1, we will prove the supershift property for function Y n • P Z n (x, a) and if h and g satisfy additional conditions, then the sequence Y n • P Z n (x, a) is, in particular, superoscillating.

Theorem 2.7
Assume that x is a fixed real parameter and let ξ ∈ C. We define the formal infinite order differential operator: where the operator A(∂ ξ ) is defined replacing z by i −1 ∂ ξ in the entire function Proof First we consider the action of the operators The proof of the above formula is by induction. First we observe that for m = 1 we have Thus the formula (12) is true for m = 1. If we suppose that the formula (12) holds for m = n then for m = n + 1 we have where in the third equality we set k = j − p 1 − · · · − p n and in the last equality we rename the index p by p n+1 . Thus the proof of the formula (12) is complete. We recall that if G ∈ A q , for q ≥ 1, by Lemma 2.2, there exist positive constants b G , C G such that where in the fourth inequality we have used that the series are absolute convergent and is the Mittag-Leffler function of order q. The previous inequality means that Observe that instead of f n → 0 we can equivalently require that C f n → 0 with b f n ≤ b 0 , for some positive constant b 0 , where the constants C f n and b f n refer where in the second equality we have used the absolute convergence for a fixed.
Now we can prove the superoscillating property of the product of functions in Definition 2.6. Theorem 2.8 Let h, g ∈ A q , for some q ≥ 1, be such that h and g map R to itself. Let Y n (x, a) and Z n (x, a), for x ∈ R and |a| > 1, be as in the Definition 2.6. Then we have Moreover, if |g(a)h(a)| < 1 for |a| ≤ 1 and |g(a)h(a)| > 1 for |a| > 1 then the sequence (Y n • P Z n )(x, a) is superoscillating.

Remark 2.9
In Theorem 2.8, if we consider the supershift property instead of the superoscillating property, we do not have to require that the functions g and h to map R to itself.
Proof Let h, g ∈ A q for some q ≥ 1. We consider the function Since h, g ∈ A q the function G(z) := h(z)g(z) belongs to A q . We define the operator U(x, ∂ ξ ) as in Theorem 2.7 and thus we obtain the relations and Thus we have = e i xG(a) .

Remark 2.10
A simple example for the superoscillatory case is g(z) = z l and h(z) = z s for l, s ∈ N.

The relativistic sum and superoscillating functions
In the previous section we have considered generalized Fourier sequences as in Definition 2.6, where the product • P involves the pointwise product of the coefficients j (n) and k j (n) of the exponentials in Y n (x, a) and Z n (x, a), respectively. Using the • P product it is clear that, if we are in the superoscillating case where | j (n)| < 1 and |k j (n)| < 1, then we have | j (n)k j (n)| < 1. But if we consider the sum j (n) + k j (n) instead of the product j (n)k j (n) we cannot guarantee that | j (n) + k j (n)| < 1. An interesting sum that overcomes this fact is the relativistic sum of velocities, see [35] where we can set c = 1 so that We denote by D the open unit disc in C. An important point in our considerations is the following result. (I) The relativistic sum of g and h is a holomorphic function on D.
Proof Point (I) follows by observing that |g(z)h(z)| < 1. Point (II) can be deduced from the observation that u ⊕ R v is the restriction to the square (−1, 1) × (−1, 1) of the Möbius map

Remark 3.2
In general, the function g ⊕ R h defined in the previous lemma, does not map D in D. For example we can consider h(z) = g(z) ≡ 1 2 i for any z ∈ D. In this case we have where ⊕ R is the relativistic sum (15).

Remark 3.4 When we consider the supershift notion in Definition 3.3, we do not have
to assume that f and g are real valued on (−1, 1).
which have series expansions: Associated with A and B we define the operators and The linear operator associated with the relativistic sum is defined as where x ∈ R is a parameter and ξ ∈ C.

Remark 3.6
The operator C(∂ ξ ), introduced in the Definition 3.5, is obtained by the formal substitution of z with ∂ ξ in the relativistic sum of g and h. For we observe that The natural space of holomorphic functions on which the operator U R (x, ∂ ξ ) acts continuously is defined as follows.
Definition 3.7 Let α be a fixed positive number. We define the class A 1,α to be the set of entire functions such that there exists C > 0 for which .
We are now ready to prove our main result on the continuity of the infinite order differential operator U R (x, ∂ ξ ) associated with the relativistic sum.

Remark 3.8
In the Theorem 3.9 below we will study the continuity of the operator U R (x, ∂ ξ ) defined in the domains A 1,α for α > 0. In the proof of Theorem 3.11 will be sufficient to consider the continuity of the operator U R (x, ∂ ξ ) defined in A 1,1 .

Theorem 3.9
Let α > 0 be fixed. Let g, h : D → D be holomorphic functions, let (a p ) p∈N 0 , (b q ) q∈N 0 be the sequences of complex numbers as in (18) and such that the conditions hold. Then, the linear operator U R (x, ∂ ξ ), for x ∈ R, defined in (21), acts continuously from A 1,α to A 1 . Moreover, we have Proof The proof is split in steps. STEP 1. We study the action of the operator C(∂ ξ ) = B(∂ ξ ) ∞ =0 (−1) A (∂ ξ ) on holomorphic functions belonging to A 1,α .
We first consider the operator A(∂ ξ ) defined in (19). Using the formula (12) for the operators A (∂ ξ ) : (24) and the operator C(∂ ξ ) : A 1,α → A 1 , defined in (20), acts as Using (24) it turns out that STEP 2. To compute C(∂ ξ ) m we need some more notation. By the formula (25), whenever we fix the index m we have the following set of indexes: . . , m. We will prove, by induction, that: (26) The first step (i.e., m = 1) is described in the formula (25). We now proceed by induction. We suppose the formula holds to be true for m > 1 and we show that it is true for m + 1: where in the last equality we have used (25). After renaming the indexes and letting μ varying from 1 to m + 1, we obtain Thus formula (26) are proved. STEP 3. We show that U R (x, ∂ ξ ), defined in (21), acts continuously from A 1,α to A 1 .
The formula (26), which gives the explicit expression of C(∂ ξ ) m , will be crucial to study the continuity of U R (x, ∂ ξ ). In fact, applying the operator U R (x, ∂ ξ ) to holomorphic functions such that | f (z)| ≤ C exp(α|z|) we have: Taking the modulus of both sides in the previous equality, we obtain where we have used Lemma 2.2 with p = 1 (see also Remark 2.3 with p = 1). After some computations we get Keeping in mind that b f = 2eα and recalling the assumptions (23) on the coefficients a p 's and b q 's, there exist two positive numbers c 1 < 1 and c 2 such that Thus we obtain and finally, setting with the bounds b n ≤ 2eα, shows that U (x, ∂ ξ ) f n (ξ ) → 0. Finally we have for |λ| < α

Remark 3.10
In the case of the relativistic sum, we have to re-scale the interval (−1, 1) in the definition of superoscillations and/or supershift property because here we work in the unit ball in C. So we take ε ∈ (0, 1) and the interval (−1, 1) is replaced by the interval (−ε, ε) when we set j (n) := g(ε(1 − 2 j/n)) and k j (n) := h(ε(1 − 2 j/n)).
We are ready to state the main result on the relativistic product.

Remark 3.12
The Theorem 3.11 also works in the case of sequences that satisfy the supershift property (in particular in this case |a| > 1) but they are not necessarily generalized Fourier sequences.
Proof For every a ∈ R and ε ∈ (0, 1) such that |εa| < 1, we consider We define the operator U R (x, ∂ ξ ) as in (21) and we consider its action over the space A 1,1 (see Remark 3.8). By Theorem 3.9 we obtain the relations and = e i x(h⊕ R g)(εa) where we have assumed |εa| < 1.

Blaschke products and superoscillations
In this section we will consider a new type of infinite order differential operators that will allow us to study generalized Fourier series defined by Blaschke products. Let (b n ) M n=1 be a finite sequence of complex numbers such that 0 < |b i | < 1 for i = 1, . . . , M. A finite Blaschke product is defined for |z| < 1 by where the last equality holds in the unit disc since |b n z| < 1. Evidently, the product of two finite Blaschke products is still a finite Blaschke product.
In the sequel we will study the limit of generalized Fourier sequences of the form: for a ∈ R such that ε|a| < 1 and B is a finite Blaschke product. In order to compute the limit for n → ∞ we need the following infinite order differential operators formally obtained by the substitution of z with ∂ ξ in the series expansion (32).
be a finite sequence of complex numbers such that 0 < |b i | < 1 for i = 1, . . . , M. Let B(z) be a finite Blaschke products as in (32). We define the formal operators and where A Proof We proceed by induction over M. The first step is trivial. We suppose that the formula (35) holds to be true for M then for M + 1 we have

Thus by induction the formula (35) holds for any M.
For the sake of simplicity, from now on we refer to the set M M as M.  for |λ| < min(α, 1).
Using Lemma 4.2 we can write We fix an element f ∈ A 1,α and s ∈ M and we observe that Note that we have used the elements of the vector s c as exponents of the b q 's in the previous formula since they appear as factors in A (0) q but not in A (1) q . Taking the absolute value of both sides in the previous formula and using the fact that | f j | ≤ C f b j j! we obtain We deduce The previous inequality means that U B (x, ∂ ξ ) f (ξ ) ∈ A 1 and also that U B (x, ∂ξ) is continuous from A 1,α to A 1 . Finally we have that where we have assumed |εa| < 1.
We conclude this paper by observing that it is also possible to combine the relativistic sum of Blaschke products and superoscillations. These leads to more complicated computations because if we consider the Blaschke products From these expressions we deduce suitable infinite order differential and the we have to study their continuity properties on the space of entire functions A 1 .

Concluding remarks
Recently in the paper [5] the authors have solved explicitly the following problem: we obtain: f n (x) = n j=0 n k=0, k = j n p 2 1 − 2 n p k − a j − k e i(1− 2 n p j)x , x ∈ R.
It is worthwhile to mention that in antenna theory the phenomenon of superoscillations was discovered by G. Toraldo di Francia in [34]. The literature on superoscillations from the physics point of view is large, but since physical aspects are outside the scope of this paper we mention here only [22] and the recent paper Roadmap on superoscillations, see [21], in which some of the most important achievements in the applications of superoscillations are well explained by the leading experts in this field.
More information on the mathematical approach to superoscillations can be found in the introductory papers [13,14,17,33].