A new method to generate superoscillating functions and supershifts
Y. Aharonov, F. Colombo, I. Sabadini, T. Shushi, D.C. Struppa, J. Tollaksen
aa r X i v : . [ m a t h - ph ] J a n A new method to generate superoscillating functionsand supershifts
Y. Aharonov ∗ , F. Colombo † , I. Sabadini † , T. Shushi ‡ , D.C. Struppa § , J. Tollaksen ∗ January 19, 2021
Abstract
Superoscillations are band-limited functions that can oscillate faster than their fastestFourier component. These functions (or sequences) appear in weak values in quantum me-chanics and in many fields of science and technology such as optics, signal processing andantenna theory. In this paper we introduce a new method to generate superoscillatory func-tions that allows us to construct explicitly a very large class of superoscillatory functions.
AMS Classification: 26A09, 41A60.
Key words : Superoscillating functions, New method to generate superoscillations, Supershift.
Superoscillating functions are band-limited functions that can oscillate faster than their fastestFourier component. Physical phenomena associated with superoscillatory functions are knownsince long time and in more recent years there has been a wide interest both from the physicaland the mathematical point of view. These functions (or sequences) appeared in weak valuesin quantum mechanics, see [2, 11, 22], in antenna theory this phenomenon was formulated in[34]. The literature on superoscillations is large, and without claiming completeness we mentionthe papers [13], [18]-[21], [28]-[30] and [32]. This class of functions has been investigated alsofrom the mathematical point of view, as function theory, but large part of the results areassociated with the study of the evolution of superoscillations by quantum fields equations withparticular attention to Schr¨odiger equation. We give a quite complete list of papers [1], [3]-[9], [14]-[16], [23]-[27], [33] where one can find an up-to-date panorama of this field. In orderto have an overview of the techniques developed in the recent years to study the evolutionof superoscillations and their function theory, we refer the reader to the introductory papers ∗ Schmid College of Science and Technology, Chapman University, Orange 92866, CA, US, [email protected], [email protected] † Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9 20133 Milano, Italy, [email protected], [email protected] ‡ Department of Business Administration, Guilford Glazer Faculty of Business and Management, Ben-GurionUniversity of the Negev, Beer-Sheva, Israel, [email protected] § The Donald Bren Distinguished Presidential Chair in Mathematics, Chapman University, Orange, USA, [email protected]
Roadmap on superoscillations , see [17], where themost recent advances in superoscillations and their applications to technology are well explainedby the leading experts in this field.A fundamental problem is to determine how large is the class of superoscillatory functions.The prototypical superoscillating function that is the outcome of weak values is given by F n ( x, a ) = (cid:16) cos (cid:16) xn (cid:17) + ia sin (cid:16) xn (cid:17)(cid:17) n = n X j =0 C j ( n, a ) e i (1 − j/n ) x , x ∈ R , (1)where a > C j ( n, a ) are given by C j ( n, a ) = (cid:18) nj (cid:19) (cid:18) a (cid:19) n − j (cid:18) − a (cid:19) j . (2)If we fix x ∈ R and we let n go to infinity, we obtain thatlim n →∞ F n ( x, a ) = e iax . Clearly the name superoscillations comes from the fact that in the Fourier’s representation ofthe function (1) the frequencies 1 − j/n are bounded by 1, but the limit function e iax hasa frequency a that can be arbitrarily larger than 1. A precise definition of superoscillatingfunctions is as follows.We call generalized Fourier sequence a sequence of the form f n ( x ) := n X j =0 X j ( n, a ) e ih j ( n ) x , j = 0 , ..., n, n ∈ N , (3)where a ∈ R , X j ( n, a ) and h j ( n ) are complex and real valued functions of the variables n, a and n , respectively. A generalized Fourier sequence of the form (3) is said to be a superoscillatingsequence if sup j,n | k j ( n ) | ≤ R , which will be called asuperoscillation set , on which f n ( x ) converges uniformly to e ig ( a ) x , where g is a continuous realvalued function such that | g ( a ) | > Y n ( x, a ) = n X j =0 C j ( n, a ) e ig (1 − j/n ) x , (4)where C j ( n, a ) are the coefficients in (2), g are given entire functions, monotone increasing in a ,and x ∈ R . We have shown that lim n →∞ Y n ( x, a ) = e ig ( a ) x under suitable conditions on g and the simplest, but important, example is Y n ( x, a ) = n X j =0 C j ( n, a ) e i (1 − j/n ) m x , for fixed m ∈ N . g .In this paper we further enlarge the class of superoscillating functions enlarging both theclass of the coefficients C j ( n, a ) and of the sequence of frequencies 1 − j/n that are bounded by1. A large class of superoscillating functions can be determined solving the following problem. Problem 1.1.
Let h j ( n ) be a given set of points in [ − , , j = 0 , , ..., n , for n ∈ N and let a ∈ R be such that | a | > . Determine the coefficients X j ( n ) of the sequence f n ( x ) = n X j =0 X j ( n ) e ih j ( n ) x , x ∈ R in such a way that f ( p ) n (0) = ( ia ) p , for p = 0 , , ..., n. Remark 1.2.
The conditions f ( p ) n (0) = ( ia ) p mean that the functions x e iax and f n ( x ) havethe same derivatives at the origin, for p = 0 , , ..., n , so they have the same Taylor polynomialof order n . Under the condition that the points h j ( n ) for j = 0 , ..., n , (often denoted by h j ) are distinctwe obtain an explicit formula for the coefficients X j ( n, a ) given by X j ( n, a ) = n Y k =0 , k = j (cid:16) h k ( n ) − ah k ( n ) − h j ( n ) (cid:17) , so the superoscillating sequence f n ( x ), that solves Problem 1.1, takes the explicit form f n ( x ) = n X j =0 n Y k =0 , k = j (cid:16) h k ( n ) − ah k ( n ) − h j ( n ) (cid:17) e ih j ( n ) x , x ∈ R , as shown in Theorem 2.2. Observe that, by construction, this function is band limited and itconverges to e iax with arbitrary | a | >
1, so it is superoscillating.Observe that different sequences X j ( n ) can be explicitly computed when we fix the points h j ( n ). See, for example, the case of the sequence h j ( n ) = 1 − j/n p for j = 0 , ..., n, n ∈ N and for fixed p ∈ N , in Section 3.We consider now the frequencies h j ( n ) = (1 − j/n ) m , for fixed m ∈ N , to explain somefacts.(I) If we consider the sequence (4), with coefficients C j ( n, a ) given by (2), we obtainlim n →∞ n X j =0 C j ( n, a ) e i (1 − j/n ) m x = e ia m x , for fixed m ∈ N . Note that, in this case, we could have used the frequencies h j ( n ) = (1 − j/n ) and the coefficients˜ C j ( n, a ) := C j ( n, a m ) to get as limit function e ia m x . Thus the same limit function e ia m x can beobtained by tuning the frequencies and the coefficients.3II) By solving Problem 1.1 with the frequences h j ( n ) = (1 − j/n ) m , we can determine thecoefficients X j ( n ) = X j ( n, a ) such that we obtain as limit function e iax , namelylim n →∞ n X j =0 X j ( n, a ) e i (1 − j/n ) m x = e iax , for fixed m ∈ N . (5)Changing the coefficients ˜ X j ( n, a ) we can get, as limit function, e ia m x .(III) The coefficients X j and C j in the procedures (I) and (II) are different form each otherbecause the two methods to generate superoscillations are different as explained in Section 3.In section 4 we will also discuss how to generalize this method to obtain analogous resultsin the case of the supershift property of functions, a mathematical concept that generalizes thenotion of superoscillating function. In this section we show the main procedure to determine the coefficients X j ( n ) and so to con-struct explicitly the superoscillating functions solving Problem 1.1. Theorem 2.1 (Existence and uniqueness of the solution of Problem 1.1) . Let h j ( n ) be a given setof points in [ − , , j = 0 , , ..., n for n ∈ N and let a ∈ R be such that | a | > . If h j ( n ) = h i ( n ) ,for every i = j , then there exists a unique solution X j ( n ) of the linear system f ( p ) n (0) = ( ia ) p , for p = 0 , , ..., n, in Problem 1.1.Proof. For the sake of simplicity, we denote h j ( n ) by h j . Observe that the derivatives of order p of f n ( x ) are f ( p ) n ( x ) = n X j =0 X j ( n )( ih j ) p e ih j x , x ∈ R , so if we require that these derivatives are equal to the derivatives of order p for p = 0 , , ..., n ofthe function x e iax at the origin we obtain the linear system n X j =0 X j ( n )( ih j ) p = ( ia ) p , p = 0 , , ..., n (6)from which we deduce n X j =0 X j ( h j ) p = a p , p = 0 , , ..., n, (7)where we have written X j instead of X j ( n ). Now we write explicitly the linear system (7) of( n + 1) equations and ( n + 1) unknowns ( X , ..., X n ) X + X + . . . + X n = 1 X h + X h + . . . + X n h n = a. . .X h n + X h n + . . . + X n h nn = a n H ( n ) X = B ( a ) (8)where H is the ( n + 1) × ( n + 1) matrix H ( n ) = . . . h h . . . h n . . . . . . . . . . . .h n h n . . . h nn (9)and X = X X . . .X n and B ( a ) = a. . .a n . (10)Observe that the determinant of H is the Vandermonde determinant, so it is given bydet( H ( n )) = Y ≤ i
In Theorem 2.1 we proved that, if h j = h i for every i = j , there exists a unique solutionof the system (8). The solution is given by X j ( n, a ) = det( H j ( n, a ))det( H ( n )) (12)for H j ( n, a ) = . . . . . . h h . . . a . . . h n . . . . . . . . . . . . . . .h n h n . . . a n . . . h nn (13)where the j th -column contains a and its powers. The explicit form of the determinant of thematrix H is given by: det ( H ( n )) = ( h − h ) · ( h − h )( h − h ) · ( h − h )( h − h )( h − h ) · ( h − h )( h − h )( h − h )( h − h ) · . . . · ( h n − h )( h n − h )( h n − h )( h n − h ) ..... ( h n − h n − ) . H j ( n, a ) is still of Vandermonde type and its determinant can be computedsimilarly. So we have that the solution ( X ( n, a ) , . . . , X n ( n, a )) is such that X ( n, a ) = ( h − a ) · ( h − a ) · ( h − a ) · ( h − a ) · . . . · ( h n − a )( h − h ) · ( h − h ) · ( h − h ) · ( h − h ) · . . . · ( h n − h )= Q nk =0 , k =0 ( h k − a ) Q nk =0 , k =0 ( h k − h ) ,X ( n, a ) = ( a − h ) · ( h − a ) · ( h − a ) · ( h − a ) · . . . · ( h n − a )( h − h ) · ( h − h ) · ( h − h ) · ( h − h ) · . . . · ( h n − h )= Q nk =0 , k =1 ( h k − a ) Q nk =0 , k =1 ( h k − h ) , and so on, up to X n ( n, a ) = 1 · · · · . . . · ( a − h )( a − h )( a − h )( a − h ) ..... ( a − h n − )1 · · · · . . . · ( h n − h )( h n − h )( h n − h )( h n − h ) ..... ( h n − h n − )= Q nk =0 , k = n ( h k − a ) Q nk =0 , k = n ( h k − h n ) . So we get the statement with the recursive formula.
Below we compare the superoscillating functions obtained by solving Problem 1.1 and the super-oscillating functions obtained via the sequence F n ( x, a ) and infinite order differential operators.For different methods see also [30].(I) Observe that the limit lim n →∞ (cid:16) cos (cid:16) xn (cid:17) + ia sin (cid:16) xn (cid:17)(cid:17) n = e iax is a direct consequence of lim n →∞ (cid:16) ia xn (cid:17) n = e iax , while the construction method to generate superoscillations in Theorem 2.2 has a different naturebecause we require that the linear system (6) in the n + 1 unknowns X j ( n ) are determined insuch a way that n X j =0 X j ( n )( ih j ) p = ( ia ) p , p = 0 , , ..., n (14)so the derivatives f ( p ) n ( x ) = n X j =0 X j ( n )( ih j ) p e ih j ( n ) x , x ∈ R , at x = 0, are equal to the derivatives of the exponential function e iax at the origin. This meansthat the sequence of functions f n ( x ) = n X j =0 X j ( n ) e ih j ( n ) x , x ∈ R n derivatives equal to the derivatives of exponential function e iax at the origin so the limitlim n →∞ f n ( x ) = e iax follows by construction of the f n ( x ).(II) In the definition of the superoscillating function (1) the derivatives are given by F ( p ) n ( x, a ) = n X j =0 C j ( n, a ) (cid:16) i (1 − j/n ) (cid:17) p e i (1 − j/n ) x , x ∈ R and it is just in the limit that we get the derivatives of order p of the exponential function e iax at the origin, namely we havelim n →∞ n X j =0 C j ( n, a ) (cid:16) i (1 − j/n ) (cid:17) p = ( ia ) p , p ∈ N . (III) With the new procedure proposed in this paper we impose the conditions f ( p ) n (0) = ( ia ) p , p = 0 , , , ..., n (where we have genuine equalities, not in the limit) and we link n with the derivatives of f n ( x )in order to determine the coefficients X j ( n ) in (3), so we have that the Taylor polynomials ofthe two functions f n ( x ) and e iax are the same up to order n , i.e., e iax = 1 + iax + ( iax )
2! + ... + ( iax ) n n ! + R n ( x ) , so we get f n ( x ) = n X j =0 X j ( n ) e ih j x = f n (0) + f (1) n (0) x + f (2) n (0) x
2! + ... + f ( n ) n (0) x n n ! + ˜ R n ( x ) , x ∈ R = 1 + iax + ( ia ) x
2! + ... + ( ia ) n x n n ! + ˜ R n ( x ) , x ∈ R , where R n ( x ) and ˜ R n ( x ) are the errors.It is now easy to generate a very large class of superoscillatory functions. We write a fewexamples to further clarify the generality of our new construction of superoscillating sequences:given the sequence h j ( n ) we determine the coefficients accordingly for the f n ( x ) = n X j =0 n Y k =0 , k = j (cid:16) h k ( n ) − ah k ( n ) − h j ( n ) (cid:17) e ih j ( n ) x , x ∈ R . Example 3.1.
Let n ∈ N and set h j ( n ) = 1 − n j where j = 0 , ..., n . We have h k ( n ) − a = 1 − n k − a nd h k ( n ) − h j ( n ) = 1 − n k − (cid:16) − n j (cid:17) = 2 n (cid:16) j − k (cid:17) . Thus, we obtain f n ( x ) = n X j =0 n Y k =0 , k = j n (cid:16) − n k − aj − k (cid:17) e i (1 − n j ) x , x ∈ R . Example 3.2.
Let n ∈ N , and set h j ( n ) = 1 − n p j where j = 0 , ..., n , for a fixed p ∈ N . We have h k ( n ) − a = 1 − n p k − a, and h k ( n ) − h j ( n ) = 1 − n p k − (cid:16) − n p j (cid:17) = 2 n p (cid:16) j − k (cid:17) . So, we obtain: f n ( x ) = n X j =0 n Y k =0 , k = j n p (cid:16) − n p k − aj − k (cid:17) e i (1 − np j ) x , x ∈ R . Example 3.3.
Let n ∈ N , and set h j ( n ) = 1 − (cid:16) jn (cid:17) p where j = 0 , ..., n, for a fixed p ∈ N . We have h k ( n ) − a = 1 − (cid:16) kn (cid:17) p − a, and h k ( n ) − h j ( n ) = 1 − (cid:16) kn (cid:17) p − (cid:16) − (cid:16) jn (cid:17) p (cid:17) = 2 p n p (cid:16) j p − k p (cid:17) . So, we obtain f n ( x ) = n X j =0 n Y k =0 , k = j n p p (cid:16) − kn − aj p − k p (cid:17) e i (1 − (2 j/n ) p ) x , x ∈ R . The procedure to define superoscillatory functions can be extended to supershift. We recall thatthe supershift property of a function extends the notion of superoscillations and it turned out tobe the crucial concept behind the study of the evolution of superoscillatory functions as initialconditions of Schr¨odinger equation or of any other field equation. We recall the definition beforeto state our result. 8 efinition 4.1 (Supershift) . Let
I ⊆ R be an interval with [ − , ⊂ I and let ϕ : I × R → R ,be a continuous function on I . We set ϕ λ ( x ) := ϕ ( λ, x ) , λ ∈ I , x ∈ R and we consider a sequence of points ( λ j,n ) such that ( λ j,n ) ∈ [ − ,
1] for j = 0 , ..., n and n = 0 , . . . , + ∞ . We define the functions ψ n ( x ) = n X j =0 c j ( n ) ϕ λ j,n ( x ) , (15) where ( c j ( n )) is a sequence of complex numbers for j = 0 , ..., n and n = 0 , . . . , + ∞ . If lim n →∞ ψ n ( x ) = ϕ a ( x ) for some a ∈ I with | a | > , we say that the function λ → ϕ λ ( x ) ,for x fixed, admits a supershift in λ . Remark 4.2.
We observe that the definition of supershift of a function given above is not themost general one, but it is useful to explain our new procedure for the supershift case. In thefollowing, we will take the interval I , in the definition of the supershift, to be equal to R . Remark 4.3.
Let us stress that the term supershift comes from the fact that the interval I canbe arbitrarily large (it can also be R ) and so also the constant a can be arbitrarily far away fromthe interval [ − , where the function ϕ λ j,n ( · ) is computed, see (15) . Remark 4.4.
Superoscillations are a particular case of supershift. In fact, for the sequence ( F n ) in (1) , we have λ j,n = 1 − j/n , ϕ λ j,n ( t, x ) = e i (1 − j/n ) x and c j ( n ) are the coefficients C j ( n, a ) defined in (2). Problem 1.1, for the supershift case, is formulated as follows.
Problem 4.5.
Let h j ( n ) be a given set of points in [ − , , j = 0 , , ..., n , for n ∈ N and let a ∈ R be such that | a | > . Suppose that for every x ∈ R the function λ G ( λx ) is holomorphicand entire in λ . Consider the function f n ( x ) = n X j =0 Y j ( n ) G ( h j ( n ) x ) , x ∈ R where λ G ( λx ) depends on the parameter x ∈ R . Determine the coefficients Y j ( n ) in such away that f ( p ) n (0) = ( a ) p G ( p ) (0) f or p = 0 , , ..., n. (16)The solution of the above problem is obtained in the following theorem. Theorem 4.6.
Let h j ( n ) be a given set of points in [ − , , j = 0 , , ..., n for n ∈ N and let a ∈ R be such that | a | > . If h j ( n ) = h i ( n ) for every i = j and G ( p ) (0) = 0 for all p = 0 , , ..., n ,then there exists a unique solution Y j ( n, a ) of the linear system (16) and it is given by Y j ( n, a ) = n Y k =0 , k = j (cid:16) h k ( n ) − ah k ( n ) − h j ( n ) (cid:17) , o that f n ( x ) = n X j =0 n Y k =0 , k = j (cid:16) h k ( n ) − ah k ( n ) − h j ( n ) (cid:17) G ( h j ( n ) x ) , x ∈ R and, by construction, it is lim n →∞ f n ( x ) = G ( ax ) , x ∈ R . Proof.
Observe that we have f ( p ) n ( x ) = n X j =0 Y j ( n )( h j ( n )) p G ( p ) ( h j ( n ) x ) , x ∈ R where G ( p ) are the derivatives of order p , for p = 0 , ..., n with respect to x of the function G ( λx ),for λ ∈ R considered as a parameter. So we get the system f ( p ) n (0) = n X j =0 Y j ( n )( h j ( n )) p G ( p ) (0) = a p G ( p ) (0) . Now, since we have assumed that G ( p ) (0) = 0 for all p = 0 , , ..., n , the system becomes n X j =0 Y j ( n )( h j ( n )) p = a p and we can solve it as in Theorem 2.2 to get Y j ( n, a ) = n Y k =0 , k = j (cid:16) h k ( n ) − ah k ( n ) − h j ( n ) (cid:17) . Finally, we get f n ( x ) = n X j =0 n Y k =0 , k = j (cid:16) h k ( n ) − ah k ( n ) − h j ( n ) (cid:17) G ( h j ( n ) x ) , x ∈ R and by construction it is lim n →∞ f n ( x ) = G ( ax ) , x ∈ R . References [1] D. Alpay, F. Colombo, I. Sabadini, D.C. Struppa,
Aharonov-Berry superoscillations in theradial harmonic oscillator potential , Quantum Stud. Math. Found., (2020), 269–283.[2] Y. Aharonov, D. Albert, L. Vaidman, How the result of a measurement of a componentof the spin of a spin-1/2 particle can turn out to be 100 , Phys. Rev. Lett., (1988),1351-1354.[3] Y. Aharonov, J. Behrndt, F. Colombo, P. Schlosser, Schr¨odinger evolution of superoscilla-tions with δ - and δ ′ -potentials , Quantum Stud. Math. Found., (2020), 293–305.104] Y. Aharonov, J. Behrndt, F. Colombo, P. Schlosser, Green’s Function for the Schr¨odingerEquation with a Generalized Point Interaction and Stability of Superoscillations , J. Differ-ential Equations, (2021), 153–190.[5] Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen,
Evolution of superoscil-lations in the Klein–Gordon field , Milan J. Math., (2020), no. 1, 171–189.[6] Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, How superoscillatingtunneling waves can overcome the step potential , Ann. Physics, (2020), 168088, 19 pp.[7] Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen,
On the Cauchy problemfor the Schr¨odinger equation with superoscillatory initial data , J. Math. Pures Appl., (2013), 165–173.[8] Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, Superoscillating se-quences in several variables , J. Fourier Anal. Appl., (2016), 751–767.[9] Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, The mathematics ofsuperoscillations , Mem. Amer. Math. Soc., (2017), no. 1174, v+107 pp.[10] Y. Aharonov, F. Colombo, D.C. Struppa, J. Tollaksen,
Schr¨odinger evolution of superoscil-lations under different potentials , Quantum Stud. Math. Found., (2018), 485–504.[11] Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed , Wiley-VCH Verlag, Weinheim, 2005.[12] Y. Aharonov, I. Sabadini, J. Tollaksen, A. Yger,
Classes of superoscillating functions , Quan-tum Stud. Math. Found., (2018), 439–454.[13] Y. Aharonov, T. Shushi, A new class of superoscillatory functions based on a generalizedpolar coordinate system , Quantum Stud. Math. Found., (2020), 307–313.[14] T. Aoki, F. Colombo, I. Sabadini, D. C. Struppa, Continuity of some operators arising inthe theory of superoscillations , Quantum Stud. Math. Found., (2018), 463–476.[15] T. Aoki, F. Colombo, I. Sabadini, D.C. Struppa, Continuity theorems for a class of convo-lution operators and applications to superoscillations , Ann. Mat. Pura Appl., (2018),1533–1545.[16] J. Behrndt, F. Colombo, P. Schlosser,
Evolution of Aharonov–Berry superoscillations inDirac δ -potential , Quantum Stud. Math. Found., (2019), 279–293.[17] M. Berry et al, Roadmap on superoscillations , 2019, Journal of Optics 21 053002.[18] M. V. Berry,
Faster than Fourier , in Quantum Coherence and Reality; in celebration ofthe 60th Birthday of Yakir Aharonov ed. J.S.Anandan and J. L. Safko, World Scientific,Singapore, (1994), pp. 55-65.[19] M. Berry,
Exact nonparaxial transmission of subwavelength detail using superoscillations ,J. Phys. A , (2013), 205203.[20] M. V. Berry, Representing superoscillations and narrow Gaussians with elementary func-tions , Milan J. Math., (2016), 217–230.1121] M. V. Berry, S. Popescu, Evolution of quantum superoscillations, and optical superresolutionwithout evanescent waves , J. Phys. A, (2006), 6965–6977.[22] M.V. Berry, P. Shukla, Pointer supershifts and superoscillations in weak measurements , J.Phys A , (2012), 015301.[23] F. Colombo, J. Gantner, D.C. Struppa, Evolution by Schr¨odinger equation of Aharonov-Berry superoscillations in centrifugal potential , Proc. A., (2019), no. 2225, 20180390,17 pp.[24] F. Colombo, I. Sabadini, D.C. Struppa, A. Yger,
Gauss sums, superoscilla-tions and the Talbot carpet , Journal de Math´ematiques Pures et Appliqu´ees,https://doi.org/10.1016/j.matpur.2020.07.011.[25] F. Colombo, I. Sabadini, D.C. Struppa, A. Yger,
Superoscillating sequences and hyperfunc-tions , Publ. Res. Inst. Math. Sci., (2019), no. 4, 665–688.[26] F. Colombo, D.C. Struppa, A. Yger, Superoscillating sequences towards approximation in S or S ′ -type spaces and extrapolation , J. Fourier Anal. Appl., (2019), no. 1, 242–266.[27] F. Colombo, G. Valente, Evolution of Superoscillations in the Dirac Field , Found. Phys., (2020), 1356–1375.[28] P. J. S. G. Ferreira, A. Kempf, Unusual properties of superoscillating particles , J. Phys. A, (2004), 12067-76.[29] P. J. S. G. Ferreira, A. Kempf, Superoscillations: faster than the Nyquist rate , IEEE Trans.Signal Processing, (2006), 3732–3740.[30] P. J. S. G. Ferreira, A. Kempf, M. J. C. S. Reis, Construction of Aharonov-Berry’s super-oscillations , J. Phys. A (2007), 5141–5147.[31] A. Kempf, Four aspects of superoscillations , Quantum Stud. Math. Found., (2018), 477–484.[32] J. Lindberg, Mathematical concepts of optical superresolution , Journal of Optics, (2012),083001.[33] B. ˇSoda, A. Kempf, Efficient method to create superoscillations with generic target behavior ,Quantum Stud. Math. Found., (2020), no. 3, 347–353.[34] G. Toraldo di Francia, Super-Gain Antennas and Optical Resolving Power , Nuovo CimentoSuppl.,9