Abstract
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.
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1 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. 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 \(\mathcal {S}\), in the space of distributions of type \(\mathcal {S}'\), 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 rigorous treatment of the evolution of superoscillations, as initial datum of the Schrödinger equation, required some sophisticated mathematical tools, as it has been shown for example in [2,3,4, 6,7,8, 18, 24,25,26] and in [11]. We also mention that evolution of superoscillations in discontinuous potentials is considered in [3, 9, 19], while the evolution of superoscillations for Klein–Gordon and Dirac equations have been studied in [12] and [32], respectively.
The most important superoscillatory function, that appears in connection with weak values in quantum mechanics, is
where \(a>1\) and the coefficients \(C_j(n,a)\) are given by
If we fix \(x \in \mathbb {R}\) and we let n go to infinity, we obtain that \( \lim _{n \rightarrow \infty } F_n(x,a)=e^{iax}. \)
Inspired by (1) we define a more general class of superoscillatory functions.
We call generalized Fourier sequence a sequence of the form
where \(a\in \mathbb {R}\), \(k_j(n)\in \mathbb {R}\) and \(E_j(n,a)\in \mathbb {C}\) for all \(j=0,...,n\) and \(n\in \mathbb {N}\).
A generalized Fourier sequence \(Y_n(x,a)\) is said to be a superoscillating sequence if it satisfy the two properties:
(P1) \(\sup _{j,n}|k_j(n)|\le 1\) for all \(j=0,...,n\) and \(n\in \mathbb {N}\).
(P2) there exists a compact subset of \(\mathbb R\), which will be called a superoscillation set, on which \( \lim _{n\rightarrow \infty }Y_n(x,a)=e^{ig(a)x} \) 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 \(\lambda \rightarrow \varphi _\lambda (x)\in \mathbb {C}\) be a continuous function in the variables \(\lambda \in \mathcal {I}\), where \(\mathcal {I}\subseteq \mathbb {R}\) is an interval, and \(x\in \Omega \), where \(\Omega \) is a domain. We consider \(x\in \Omega \) as parameter for the function \(\lambda \rightarrow \varphi _\lambda (x)\) where \(\lambda \in \mathcal {I}\). When \([-1,1]\) is contained in \(\mathcal {I}\) and \(a\in \mathcal {I}\), we define the sequence
in which \(\varphi _{\lambda }\) is computed just on the points \(1-2j/n\) 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 \(n\in \mathbb {N}\). If
for \(|a|>1\) arbitrary large (but belonging to \(\mathcal {I}\)), we say that the function \(\lambda \rightarrow \varphi _{\lambda }(x)\), for x fixed, admits a supershift in \(\lambda \).
If we set \(\varphi _\lambda (x)=e^{i\lambda x}\), we obtain the superoscillating sequence described above as a particular case of the supershift. In fact, in this case, we have \(\psi _n(x)= F_n(x,a)\), where \(F_n(x,a)\) is defined in (1). The name supershift is due to the fact that we are able to obtain \(\varphi _{a},\) for \(|a|>1\) arbitrarily large, by simply calculating the function \(\lambda \rightarrow \varphi _{\lambda }\) 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 \(\ell _j(n)\) and \(k_j(n)\)), denoted by \(\circ _P\), as
In particular, we can set \(\ell _j(n):=g(1-2j/n)\) and \( k_j(n):=h(1-2j/n)\) 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 \(|g(z)|\le Ae^{B|z|^p}\) and \(|h(z)|\le Ae^{B|z|^p}\) for some \(p\in [1,\infty )\). 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 \(Y_n\circ _PZ_n(x,a)\), see Theorem 2.7. When g and h are such that \(g([-1,1])\subseteq \mathbb {R}\), \(h([-1,1])\subseteq \mathbb {R}\), with the assumption that for \(a\in \mathbb {R}\) we have \(|g(a)h(a)|<1\) for \(|a|\le 1\) and \(|g(a)h(a)|>1\) for \(|a|>1\), then the sequence \(F_n\circ _PY_n(x,a)\) 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 \(p,q\in \mathbb {N}\).
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 \(|u\oplus _Rv|< 1\), 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 \(\ell _j(n)\) and \(k_j(n)\) are given by \(\ell _j(n)=g(\varepsilon (1-2j/n))\) and \(k_j(n)=h(\varepsilon (1-2j/n))\), for \(\varepsilon \in (0,1)\), where g and h are holomorphic functions defined on the unit ball \(\mathbb {D}\) in \(\mathbb {C}\) that satisfy suitable conditions. Then we prove that
for any \(\varepsilon a < 1\) 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 \(\mathcal {U}(x,\partial _\xi )\) that are applied to the fundamental superoscillating function (1). We then define
where \(|_{\xi =0}\) denotes the restriction to \(\xi =0\) with respect to the auxiliary complex variable \(\xi \). A crucial fact in the strategy of generating superoscillating functions is the study of the continuity of the operator \(\mathcal {U}(x,\partial _\xi )\) 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
where
and the coefficients of the \((a_p)_{p\in \mathbb {N}_0}\) and \((b_p)_{p\in \mathbb {N}_0}\) in the formal operators \(A(\partial _\xi )\) and \(B(\partial _\xi )\) 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.
2 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 \(p\ge 1\). We denote by \(\mathcal {A}_p\) 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
Let \((f_n)_{n\in \mathbb {N}}\), \(f_0\in \mathcal {A}_p\). Then \(f_n \rightarrow f_0\) in \(\mathcal {A}_p\) if there exists some \(B > 0\) such that
The following result gives a characterization of functions in \(\mathcal {A}_p\) in terms of their Taylor coefficients.
Lemma 2.2
(Lemma 2.2 in [18]) Let \(p\ge 1\). A function \(f(z)=\sum _{j=0}^\infty f_jz^j\) belongs to \(\mathcal {A}_p\), i.e., there exist positive constants \(B,\, C>0\) such that \(|f(z)|\le Ce^{B|z|^p}\), if and only if there exist constants \(C_f,b_f>0\) such that
where the constant \(b_f\) is given by
Furthermore, a sequence \(f_n\) in \(\mathcal {A}_p\) tends to zero if and only if \(C_{f_n}\rightarrow 0\) and \(b_{f_n} < b\) for some \(b>0\).
Remark 2.3
The constant \(C>0\) in the estimate (6), i.e., \( |f(z)|\le C e^{B|z|^p}, \) and the constant \(C_f>0\) in estimate (8), i.e., \( |f_j|\le C_f \frac{b_f^j}{\Gamma (\frac{j}{p}+1)}, \) 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., \( b_f=(2^pBp e)^{1/p}, \) where e is the Neper number.
The holomorphic extension of \(F_n(x,a)\) as in (1) converges in \(\mathcal {A}_1\) as the following lemma, proved in [28], shows.
Lemma 2.4
Let \(a\in \mathbb {C}\), for any \(z\in \mathbb {C}\), consider
with \(z, a\in \mathbb {C}\). Then, for every \(n\in \mathbb {N}\) and \(z\in \mathbb {C}\), one has
Now we will consider generalized Fourier sequences \((\sum _{j=0}^n C_j(n,a)e^{ik_j(n)x})_n\) in which the exponential sequences \((k_j(n))_{j,n}\) are defined by holomorphic functions in \(\mathcal A_q\) for some \(q\ge 1\).
Definition 2.5
(Generalized Fourier sequences depending on an entire function) Let \(a\in \mathbb {R}\) and \(C_j(n,a)\in \mathbb {C}\) be as in (2) for all \(j=0,...,n\) and \(n\in \mathbb {N}\). Let \(h \in \mathcal A_q\), for some \(q\ge 1\), be such that h maps \(\mathbb {R}\) to itself, we define the generalized Fourier series
where \(k_j(n):=h(1-2j/n)\) for all \(j=0,\dots , n\) and \(n\in \mathbb N\).
Definition 2.6
(The product \(\circ _P\) 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 \(C_j(n,a)\in \mathbb {C}\) be as in (2) for all \(j=0,...,n\) and \(n\in \mathbb {N}\). Let \(h,\, g\in \mathcal A_q\), for some \(q\ge 1\), be such that h and g map \(\mathbb {R}\) to itself, \(\ell _j(n):=h(1-2j/n)\) and \(k_j(n):=g(1-2j/n)\). We define the product \(\circ _P\) of the sequences \(Y_n(x,a)\) and \(Z_n(x,a)\) by
In order to compute the limit \(\lim _{n\rightarrow \infty }Y_n\circ _PZ_n(x,a)\) we need to define a suitable infinite order differential operator acting on \(\mathcal {A}_1\) and to prove the continuity of such operator on \(\mathcal {A}_1\). Thanks to the assumptions on the functions \(h,\, g\in \mathcal A_q\), for some \(q\ge 1\), we will prove the supershift property for function \(Y_n\circ _PZ_n(x,a)\) and if h and g satisfy additional conditions, then the sequence \(Y_n\circ _PZ_n(x,a)\) is, in particular, superoscillating.
Theorem 2.7
Assume that x is a fixed real parameter and let \(\xi \in \mathbb C\). We define the formal infinite order differential operator:
where the operator \(A(\partial _\xi )\) is defined replacing z by \(i^{-1}\partial _\xi \) in the entire function
i.e.,
Then \(\mathcal {U}(x,\partial _\xi ): \mathcal A_1 \rightarrow \mathcal A_1\) is a continuous linear operator. Moreover, we have
Proof
First we consider the action of the operators \(A^m(\partial _\xi )\), \(m\in \mathbb {N}\), on functions \(f\in \mathcal A_1\), \(f(\xi )=\sum _{j=0}^{\infty } f_j\xi ^j\), \(f_j\in \mathbb C\). We claim that
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 \(k=j-p_1-\dots -p_n\) 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 \(G\in \mathcal A_q\), for \(q\ge 1\), by Lemma 2.2, there exist positive constants \(b_G\), \(C_{G}\) such that
and in particular if \(f\in \mathcal A_1\), \(f(z)=\sum _{j=0}^{\infty } f_jz^j\), then \( |f_j|\le C_f\frac{b_f^j}{j!}\) for some positive constants \(b_f\), \(C_f\). Using these estimates we have
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 \(\mathcal {U}(x,\partial _{\xi }) f(\xi )\in \mathcal A_1\) and also that \(\mathcal {U}(x,\partial _{\xi })\) is continuous over \(\mathcal A_1\), i.e,. \(\mathcal {U}(x,\partial _{\xi }) f_n(\xi )\rightarrow 0\) as \(f_n\rightarrow 0\). Observe that instead of \(f_n\rightarrow 0\) we can equivalently require that \(C_{f_n}\rightarrow 0\) with \(b_{f_n}\le b_0\), for some positive constant \(b_0\), where the constants \(C_{f_n}\) and \(b_{f_n}\) refer to \( |f_{n,j}|\le C_{f_n}\frac{b_{f_n}^j}{j!}\). Moreover, we have that
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 \(h,\, g\in \mathcal A_q\), for some \(q\ge 1\), be such that h and g map \(\mathbb {R}\) to itself. Let \(Y_n(x,a)\) and \(Z_n(x,a)\), for \(x\in \mathbb {R}\) and \(|a|>1\), be as in the Definition 2.6. Then we have
Moreover, if \(|g(a)h(a)|<1\) for \(|a|\le 1\) and \(|g(a)h(a)|>1\) for \(|a|>1\) then the sequence \((Y_n\circ _PZ_n)(x,a)\) 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 \(\mathbb {R}\) to itself.
Proof
Let \(h,\, g\in \mathcal A_q\) for some \(q\ge 1\). We consider the function
Since \(h,g\in \mathcal A_q\) the function \(G(z):= h(z) g(z)\) belongs to \(\mathcal A_q\). We define the operator \(\mathcal {U}(x,\partial _\xi )\) as in Theorem 2.7 and thus we obtain the relations
and
Thus we have
Remark 2.10
A simple example for the superoscillatory case is \(g(z)=z^{l}\) and \(h(z)=z^{s}\) for \(l,\, s\in \mathbb {N}\).
3 The relativistic sum and superoscillating functions
In the previous section we have considered generalized Fourier sequences as in Definition 2.6, where the product \(\circ _P\) involves the pointwise product of the coefficients \(\ell _j(n)\) and \(k_j(n)\) of the exponentials in \(Y_n(x,a)\) and \(Z_n(x,a)\), respectively. Using the \(\circ _P\) product it is clear that, if we are in the superoscillating case where \(|\ell _j(n)|< 1\) and \(|k_j(n)|< 1\), then we have \(|\ell _j(n)k_j(n)|< 1\). But if we consider the sum \(\ell _j(n)+k_j(n)\) instead of the product \(\ell _j(n)k_j(n)\) we cannot guarantee that \(|\ell _j(n)+k_j(n)|< 1\). 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 \(\mathbb {D}\) the open unit disc in \(\mathbb {C}\). An important point in our considerations is the following result.
Lemma 3.1
Let g, \(h:\mathbb {D}\rightarrow \mathbb {D}\) be two holomorphic functions.
(I) The relativistic sum of g and h
is a holomorphic function on \(\mathbb {D}\).
(II) If u, \(v\in (-1,1)\) then \(|u\oplus _Rv|< 1\).
Proof
Point (I) follows by observing that \(|g(z)h(z)|<1\). Point (II) can be deduced from the observation that \(u\oplus _R v\) is the restriction to the square \((-1,1)\times (-1,1)\) of the Möbius map
that sends \(\mathbb D\times \mathbb D\) to \(\mathbb D\) where \(z_1=u+il\) and \(z_2=v+is\).
Remark 3.2
In general, the function \(g\oplus _R h\) defined in the previous lemma, does not map \(\mathbb D\) in \(\mathbb D\). For example we can consider \(h(z)=g(z)\equiv \frac{1}{2} i\) for any \(z\in \mathbb D\). In this case we have
Definition 3.3
(The relativistic product \(\circ _R\) of generalized Fourier sequences) Let us consider two generalized Fourier sequences \(Y_n(x,a)\) and \(Z_n(x,a)\) defined by
with \(\ell _j(n):=g(1-2j/n)\) and \(k_j(n):=h(1-2j/n)\) for h, \(g:\mathbb {D}\rightarrow \mathbb {D}\). Moreover, we assume that h and g satisfy the conditions \(h((-1,1))\subset (-1,1)\) and \(g((-1,1))\subset (-1,1)\). The relativistic product of \(Y_n(x,a)\) and \(Z_n(x,a)\), denoted by \(\circ _R\), is defined as
where \(\oplus _R \) 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 \((-1,1)\).
Definition 3.5
(The operator associated with the relativistic sum) Let g and h be holomorphic functions form \(\mathbb D\) to \(\mathbb D\). Let A and B be holomorphic functions from \(\mathbb D\) to \(\mathbb {C}\) defined by
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 \(x\in \mathbb {R}\) is a parameter and \(\xi \in \mathbb {C}\).
Remark 3.6
The operator \(C(\partial _\xi )\), introduced in the Definition 3.5, is obtained by the formal substitution of z with \(\partial _\xi \) in the relativistic sum of g and h. For we observe that
The natural space of holomorphic functions on which the operator \(\mathcal {U}_R(x,\partial _\xi )\) acts continuously is defined as follows.
Definition 3.7
Let \(\alpha \) be a fixed positive number. We define the class \(\mathcal {A}_{1,\alpha }\) to be the set of entire functions such that there exists \(C>0\) for which
Let \((f_n)_{n\in \mathbb {N}}\), \(f_0\in \mathcal {A}_{1,\alpha }\). Then \(f_n \rightarrow f_0\) in \(\mathcal {A}_{1,\alpha }\) if
We are now ready to prove our main result on the continuity of the infinite order differential operator \(\mathcal {U}_R(x,\partial _\xi )\) associated with the relativistic sum.
Remark 3.8
In the Theorem 3.9 below we will study the continuity of the operator \(\mathcal {U}_R(x,\partial _\xi )\) defined in the domains \(\mathcal A_{1,\alpha }\) for \(\alpha >0\). In the proof of Theorem 3.11 will be sufficient to consider the continuity of the operator \(\mathcal {U}_R(x,\partial _\xi )\) defined in \(\mathcal A_{1,1}\).
Theorem 3.9
Let \(\alpha >0\) be fixed. Let \(g,h:\mathbb D\rightarrow \mathbb D\) be holomorphic functions, let \((a_p)_{p\in \mathbb {N}_0}\), \((b_q)_{q\in \mathbb {N}_0}\) be the sequences of complex numbers as in (18) and such that the conditions
hold. Then, the linear operator \(\mathcal {U}_R(x,\partial _\xi )\), for \(x\in \mathbb {R}\), defined in (21), acts continuously from \(\mathcal {A}_{1,\alpha }\) to \(\mathcal {A}_1\). Moreover, we have
for \(|\lambda |<\alpha \).
Proof
The proof is split in steps.
STEP 1. We study the action of the operator \(C(\partial _\xi )=B(\partial _\xi ) \sum _{\ell =0}^\infty (-1)^\ell A^\ell (\partial _\xi )\) on holomorphic functions belonging to \(\mathcal {A}_{1,\alpha }\).
We first consider the operator \(A(\partial _\xi )\) defined in (19). Using the formula (12) for the operators \(A^\ell (\partial _\xi ): \mathcal A_{1,\alpha }\rightarrow \mathcal A_1\), we have
and the operator \(C(\partial _\xi ):\, \mathcal {A}_{1,\alpha }\rightarrow \mathcal {A}_1\), defined in (20), acts as
Using (24) it turns out that
STEP 2. To compute \(C(\partial _\xi )^m\) we need some more notation. By the formula (25), whenever we fix the index m we have the following set of indexes: \(q_\mu ,\, \ell _\mu ,\, p^{(\mu )}_1,\,\dots ,\, p^{(\mu )}_{\ell _i}\) for \(\mu =1,\dots , m\). We will prove, by induction, that:
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 \(\mu \) varying from 1 to \(m+1\), we obtain
Thus formula (26) are proved.
STEP 3. We show that \(\mathcal {U}_R(x,\partial _\xi )\), defined in (21), acts continuously from \(\mathcal {A}_{1,\alpha }\) to \(\mathcal {A}_1\).
The formula (26), which gives the explicit expression of \(C(\partial _\xi )^m\), will be crucial to study the continuity of \(\mathcal {U}_R(x,\partial _\xi )\). In fact, applying the operator \(\mathcal {U}_R(x,\partial _\xi )\) to holomorphic functions such that \(|f(z)|\le C\exp (\alpha |z|)\) we have:
Taking the modulus of both sides in the previous equality, we obtain
where we have used the estimate
Since \(f\in \mathcal {A}_{1,\alpha }\) we have that
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 \(b_f=2e \alpha \) 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
we have
where \(b_f=2e\alpha \). The previous inequality means that \(\mathcal U_R(x,\partial _{\xi }) f(\xi )\in \mathcal {A}_1\) and also that \(\mathcal U_R(x,\partial _{\xi })\) is continuous from \(\mathcal {A}_{1,\alpha }\) to \(\mathcal {A}_1\). Precisely, taking \(f_n\) such that \(f_n\rightarrow 0\) the estimate
with the bounds \(b_n\le 2e\alpha \), shows that \(U(x,\partial _{\xi }) f_n(\xi )\rightarrow 0\). Finally we have for \(|\lambda |<\alpha \)
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 \(\mathbb {C}\). So we take \(\varepsilon \in (0,1)\) and the interval \((-1,1)\) is replaced by the interval \((-\varepsilon ,\varepsilon )\) when we set \(\ell _j(n):=g(\varepsilon (1-2j/n))\) and \(k_j(n):=h(\varepsilon (1-2j/n))\).
We are ready to state the main result on the relativistic product.
Theorem 3.11
Let \(Y_n(x,a)\) and \(Z_n(x,a)\) be two generalized Fourier sequences for \(x, a\in \mathbb {R}\), with \(\ell _j(n):=g(\varepsilon (1-2j/n))\) and \(k_j(n):=h(\varepsilon (1-2j/n))\) as in Definition 3.3 for \(\varepsilon \in (0,1)\) and such that conditions (23) of Theorem 3.9 are satisfied for \(\alpha =1\). Let \((Y_n\circ _RZ_n)(x, a)\) be the relativistic product. Then we have
for any \(|\varepsilon a|<1\).
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 \(a\in \mathbb {R}\) and \(\varepsilon \in (0,1)\) such that \(|\varepsilon a|< 1\), we consider
We define the operator \(\mathcal {U}_R(x,\partial _\xi )\) as in (21) and we consider its action over the space \(\mathcal A_{1,1}\) (see Remark 3.8). By Theorem 3.9 we obtain the relations
and
Thus we have
where we have assumed \(|\varepsilon a|< 1\).
4 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 \((b_n)_{n=1}^M\) be a finite sequence of complex numbers such that \(0<|b_i|< 1\) for \(i=1,\dots , M\). A finite Blaschke product is defined for \(|z|<1\) by
where the last equality holds in the unit disc since \(|\overline{b}_nz|<1\). 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 \(a\in \mathbb {R}\) such that \(\varepsilon |a|<1\) and B is a finite Blaschke product. In order to compute the limit for \(n\rightarrow \infty \) we need the following infinite order differential operators formally obtained by the substitution of z with \(\partial _\xi \) in the series expansion (32).
Definition 4.1
Let \((b_i)_{i=1}^M\) be a finite sequence of complex numbers such that \(0<|b_i|< 1\) for \(i=1,\dots , M\). Let B(z) be a finite Blaschke products as in (32). We define the formal operators
and
We set \(\mathcal M_M:=\{0,1\}^M\). Given an element \(s\in \mathcal M_M\) we denote by \(s_i\) the element in the i-th position of the vector s and we denote by \(s^c\) the vector in \(\mathcal M_M\) defined by \((s^c)_i=1-s_i\).
Lemma 4.2
Let \((\mathsf R,+,\cdot )\) be a ring. Then for any positive integer M we have
where \(A^{(s_i)}_i\in \mathsf R\) and \(|s|=\sum _{i=1}^M s_i\).
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 \(\mathcal M_M\) as \(\mathcal M\).
Theorem 4.3
Let \((b_i)_{i=1}^M\) be a finite sequence of complex numbers such that \(0<|b_i|< 1\) for \(i=1,\dots , M\). Let B(z) be the finite Blaschke product (32). Let b be a positive constant such that \(b<\min _{j=1,\dots , M} \frac{1}{|b_j|}\). Then the operator
in Definition 4.1 acts continuously from \(\mathcal {A}_{1,\alpha }\) to \(\mathcal {A}_1\) where \(\alpha =\dfrac{b}{2e}\). Moreover, we have that
for \(|\lambda |<\min (\alpha , 1)\).
Proof
We observe that \(\mathcal {V}_q(\partial _\xi )=A^{(0)}_q(\partial _\xi )+A^{(1)}_q(\partial _\xi )\) where
Using Lemma 4.2 we can write
We fix an element \(f\in A_{1,\alpha }\) and \(s\in \mathcal M\) 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 \(A_q^{(0)}\) but not in \(A_q^{(1)}\). Taking the absolute value of both sides in the previous formula and using the fact that \( |f_j|\le C_f \frac{b^j}{j!} \) we obtain
where in the third inequality we have used the fact that b is in the set of the convergence of the series \(\sum _{p=0}^{\infty } |b_{q_i}|^p x^p\) for any \(i=1,\dots , M\). Thus we obtain
Moreover, we have
We deduce
The previous inequality means that \(\mathcal {U}_B(x,\partial _{\xi }) f(\xi )\in \mathcal A_1\) and also that \(\mathcal {U}_B(x,\partial {\xi })\) is continuous from \(\mathcal A_{1,\alpha }\) to \(A_1\). Finally we have that
for \(|\lambda |\le \min (\alpha ,1)\).
Theorem 4.4
Let \((b_i)_{i=1}^M\) be a finite sequence of complex numbers such that \(0<|b_i|< 1\) for \(i=1,\dots , M\). Let B(z) be the finite Blaschke product (32) and set
Then we have
for \(a\in \mathbb {R}\) such that \(\varepsilon |a|<1\).
Proof
We define the operator \(\mathcal {U}_B(x,\partial _\xi )\) as in (33). By Theorem 4.3 we obtain the relations
and
Thus we have
where we have assumed \(|\varepsilon a|< 1\).
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
we have to deal with terms such as
in the expansion
From these expressions we deduce suitable infinite order differential and the we have to study their continuity properties on the space of entire functions \(\mathcal {A}_1\).
5 Concluding remarks
Recently in the paper [5] the authors have solved explicitly the following problem:
Problem 5.1
Let \(h_j(n)\) be a given set of points in \([-1,1]\), \(j=0,1,...,n\), for \(n\in \mathbb {N}\) and let \(a\in \mathbb {R}\) be such that \(|a|>1\). Determine the coefficients \(X_j(n)\) of the sequence
in such a way that
Remark 5.2
The conditions \(f_n^{(p)}(0)=(ia)^p\) mean that the functions \(x\mapsto e^{iax}\) and \(f_n(x)\) have the same derivatives at the origin, for \(p=0,1,...,n\), so they have the same Taylor polynomial of order n.
Under the condition that the points \(h_j(n)\) for \(j=0,...,n\), (often denoted by \(h_j\)) are distinct we obtain an explicit formula for the coefficients \(X_j(n,a)\) given by
so the superoscillating sequence \(f_n(x)\), that solves Problem 5.1, takes the explicit form
Observe that, by construction, this function is band limited and it converges to \(e^{iax}\) with arbitrary \(|a|>1\), so it is superoscillating.
Different sequences \(X_j(n)\) can be explicitly computed when we fix the configurations of the points \(h_j(n)\). For example, with the configuration of points
we obtain:
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].
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The authors would like to thank both the referees for carefully reading the manuscript and for their comments. The second author is partially supported by the PRIN project Direct and inverse problems for partial differential equations: theoretical aspects and applications.
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Alpay, D., Colombo, F., Pinton, S. et al. Holomorphic functions, relativistic sum, Blaschke products and superoscillations. Anal.Math.Phys. 11, 139 (2021). https://doi.org/10.1007/s13324-021-00572-7
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DOI: https://doi.org/10.1007/s13324-021-00572-7