Fractional Paley–Wiener and Bernstein spaces

We introduce and study a family of spaces of entire functions in one variable that generalise the classical Paley–Wiener and Bernstein spaces. Namely, we consider entire functions of exponential type a whose restriction to the real line belongs to the homogeneous Sobolev space W˙s,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{W}^{s,p}$$\end{document} and we call these spaces fractional Paley–Wiener if p=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2$$\end{document} and fractional Bernstein spaces if p∈(1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (1,\infty )$$\end{document}, that we denote by PWas\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PW^s_a$$\end{document} and Bas,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}^{s,p}_a$$\end{document}, respectively. For these spaces we provide a Paley–Wiener type characterization, we remark some facts about the sampling problem in the Hilbert setting and prove generalizations of the classical Bernstein and Plancherel–Pólya inequalities. We conclude by discussing a number of open questions.


Introduction and statement of the main results
A renowned theorem due to Paley and Wiener [21] characterizes the entire functions of exponential type a > 0 whose restriction to the real line is square-integrable in terms of the support of the Fourier transform of their restriction to the real line. An analogous characterization holds for entire functions of exponential type a whose restriction to to the real line belongs to some L p space, p ≠ 2 [9]. To be precise, let E a be the space of entire functions of exponential type a, Then, for any p ∈ (1, ∞) , the Bernstein space B p a is defined as where f 0 ∶= f | ℝ denotes the restriction of f to the real line and L p is the standard Lebesgue space. In the Hilbert setting p = 2 , the Bernstein space B 2 a is more commonly known as the Paley-Wiener space and we will denote it by PW a in place of B 2 a . Let S and S ′ denote the space of Schwartz functions and the space of tempered distributions, resp. For f ∈ S we equivalently denote by f or Ff the Fourier transform given by The Fourier transform F is an isomorphism of S onto itself with inverse given by By Plancherel Theorem, the operator F extends to a surjective isometry F ∶ L 2 (ℝ) → L 2 (ℝ).
We now recall the classical Paley-Wiener characterization of the space PW a .
In particular, the Fourier transform F induces a surjective isometry between the spaces L 2 ([−a, a]) and PW a . We shall write L 2 a instead of L 2 ([−a, a]) for short. (1) E a = f ∈ Hol(ℂ) ∶ for every > 0 there exists C > 0 such that |f (z)| ≤ C e (a+ )|z| .
A similar characterization holds true for the Bernstein spaces B p a , 1 < p < +∞ . We refer the reader, for instance, to [4,Theorem 4]. We shall denote by ℕ 0 the set of nonnegative integers.
Theorem (Characterizations of B p a ) Let 1 < p < ∞. Then, the following conditions on a function h defined on the real line are equivalent.
(i) The function h is the restriction of an entire function f ∈ B p a to the real line, that is, h = f 0 ; (ii) h ∈ L p (ℝ) and supp � h ⊆ [−a, a]; (iii) h ∈ C ∞ , h (n) ∈ L p for all n ∈ ℕ 0 and ‖h (n) ‖ L p ≤ a n ‖h‖ L p.
The above theorem holds in the limit cases p = 1 and p = +∞ as well, but in this paper we only focus on the range 1 < p < +∞.
We remark that in the Paley-Wiener characterization of the Bernstein spaces, the Fourier transform of f 0 ∈ L p (ℝ) is to be understood in the sense of tempered distributions. Namely, L p (ℝ) ⊆ S � , and the Fourier transform extends to a isomorphism of S ′ onto itself, where f is defined by the formula The Paley-Wiener and Bernstein spaces are classical and deeply studied for several reasons. A well-studied problem for these spaces, for instance, is the sampling problem and we refer the reader to [19,24] and references therein. Moreover, the Paley-Wiener space PW a is the most important example of a de Branges space, which are spaces of entire functions introduced by de Branges in [13]. They have deep connections with canonical systems and have been extensively studied in the recent years. For an overview of de Branges spaces and canonical systems we refer the reader, for instance, to [23].
In this paper we introduce a family of spaces which generalizes the classical Paley-Wiener and Bernstein spaces; we deal with spaces of entire functions of exponential type a whose restriction to the real line belongs to some homogeneous Sobolev space and we call these spaces fractional Paley-Wiener and Bernstein spaces. In the present work we start such investigation: we introduce the spaces, we study some of their structural properties, we prove a Paley-Wiener type characterization and generalizations of the classical Bernstein and Plancherel-Pólya inequalities. We also point out that classical results such as sampling theorems for the Paley-Wiener space do not necessarily extend to the fractional setting (Sect. 5). Finally we mention the papers [20,22] in which the authors studied other generalizations of the Paley-Wiener spaces. We also point out that the investigation of these spaces has natural counterparts in the several variable setting. We refer the reader to [3] and the upcoming manuscript [17] where we present the development of this theory in the setting of several variables.
We now precisely define the function spaces we are interested in. Given a function f ∈ S and s > 0 , we define its fractional Laplacian Δ We remark that for f ∈ S the fractional Laplacian Δ s 2 f is a well-defined function and that ‖ ⋅ ‖ s,p is a norm on the Schwartz space (see, for instance, [18]). Therefore, we define the homogeneous Sobolev space Ẇ s,p as the closure of S with respect to ‖ ⋅ ‖ s,p , i.e., As described in [18], the space Ẇ s,p turns out to be a quotient space of tempered distributions modulo polynomials of degree m = ⌊s − 1∕p⌋ , where we denote by ⌊x⌋ the integer part of x ∈ ℝ and by P m the set polynomials of degree at most m, where m ∈ ℕ 0 . In [18,Corollary 3.3] we prove that f ∈Ẇ s,p if and only if If f ∈Ẇ s,p we then set where the limit is to be understood as a limit in the L p norm.
In order to avoid working in a quotient space, instead of considering the spaces Ẇ s,p , we consider the realization spaces E s,p , see [18,Corollary 3.2]. Inspired by the works of G. Bourdaud [10][11][12] We recall the definition of the homogeneous Lipschitz space Λ , for > 0 , for details see e.g. [15] and [18]. For k ∈ ℕ and h ∈ ℝ ⧵ {0} , let D k h , the difference operator of increment h and of order k, be defined as follows.
The homogeneous Lipschitz space is defined as For a non-negative integer m, we denote by C m the space of continuously differentiable functions of order m. The next result describes the realization spaces E s,p that we will need.
(i) Let 0 < s < 1 p , and let p * ∈ (1, ∞) given by 1 . The case s − 1 p ∈ ℕ 0 could be thought to be the critical case, as in the Sobolev embedding theorem. All the proofs break down for these values of s and p, although we believe that all the results in this paper extend also to case s − 1 p ∈ ℕ 0 . Thus, the case s − 1 p ∈ ℕ 0 remains open and is, in our opinion, of considerable interest. We will add some comments on this problem in the final Sect. 8. Remark 1. 3 We point out that from the results in the present work we can easily deduce analogous results for the homogeneous fractional Bernstein spaces Ḃ s,p a , defined as above, but without requiring that P f 0 ;m;0 = 0 . In this way, we obtain spaces of entire functions of exponential type modulo polynomials of degree m = ⌊s − 1 p ⌋.
We first consider the spaces PW s a , s > 0 , and we prove some Paley-Wiener type theorems assuming that s − 1 2 ∉ ℕ 0 . For any s > 0 let L 2 a (| | 2s ) be the weighted L 2 -space We prove the following Paley-Wiener type theorems. We distinguish the case 0 < s < 1 2 from the case s > 1 2 .
We denote by D ′ c the space of distributions with compact support, which is the dual of C ∞ .
) and define f by setting Then, f ∈ PW s a and ‖f ‖ PW s a = ‖g‖ L 2 a (� � 2s ) .
Observe that in particular Theorem 1 says that, if 0 < s < 1 2 , the Fourier transform F ∶ PW s a → L 2 a (| | 2s ) is a surjective isomorphism, as in the case s = 0 . On the other hand, denotes the image of L 2 a (| | 2s ) via the operator T, endowed with norm ‖Tg‖ ∶= ‖g‖ L 2 a (� � 2s ) . As a consequence of the above theorems we obtain that the spaces PW s a are reproducing kernel Hilbert spaces and we are able to make some interesting remarks concerning reconstruction formulas and sampling in PW s a for 0 < s < 1 2 . In particular, we obtain that the spaces PW s a are not de Branges spaces. We refer the reader to Sect. 5 below for more details.
Then we turn our attention to the fractional Bernstein spaces B s,p a . and y ∈ ℝ given, define F(w) = f (w + iy) − P f (⋅+iy);m;0 (w) , w ∈ ℂ . Then, F ∈ B s,p a and (iii) h ∈ C ∞ and it is such that h (n) ∈ E s,p for all n ∈ ℕ 0 and ‖h (n) ‖ E s,p ≤ a n ‖h‖ E s,p.
Finally, the spaces PW s a are closed subspaces of the Hilbert spaces E s,2 , and thus there exists a Hilbert space projection operator s ∶ E s,2 → PW s a . It is natural to study the mapping property of the operator s with respect to the L p norm. We prove the following result. The paper is organized as follows. After recalling some preliminary results in Sect. 2, we prove Theorem 1 and 2 in Sect. 3. In Sect. 4 we investigate the fractional Bernstein spaces proving Theorems 3 and 4, whereas in Sect. 5 we shortly discuss the sampling problem for the fractional Paley-Wiener spaces. Finally, we prove prove Theorem 5 in Sect. 6, and conclude with further remarks and open questions in Sect. 8.

Preliminaries
In this section we recall some results of harmonic analysis we will need in the remaining of the paper. We omit the proofs of the results and we refer the reader, for instance, to [25]. We do not recall the results in their full generality, but only in the version we need them.
Let 0 < s < 1 so that the function → | | −s is locally integrable. Then, the Riesz potential operator I s is defined on S as We recall that the definition of H p is independent of the choice of Φ and that, when p ∈ (1, ∞) , H p coincides with L p , with equivalence of norms.
The Riesz potential operator extends to a bounded operator on H p , 0 < p < ∞ , according to the following theorem. Part (ii) is due to Adams, see [2].

Definition 2.2 For M a nonnegative integer, define
and We recall that, as it is elementary to verify, ∈ S M , if and only if there exists Φ ∈ S such that = Φ (M) . We will use this fact several times. We also recall that S ∞ is dense in H p for all p ∈ (0, ∞) , see [25,Ch.II,5.2]. For these and other properties of the Hardy spaces see e.g. [25] or [15].
Notice that the Riesz potential operator I s is also well-defined on S ∞ for any

Fractional Paley-Wiener spaces
In this section we first prove Theorems 1 and 2, we deduce that the space PW s a is a reproducing kernel Hilbert space for every s > 0, s − 1 2 ∉ ℕ 0 , and explicitly compute its reproducing kernel. We conclude this section by proving that the classical Paley-Wiener space PW a and PW s a are actually isometric.
Proof It is clear from the description of the realization spaces E s,2 that f 0 ∈ S � , hence, once we prove that supp and we claim that f ∈ E a and (f ) 0 = f 0 * ∈ L 2 ; where the symbol * denotes the standard convolution on the real line. The function f is clearly entire and, for every > 0, where the last integral converges since is compactly supported. Hence, f ∈ E a . Moreover, since ∈ S M , then * ∈ S M as well for any ∈ S . Therefore, if The last equality follows using the Parseval identity, since ∈ S M , hence * ∈ S M as well, so that where R denotes the Riesz transform, and = ⌊s⌋ . Then, Notice that p could be either greater or smaller than 1. [25]. Therefore, for any ∈ S . In particular f 0 * = Δ . For > 0 , we adopt here and throughout the paper the notation This equality holds for all 0 < ≤ 1 and we observe that, since f 0 is of moderate growth, both I and II are given by absolutely convergent integrals. Let M > 0 be such that On the other hand, using Lebesgue's dominated convergence theorem it is easy to see that, as → 0, since P f 0 ;m;0 = 0 and Q ∈ P m . Hence, f 0 (Q) = 0 and we are done. ◻ We now prove our first main theorem.

Proof of Theorem 1
We start proving the second part of the statement. Let g ∈ L 2 a (| | 2s ) and define f as in (4). Then, since 0 < s < 1 2 , for z = x + iy, Therefore, f is well-defined, is clearly entire and belongs to E a . We wish to show that f 0 ∈ E s,2 . Observing that Now, let f ∈ PW s a . Lemma 3.1 guarantees that f 0 ∈ L 2 a (| | 2s ) and in particular is compactly  a] ( ) and the conclusion follows. ◻ Next, we consider the case s > 1 2 .
for any ∈ S . Then, Tg is well-defined and T ∶ L 2 (| | 2s ) → S � is a continuous operator.
Proof By Hölder's inequality we have where C denotes a finite positive constant bounded by some Schwartz seminorm of . Moreover, From these estimates it is clear that T ∶ L 2 (| | 2s ) → S � is a continuous operator as we wished to show. ◻ Proof By the results in [18], since f 0 ∈ E s,2 , there exists a sequence { n } ⊆ S such that {Δ s 2 n } is a Cauchy sequence in L 2 , and n → f 0 in S � ∕P m , where m = ⌊s − 1 2 ⌋ , that is, ⟨ n , ⟩ → ⟨f 0 , ⟩ = ⟨f 0 ,̂ ⟩ , as n → ∞ , for all ∈ S m . Therefore, as n → ∞ , for all ∈Ŝ m = S ∩ { ∈ S ∶ P ;m;0 } = 0 . Moreover, there exists a unique g ∈ L 2 (| | 2s ) such that ̂ n → g in L 2 (| | 2s ) . Since T ∶ L 2 (| | 2s ) → S � is continuous, we also (12) have that T̂ n → Tg in S ′ . We now prove that it holds also T̂ n →f 0 in (Ŝ m ) � In fact, given ∈Ŝ m , we have Before proving the next lemma, we need the following definition. Given s > 0 and ∈Ŝ ∞ , notice that | | s ∈Ŝ ∞ . Then, given U in S ′ , for any we define | | s U by setting We now prove the following simple, but not obvious, lemma.
By the density of Ŝ ∞ in L 2 , we conclude that | | sf 0 ∈ (L 2 ) � , that is, | | sf 0 ∈ L 2 a as we wished to show. Now, since f 0 ∈ E s,2 , there exists { n } ⊆ S such that n → f 0 in S � ∕P m and {Δ s 2 n } is a Cauchy sequence in L 2 . Then, for ∈ S ∞ , which is dense in L 2 , we have The conclusion follows from the density of S ∞ ⊆ L 2 . ◻

Proof of Theorem 2
We first prove the second part of the theorem. Recall that m = ⌊s − 1 2 ⌋ is the integer part of s − 1 2 . Given f defined as in (7), we see that for every > 0 since 2(m − s + 1) > −1 and where we have used the inequality ∑ +∞ j=0 r j ∕(j + m + 1)! ≤ e r , for r > 0 . Hence, f is well-defined, clearly entire and it belongs to E a . Since it is clear that P f ;m;0 = 0 , it remains to show that f 0 ∈ E s,2 . We have so that, and we conclude that f 0 ∈Λ s− 1 2 as we wished to show (see [15,Proposition 1.4.5]). Next, we need to show that Δ s 2 f 0 ∈ L 2 and ‖Δ s 2 f 0 ‖ L 2 = ‖g‖ L 2 a (� � 2s ) . To this end, let { n } ⊆ S be such that n → g in L 2 a (| | 2s ) and define n as in (7), that is, where, we recall, m = ⌊s − 1 2 ⌋ . Observe that, by (12), n (x) = ⟨T n , e ix(⋅) ⟩ . Given ∈ S , using Lemma 3.3, we have that all integrals in the equalities that follow converge absolutely and we have that Therefore, n → f 0 in S ′ . Moreover, we have that D m+1 n = F −1 (i ) m+1 n and setting s � ∶= s − (m + 1) ∈ (− . Therefore, f and f coincide up to a polynomial of degree at most m. Since P f ;m;0 = Pf ;m;0 ≡ 0 , we get f = f ; in particular, and ‖Δ Notice that, since K z = K(⋅, z) ∈ PW s a , P K z ;m;0 = 0 , that is, K z vanishes of order m at the origin, where m = ⌊s − 1 2 ⌋.
Proof From the previous theorem we know that the Fourier transform is a surjective isometry from PW s a onto T(L 2 a (| | 2s )) , the closed subspace of S ′ endowed with norm ‖Tg‖ ∶= ‖g‖ L 2 a (� � 2s ) . Therefore, PW s a are Hilbert spaces. Similarly to the proof of Corollary 3.2 we deduce from the representation formula (6) that the spaces PW s a are reproducing kernel Hilbert spaces. Then, and therefore, From this identity and (6), the conclusion follows. ◻ The following lemma is obvious and we leave the details to the reader (or see the proof of Lemma 4.1).

Lemma 3.7 The space
We now show that the fractional Laplacian Δ s 2 induces a surjective isometry from PW s a onto PW a .
Theorem 3.8 Let s > 0 and assume s − 1 2 ∉ ℕ 0 . Then, the operator Δ s 2 ∶ PW s a → PW a is a surjective isometry, whose inverse is I s if 0 < s < 1 2 , whereas if s > 1 2 the inverse is given by Proof We only need to prove the theorem in the case s > 1 2 , s − 1 2 ∉ ℕ . We recall that from Hence, the map f ↦ Δ We observe that, since n ∈ S ∞ , we can write since both the integrals converge absolutely. We are going to show that Φ n ∈ PW s a , {Φ n } is a Cauchy sequence in PW s a , and that Δ s 2 Φ n → g in L 2 . From these facts the surjectivity follows at once. As in the proofs of Theorems 1 and 2, we see that Φ n ∈ PW s a . Moreover, using [18,Corollary 3.4] and the fact that n ∈ S ∞ , we see that Hence, Δ s 2 Φ n → g in L 2 , and the surjectivity follows. In order to show that the inverse of Δ s 2 has the expression (13), we observe that ĥ 0 ( )| | −s ∈ L 2 a (| | 2s ) , so that arguing as in the proof of Theorem 2, we see that = lim n→∞ �t� s g n (t) =ĥ 0 (t).

Fractional Bernstein spaces
In this section we study the fractional Bernstein spaces and we first show that the spaces B s,p a are isometric to the classical Bernstein spaces B p a . The proof is similar to the Hilbert case, but we have to overcome the fact that Plancherel and Parseval's formulas are no longer available.
We need the following density lemma.
The proof of such lemma is somewhat elementary but not immediate and it is postponed to Sect. 7. Proof We first notice that Δ s 2 is injective on B s,p a since these spaces are defined using the realizations E s,p of the homogeneous Sobolev spaces Ẇ s,p .
We now prove that Δ n } is a Cauchy sequence in L p . We now argue as in (11). Since ∈ S M , let Φ ∈ S be such that = Φ (M) , so that I s = R I s− Φ (M− ) , where R denotes the Riesz transform. Then we have q − (s − ) , choosing = ⌊s⌋ in such a way that q ≤ 1 . In particular, we get that { n * } is a Cauchy sequence in L p . Since n * → f 0 * in S � ∕P m and S m is dense in L p ′ , we see that n * → f 0 * = (f ) 0 in L p . Therefore, f is an exponential function of type a whose restriction to the real line is L p -integrable. Hence, f ∈ B p a and supp � a] . From the arbitrariness of we conclude that supp � f 0 ⊆ [−a, a]. We now argue as in the proof of Lemma 3.5, to show that also supp � Δ , Φ n →f 0 in S ′ , and setting n = F −1 Φ n we have n ∈ S ∞ . Now, for ∈ S , as n → ∞ we have where the pairings are in S ′ . Thus, n → f 0 in S ′ , which implies that Δ Then, F n ∈ E a , Δ s 2 (F n ) 0 = (h n ) 0 and {F n } is a Cauchy sequence in B s,p a since ‖F n ‖ B s,p a = ‖h n ‖ B p a . In particular, this means that {(F n ) 0 } is a Cauchy sequence in E s,p . Hence, there exists a limit function F ∈ E s,p . We need to prove that F is the restriction to the real line of a function in B s,p a . , belongs to B q a for any q ∈ (1, ∞) . Therefore, by the classical Plancherel-Pólya Inequality, we obtain where y = Imz . In conclusion, a , we just proved that the B p a -convergence of {h n } implies the uniform convergence on compact subsets of ℂ of {F n } to a function F of exponential type a. Necessarily, F| ℝ =F as we wished to show. Notice that we also have that that is, the inverse is given by equation (13).
where m = ⌊s − 1∕p⌋ . Then, F n ∈ E a and Δ s 2 (F n ) 0 = (h n ) 0 by [18,Corollary 3.4]. Thus, {F n } is a Cauchy sequence in B s,p a , that is, {(F n ) 0 } is a Cauchy sequence in E s,p , hence there exists a limit function F ∈ E s,p . We need to prove that F is the restriction of some entire function of exponential type a.
Differentiating m + 1 times, since s � ∶= m + 1 − s ∈ (−1∕p, 1∕p � ) the integrals below converge absolutely so that Then, if − 1 p < s � < 0 , the term on the right hand side in (15) equals and by Theorem 2.1 we obtain where 1 q = 1 p � − s � , by the classical Plancherel-Pólya inequality. If 0 ≤ s ′ < 1 p ′ , we repeat the same argument with s � − 1 in place of s ′ , observing that the the term on the right hand side in (15) equals and using the classical Bernstein inequality as well.
Therefore, the convergence of {h n } in B p a implies the uniform convergence on compact subsets of ℂ of {F (m+1) n } and, in particular, the limit function G m+1 is of exponential type a. Then, F is the anti-derivative of G m+1 such that P F;m,0 = 0 and F| ℝ =F , as we wished to show. This shows that the inverse is as in (13) and concludes the proof of the theorem. ◻ Proof Let f ∈ B s,p a . From the identity, (13) with h ∈ B p a , arguing as in (16), we obtain that Since P f ;m;0 = 0 , it follows that for any compact K ⊆ ℂ,

◻
We are now ready to prove Theorems 3 and 4.

Proof of Theorem 3
We observe that the completeness follows from the above corollary, or from the surjective isometry between B s,p a and B p a . For the second part of the theorem we argue as follows. Let h ∈ T , and as in the proof of Theorem 4.2 define and therefore Hence, Hence, from (13), we conclude that F ∈ B s,p a since, by the classical Plancherel-Pólya inequality ( [26]), h(⋅ + iy) ∈ B p a and by Theorem 4.2 we obtain By applying the classical characterization of Bernstein spaces to Δ is an orthonormal basis for PW a . The conclusion follows from Theorem 3.8.
We have the following consequences.

Corollary 5.2 For every f ∈ PW s a we have the orthogonal expansion
where the series converges in norm and uniformly on compact subsets of ℂ . Moreover,  (17), but we cannot really improve (18). For simplicity, we now restrict ourselves to the case 0 < s < 1 2 .
where the series converges absolutely and uniformly on compact subsets of ℂ.
Proof Let f ∈ PW s a and let {f k } ⊆ PW a be a sequence such that f k → f in PW s a . Then, by the Shannon-Kotelnikov theorem, we have that where the series converges absolutely and in PW a -norm. However, norm convergence in PW a implies uniform convergence on compact subsets of ℂ . Thus, we obtain ◻ We now point out that, in general, we cannot improve (17) with point evaluations of f instead of its fractional Laplacian. More generally, we would like to know if it is possible to have a real sampling sequence in PW s a . We will see that, at least in the case 0 < s < 1 2 , this is not the case. Proof Assume that {e −īn | | −2s } n ∈Λ is a frame for L 2 a (| | 2s ) and let f be a function in PW a . Then, hence, {e −īn | | −s } n ∈Λ is a frame for L 2 a (| | 2s ) . The reverse implication is similarly proved. ◻ Therefore, the sampling problem for PW s a , 0 < s < 1 2 , is equivalent to study windowed frames for L 2 a . The following result, due to [16] (see also [14]) implies that we cannot have real sampling sequences for PW s a . Hence, we cannot obtain an analogue of (18) with point evaluations of the function instead of point evaluations of its fractional Laplacian.

Definition 5.4 Let
Theorem [14,16] The family g( )e i n n ∈Λ is a frame for L 2 a for some sequence of points Λ = { n } n∈ℤ ⊆ ℝ if and only if there exist positive constants m, M such that m ≤ g( ) ≤ M .
We conclude the section with one last comment about fractional Paley-Wiener spaces and de Brange spaces. These latter spaces were introduced by de Branges [13] and have been extensively studied in the last years. Among others, we recall the papers [1, 5-8, 19]. The space PW a is the model example of a de Branges space. A classical result, see [13], states that de Brange spaces always admit a real sampling sequence, or, equivalently, always admit a Fourier frame of reproducing kernels. The above discussion proves that this is not the case for the spaces PW s a , 0 < s < 1 2 . Therefore, the following result holds. assuming that s − 1 2 ∉ ℕ 0 , s − 1 p ∉ ℕ 0 and ⌊s − 1 2 ⌋ = ⌊s − 1 p ⌋ . This is the content of Theorem 5 which we now prove. 5 We first assume 0 < s < 1 2 . Let f be a function in E s,p ∩ E s,2 . By definition of E s,p ∩ E s,2 , we can assume f to be in the Schwartz space S . Then, the projection s f is given by (19). The function s f clearly extends to an entire function of exponential type a, which we still denote by s f . Moreover, we assumed f ∈ S , so that, for instance, s f is a well-defined L 2 function with a well-defined Fourier transform. Thus, Hence where the inequality holds since [−a,a] is an L p -Fourier multiplier for any 1 < p < +∞ . Therefore, s extends to a bounded operator s ∶ E s,p → B s,p a when 0 < s < 1 2 . Assume now s > 1 2 . Then, given f ∈ E s,p ∩ E s,2 ∩ S , the projection s f is given by (20), that is,   Naturally, another question that remains open is the boundedness of the orthogonal projection ∶ E s,p → B s,p a in the cases s − 1 p ∈ ℕ 0 . Such boundedness would allow one to explicitly describe the dual space of B s,p a , for the whole scale s > 0 and p ∈ (1, ∞). The Paley-Wiener space is a very special instance of a de Branges spaces. These spaces where introduced by de Branges also in connection with the analysis of the canonical systems, see e.g. [13,23]. It would be interesting to determine whether the fractional Paley-Wiener spaces PW s a also arise to the solution of a canonical system defined in terms of the fractional derivative.

Proof of Theorem
In [7] it is shown that the Paley-Wiener space, and, more generally, any de Branges space, coincides as set with a Fock-type space with non-radial weight. The Paley-Wiener (or de Branges) norm given by an integral on the real line is replaced by an equivalent weighted integral on the complex plane. We wonder if an analogous result holds true for the fractional Paley-Wiener spaces.
Another important fact about the classical Paley-Wiener space is that, up to a multiplication by an inner function, it admits a representation as a model space of H 2 (ℂ + ) , the Hardy space of the upper half-plane. We recall that a model subspace of H 2 (ℂ + ) is defined as K Θ = H 2 (ℂ + ) ⊖ ΘH 2 (ℂ + ) where Θ is an inner function in ℂ + . By means of the Weyl-Titchmarsh transform, it is possible to interpret the completeness problem for eigenfunctions of the Schrödinger equations as the completeness problem for a system of reproducing kernel {k } in some model space K Θ . Thus, there exists a close link between model spaces, in particular the Paley-Wiener spaces. It would be interesting to understand if an analogous link exists between the fractional canonical systems and the fractional Paley-Wiener spaces.
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