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 $\dot{W}^{s,p}$ and we call these spaces fractional Paley-Wiener if $p=2$ and fractional Bernstein spaces if $p\in(1,\infty)$, that we denote by $PW^s_a$ and $\mathcal B^{s,p}_a$, 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-P\'olya inequalities. We conclude by discussing a number of open questions.


Introduction and statement of the main results
A renowned theorem due to R. Paley and N. Wiener [PW34] characterizes the entire functions of exponential type a ą 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 [Ber23]. To be precise, let E a be the space of entire functions of exponential type a, E a " ! f P HolpCq : for every ε ą 0 there exists C ε ą 0 such that |f pzq| ď C ε e pa`εq|z| . (1) Then, for any p P p1, 8q, the Bernstein space B p a is defined as B p a " f P E a : f 0 P L p , }f } B p a " }f 0 } L p ( 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 P W a in place of B 2 a . Let S and S 1 denote the space of Schwartz functions and the space of tempered distributions, resp. For f P S we equivalently denote by p 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 pRq Ñ L 2 pRq. We now recall the classical Paley-Wiener characterization of the space P W a . Theorem ( [PW34]). Let f P P W a , then supp p f 0 Ď r´a, as, ż á a p f 0 pξqe izξ dξ and }f } P Wa " } p f 0 } L 2 pr´a,asq . Conversely, if g P L 2 pr´a, asq and we define ż á a gpξqe izξ dξ , then f P P W a , p f 0 " g and }f } P Wa " }g} L 2 pr´a,asq .
In particular, the Fourier transform F induces a surjective isometry between the spaces L 2 pr´a, asq and P W a . We shall write L 2 a instead of L 2 pr´a, asq for short. A similar characterization holds true for the Bernstein spaces B p a , 1 ă p ă`8. We refer the reader, for instance, to [And14,Theorem 4]. We shall denote by N 0 the set of nonnegative integers.
Theorem (Characterizations of B p a ). Let 1 ă p ă 8. 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 P B p a to the real line, that is, h " f 0 ; (ii) h P L p pRq and supp p h Ď r´a, as; (iii) h P C 8 , h pnq P L p for all n P N 0 and }h pnq } L p ď a n }h} L p .
The above theorem holds in the limit cases p " 1 and p "`8 as well, but in this paper we only focus on the range 1 ă p ă`8.
We remark that in the Paley-Wiener characterization of the Bernstein spaces, the Fourier transform of f 0 P L p pRq is to be understood in the sense of tempered distributions. Namely, L p pRq Ď S 1 , and the Fourier transform extends to a isomorphism of S 1 onto itself, where p f is defined by the formula x p f , ϕy " xf, p ϕy, f P S 1 , ϕ P S . 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 [Sei04], [OCS02] and references therein. Moreover, the Paley-Wiener space P W a is the most important example of a de Branges space, which are spaces of entire functions introduced by L. de Branges in [dB68]. 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 [Rom14].
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. The investigation of these spaces is not only motivated from the mere will to extend some classical results, but from the fact that these spaces arise very naturally in the several variable setting. In order to recover some classical 1-dimensional results in higher dimension, such as a Shannon-type sampling theorem, it is necessary to work with suitable defined fractional Paley-Wiener spaces on C n`1 . We refer the reader to [AMPS19,MPS20b] for details and results in the several variable setting. 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 (Section 5). Finally we mention the papers [Pes01,PZ09] in which the authors studied other generalizations of the Paley-Wiener spaces.
We now precisely define the function spaces we are interested in. Given a function f P S and s ą 0, we define its fractional Laplacian ∆ 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 9 W s,p , we consider the realization spaces E s,p , see [MPS20a,Corollary 3.2]. Inspired by the works of G. Bourdaud [Bou88,Bou11,Bou13], if m P N 0 Y t8u and 9 X is a given subspace of S 1 {P m which is a Banach space, such that the natural inclusion of 9 X into S 1 {P m is continuous, we call a subspace E of S 1 a realization of 9 X if there exists a bijective linear map Rrus ‰ " rus for every equivalence class rus P 9 X. We endow E of the norm given by }Rrus} E " }rus} 9 X . For γ ą 0, we denote by 9 Λ γ the homogeneous Lipschitz space of order γ. Given a locally integrable function, and an interval Q, we denote by f Q the average of f over Q, and denote by BMO the standard space of functions (modulo constants) of bounded mean oscillation. Finally, for a sufficiently smooth function f , we denote by P f ;m;0 the Taylor polynomial of f of order m at the origin. The next result describes the realization spaces E s,p .
Theorem ( [MPS20a]). For s ą 0 and p P p1,`8q, let m " ts´1 p u. Then, 9 W s,p Ď S 1 {P m . We define the spaces E s,p as follows.
If m " s´1 p is a positive integer, define ) .
Then, the space E s,p is a realization space for 9 W s,p .
When restricted to E s,p , }¨} s,p is no longer a semi-norm, but a genuine norm. In particular, the fractional Laplacian on E s,p is injective.
We are now ready to define the fractional Bernstein spaces.
Definition 1.1. For a, s ą 0 and 1 ă p ă`8 the fractional Bernstein space B s,p a is defined as B s,p a " f P E a : f 0 P E s,p and, if s ě 1{p and m " ts´1{pu, P f 0 ;m;0 " 0 ( .
We endow the space B s,p a with the norm }f } B s,p a :" }f 0 } E s,p .
Remark 1.2. In this paper we restrict ourselves to the case s´1 p R N 0 , s ą 0, p P p1, 8q. The case s´1 p 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 P N 0 . Thus, the case s´1 p P N 0 remains open and is, in our opinion, of considerable interest. We will add some comments on this problem in the final Section 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 9 B 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 " ts´1 p u.
We first consider the spaces P W s a , s ą 0, and we prove some Paley-Wiener type theorems assuming that s´1 2 R N 0 . For any s ą 0 let L 2 a p|ξ| 2s q be the weighted L 2 -space L 2 a p|ξ| 2s q " " f : r´a, as Ñ C such that ż á a |f pξq| 2 |ξ| 2s dξ ă 8 * .
We prove the following Paley-Wiener type theorems. We distinguish the case 0 ă s ă 1 2 from the case s ą 1 2 .
Theorem 1. Let 0 ă s ă 1 2 and let f P P W s a . Then, supp p f 0 Ď r´a, as, p f 0 P L 2 a p|ξ| 2s q and Moreover, }f } P W s a " } p f 0 } L 2 a p|ξ| 2s q . Conversely, let g P L 2 a p|ξ| 2s q, and define f by setting Then, f P P W s a , p f 0 " g and }f } P W s a " }g} L 2 a p|ξ| 2s q . Definition 1.4. Given s ą 1 2 , let m " ts´1 2 u, for any g P L 2 p|ξ| 2s q we define T g by setting, for ψ P S, xT g, ψy :" 1 ? 2π ż R gpξq`ψpξq´P ψ;m;0 pξq˘dξ.
As we will see, Lemma 3.3, T g is well-defined for any ψ P S, in particular T g P S 1 , and T : L 2 p|ξ| 2s q Ñ S 1 is a continuous operator.
We denote by D 1 c the space of distributions with compact support, which is the dual of C 8 . Theorem 2. Let s ą 1 2 , m " ts´1 2 u, assume that s´1 2 R N and set P m pizξq " ř m j"0 pizξq j {j!. Let f P P W s a , then supp p f 0 Ď r´a, as and there exists g P L 2 a p|ξ| 2s q such that p f 0 " T g in D 1 c , and Moreover, }f } P W s a " }g} L 2 a p|ξ| 2s q . Conversely, let g P L 2 a p|ξ| 2s q and define f by setting f pzq " xT g, e izp¨q y " 1 ? 2π ż á a gpξq`e izξ´P m pizξq˘dξ.
Then, f P P W s a and }f } P W s a " }g} L 2 a p|ξ| 2s q .
Observe that in particular Theorem 1 says that, if 0 ă s ă 1 2 , the Fourier transform F : P W s a Ñ L 2 a p|ξ| 2s q is a surjective isomorphism, as in the case s " 0. On the other hand, if s ą 1 2 , F : P W s a Ñ T pL 2 a p|ξ| 2s qq is a surjective isomorphism, where T pL 2 a p|ξ| 2s qq Ď D 1 c denotes the image of L 2 a p|ξ| 2s q via the operator T , endowed with norm }T g} :" }g} L 2 a p|ξ| 2s q . As a consequence of the above theorems we obtain that the spaces P W s a are reproducing kernel Hilbert spaces and we are able to make some interesting remarks concerning reconstruction formulas and sampling in P W s a for 0 ă s ă 1 2 . In particular, we obtain that the spaces P W s a are not de Branges spaces. We refer the reader to Section 5 below for more details.
Then we turn our attention to the fractional Bernstein spaces B s,p a . Theorem 3. Let s ą 0, 1 ă p ă 8 be such that s´1 p R N. Then, the fractional Bernstein spaces B s,p a are Banach spaces and the following Plancherel-Pólya estimates hold. If 0 ă s ă 1 p , for f P B s,p a and y P R we have }f p¨`iyq} B s,p a ď e a|y| }f } B s,p a . If s ą 1 p and s´1 p R N 0 , for f P B s,p a and y P R given, define F pwq " f pw`iyq´P f p¨`iyq;m;0 pwq, w P C. Then, F P B s,p a and }F } B s,p a ď e a|y| }f } B s,p a .
Theorem 4. Let s ą 0 and 1 ă p ă 8 such that s´1 p R N 0 . Given a function h on the real line, the following conditions are equivalent.
(i) The function h is the restriction of an entire function f P B s,p a to the real line, that is, h " f 0 ; (ii) h P E s,p and supp p h Ď r´a, as; (iii) h P C 8 and it is such that h pnq P E s,p for all n P N 0 and }h pnq } E s,p ď a n }h} E s,p .
Finally, the spaces P W s a are closed subspaces of the Hilbert spaces E s,2 , and thus there exists a Hilbert space projection operator P s : E s,2 Ñ P W s a . It is natural to study the mapping property of the operator P s with respect to the L p norm. We prove the following result.
Theorem 5. Let s ą 0 and 1 ă p ă 8 such that s´1 2 R N 0 , s´1 p R N 0 and ts´1 2 u " ts´1 p u. Then, the Hilbert space projection operator P s : E s,2 Ñ P W s a densely defined on E s,p X E s,2 extends to a bounded operator P s : E s,p Ñ B s,p a for all s ą 0 and 1 ă p ă`8.
The paper is organized as follows. After recalling some preliminary results in Section 2, we prove Theorem 1 and 2 in Section 3. In Section 4 we investigate the fractional Bernstein spaces proving Theorems 3 and 4, whereas in Section 5 we shortly discuss the sampling problem for the fractional Paley-Wiener spaces. Finally, we prove prove Theorem 5 in Section 6, and conclude with further remarks and open questions in Section 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 [Ste93]. We do not recall the results in their full generality, but only in the version we need them.
Let 0 ă s ă 1 so that the function ξ Ñ |ξ|´s is locally integrable. Then, the Riesz potential operator I s is defined on S as For p P p0, 8q we denote by H p , the Hardy space on R. Having fixed Φ P S with ş Φ " 1, then where }f } H p " }f˚} L p . We recall that the definition of H p is independent of the choice of Φ and that, when p P p1, 8q, H p coincides with L p , with equivalence of norms.
The Riesz potential operator extends to a bounded operator on H p , 0 ă p ă 8, according to the following theorem. Part (ii) is due to Adams, see [Ada75].
We recall that, as it is elementary to verify, ϕ P S M , if and only if there exists Φ P S such that ϕ " Φ pM q . We will use this fact several times. We also recall that S 8 is dense in H p for all p P p0, 8q, see [Ste93, Ch.II,5.2]. For these and other properties of the Hardy spaces see e.g. [Ste93] or [Gra14].
Notice that the Riesz potential operator I s is also well-defined on S 8 for any s ě 0, since if f P S 8 , then p f vanishes of infinite order at the origin. Moreover, for all s ą 0 are both surjective bounded isomorphisms and in fact one the inverse of the other one; see e.g. [Gra14, Chapter 1].

Fractional Paley-Wiener spaces
In this section we first prove Theorems 1 and 2, we deduce that the space P W s a is a reproducing kernel Hilbert space for every s ą 0, s´1 2 R N 0 , and explicitly compute its reproducing kernel. We conclude this section by proving that the classical Paley-Wiener space P W a and P W s a are actually isometric.
Proof. It is clear from the description of the realization spaces E s,2 that f 0 P S 1 , hence, once we prove that supp p f 0 Ď r´a, as, it immediately follows that p and we claim that f ϕ P E a and pf ϕ q 0 " f 0˚ϕ P L 2 ; where the symbol˚denotes the standard convolution on the real line. The function f ϕ is clearly entire and, for every ε ą 0, where the last integral converges since ϕ is compactly supported. Hence, f ϕ P E a . Moreover, since ϕ P S M , then ϕ˚η P S M as well for any η P S. Therefore, if tϕ n u Ď S is such that 1 We warn the reader that, we shall denote with the same symbol x¨,¨y different bilinear pairings of duality, such as xS 1 , Sy, xD 1 c , C 8 y, xL p 1 , L p y, etc. The actual pairing of duality should be clear from the context and there should not be any confusion.
The last equality follows using the Parseval identity, since ϕ P S M , hence ϕ˚η P S M as well, so that I s pϕ˚ηq P L 2 . Moreover, if Ψ P S is such that ϕ˚η " Ψ pM q , we have I s pϕ˚ηq " I s´ℓ`R ℓ Ψ pM´ℓq˘, where R denotes the Riesz transform, and ℓ " tsu. Then, where we have used Theorem 2.1 (i) with 1 2 " 1 p´p s´ℓq. Notice that p could be either greater or smaller than 1. If p ą 1, then }Ψ pM´ℓq } H p « }Ψ pM´ℓq } L p ă 8 since Ψ P S, if p ă 1, the fact that }Ψ pM´ℓq } H p is finite if M is sufficiently large is a well-known fact, see e.g. [Ste93]. Therefore, for M sufficiently large, where we have used Theorem 2.1 (i) with 1 " 1 p´p s´ℓq. This shows that pf ϕ q 0 P L 2 , and therefore, f ϕ P P W a . Thus, suppp { f 0˚ϕ q " suppp p f 0¨p ϕq Ď r´a, as. Since for every ξ 0 ‰ 0, ξ 0 P R, there exists ϕ P S M such that p ϕpξ 0 q ‰ 0, we conclude that supp p f 0 Ď r´a, as as we wished to show.
Let now s ą 1 2 , s´1 2 R N, and let m " ts´1 2 u. Fix χ P C 8 c , χ ě 0, χ " 1 on r´a, as. For ε ą 0, we adopt here and throughout the paper the notation This equality holds for all 0 ă ε ď 1 and we observe that, since f 0 is of moderate growth, both I ε and II ε are given by absolutely convergent integrals. Let M ą 0 be such that |f 0 pxq| ď Cp1`|x|q M , for some C ą 0. We have for any N ą 0. On the other hand, using Lebesgue's dominated convergence theorem it is easy to see that, as ε Ñ 0, since P f 0 ;m;0 " 0 and Q P P m . Hence, p f 0 pQq " 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 P L 2 a p|ξ| 2s q and define f as in (4). Then, since 0 ă s ă 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 P E s,2 .
Observing that and since I s : Lemma 3.1 guarantees that p f 0 P L 2 a p|ξ| 2s q and in particular is compactly supported in r´a, as. From the first part of the theorem, we know that the function is a well-defined function in P W s a and r f 0 " f 0 . Hence, f and r f coincide everywhere as we wished to show.
Corollary 3.2. The spaces P W s a , 0 ă s ă 1 2 , are reproducing kernel Hilbert spaces with reproducing kernel Proof. From (3) we deduce that point-evaluations are bounded on P W s a . In fact, q . This easily implies that P W s a is complete, hence a reproducing kernel Hilbert space. For z P C, the kernel function K z satisfies 2 1 ? 2π Therefore, { pK z q 0 pξq " 1 ? 2π e´iz ξ |ξ|´2 s χ r´a,as pξq and the conclusion follows. l Next, we consider the case s ą 1 2 .
From these estimates it is clear that T : L 2 p|ξ| 2s q Ñ S 1 is a continuous operator as we wished to show.
Lemma 3.4. Let s ą 1 2 , s´1 2 R N and m " ts´1 2 u. Given f P P W s a there exists a unique g P L 2 a p|ξ| 2s q such that p f 0 " T g in S 1 , that is, ż á a gpξq`ψpξq´P ψ;m;0 pξq˘dξ for all ψ P S.
Proof. By the results in [MPS20a], since f 0 P E s,2 , there exists a sequence tϕ n u Ď S such that t∆ s 2 ϕ n u is a Cauchy sequence in L 2 , and ϕ n Ñ f 0 in S 1 {P m , where m " ts´1 2 u, that is, xϕ n , ψy Ñ xf 0 , ψy " x p f 0 , p ψy, as n Ñ 8, for all ψ P S m . Therefore, x p ϕ n , ηy Ñ x p f 0 , ηy as n Ñ 8, for all η P x S m " S X tη P S : P η;m;0 u " 0. Moreover, there exists a unique g P L 2 p|ξ| 2s q such that p ϕ n Ñ g in L 2 p|ξ| 2s q. Since T : L 2 p|ξ| 2s q Ñ S 1 is continuous, we also have that T p ϕ n Ñ T g in S 1 . We now prove that it holds also T p ϕ n Ñ p f 0 in p x S m q 1 In fact, given ψ P x S m , we have In particular, this implies that supp T g Ă r´a, as, hence, T g P D 1 c and supp g Ď r´a, as. We now prove that QpDqpδq " 0. Let P P P m and let η P C 8 c such that η " 1 on r´a, as so that ηP P S. Since p f 0 is supported in r´a, as from Lemma 3.1 we get x p f 0 , ηP y " x p f 0 , P y " 0 and, since T g is supported in r´a, as as well, xT g, ηP y " ż á a gpξq`pηP qpξq´P ηP,m;0 pξq˘dξ " 0 since P ηP,m;0 " ηP on r´a, as. Therefore, we obtain that xQpDqpδq, ηP y " 0 as well and, by the arbitrariness of ηP , we conclude that QpDqpδq " 0 as we wished to show. Thus, p f 0 " T g in S 1 .
Before proving the next lemma, we need the following definition. Given s ą 0 and ψ P x S 8 , notice that |ξ| s ψ P x S 8 . Then, given U in S 1 , for any we define |ξ| s U by setting x|ξ| s U, ψy " xU, |ξ| s ψy .
We now prove the following simple, but not obvious, lemma.
Lemma 3.5. Let s ą 1 2 , s´1 2 R N and let f P P W s a . Then, Fp∆ s 2 f 0 q " |ξ| s p f 0 , with equality in L 2 a .
By the density of x S 8 in L 2 , we conclude that |ξ| s p f 0 P pL 2 q 1 , that is, |ξ| s p f 0 P L 2 a as we wished to show. Now, since f 0 P E s,2 , there exists tϕ n u Ď S such that ϕ n Ñ f 0 in S 1 {P m and t∆ s 2 ϕ n u is a Cauchy sequence in L 2 . Then, for ψ P S 8 , which is dense in L 2 , we have x p ϕ n , |ξ| s p ψy " lim nÑ`8 xϕ n , F´1`|ξ| s p ψ˘y " xf 0 , F´1`|ξ| s p ψ˘y " x p f 0 , |ξ| s p ψy " x|ξ| s p f 0 , ψy.
The conclusion follows from the density of S 8 Ď L 2 .
Proof of Theorem 2. We first prove the second part of the theorem. Recall that m " ts´1 2 u is the integer part of s´1 2 . Given f defined as in (7), we see that for every ε ą 0ˇˇˇ1 ? 2π ď C ε e pa`εq|z| since 2pm´s`1q ą´1 and where we have used the inequality ř`8 j"0 r j {pj`m`1q! ď e r , for r ą 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 P E s,2 . We have and we conclude that f 0 P 9 Λ s´1 2 as we wished to show (see [Gra14, Proposition 1.4.5]). Next, we need to show that ∆ s 2 f 0 P L 2 and }∆ s 2 f 0 } L 2 " }g} L 2 a p|ξ| 2s q . To this end, let tψ n u Ď S be such that ψ n Ñ g in L 2 a p|ξ| 2s q and define ϕ n as in (7), that is, where, we recall, m " ts´1 2 u. Observe that, by (12), ϕ n pxq " xT ψ n , e ixp¨q y. Given η P S, using Lemma 3.3, we have that all integrals in the equalities that follow converge absolutely and we have that lim nÑ8 xϕ n , ηy " lim nÑ8 1 ? 2π ż á a ψ n pξq`p ηpξq´P m;p η;0 pξq˘dξdx " lim nÑ8 xT ψ n , p ηy " xT g, p ηy " xf 0 , ηy .
Let us consider now f P P W s a . From Lemmas 3.1 and 3.4 we know that supp p f 0 Ď r´a, as, and that there exists a unique g P L 2 a p|ξ| 2s q such that p f 0 " T g in S 1 . Hence, the function is a well-defined function in P W s a by the first part of the proof. Moreover, ż á a piξq m`1 gpξqe ixξ dξ, so that FpD m`1f 0 q " piξq m`1 g. On the other hand, we also have that Fpf pm`1q 0 q " piξq m`1 p f 0 " piξq m`1 T g.
Now, for ψ P S, xpiξq m`1 T g, ψy " xT g, piξq m`1 ψy . Therefore, f and r f coincide up to a polynomial of degree at most m. Since P f ;m;0 " P r f ;m;0 " 0, we get r f " f ; in particular, and }∆ s 2 f 0 } L 2 " }g} L 2 a p|ξ| 2s q , as we wished to show. l As in the case s ă 1 2 , we have the following Corollary 3.6. For s ą 1 2 , s´1 2 R N, the spaces P W s a , are reproducing kernel Hilbert spaces with reproducing kernel Notice that, since K z " Kp¨, zq P P W s a , P Kz;m;0 " 0, that is, K z vanishes of order m at the origin, where m " ts´1 2 u.
Proof. From the previous theorem we know that the Fourier transform is a surjective isometry from P W s a onto T pL 2 a p|ξ| 2s qq, the closed subspace of S 1 endowed with norm }T g} :" }g} L 2 a p|ξ| 2s q . Therefore, P W s a are Hilbert spaces. Similarly to the proof of Corollary 3.2 we deduce from the representation formula (6) that the spaces P W s a are reproducing kernel Hilbert spaces. Then, 1 ? 2π From this identity and (6), the conclusion follows. l 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 tf P P W a : f 0 P S 8 u is dense in P W a .
We now show that the fractional Laplacian ∆ s 2 induces a surjective isometry from P W s a onto P W a .
Theorem 3.8. Let s ą 0 and assume s´1 2 R N 0 . Then, the operator ∆ s 2 : P W s a Ñ P W a is a surjective isometry, whose inverse is I s if 0 ă s ă 1 2 , whereas if s ą 1 2 the inverse is given bỳ with h P P W a .
Proof. We only need to prove the theorem in the case s ą 1 2 , s´1 2 R N. We recall that from Lemma 3.5, if f P P W s a , Fp∆ Hence, the map f Þ Ñ ∆ s 2 f 0 is clearly an isometry, and supp`Fp∆ s 2 f 0 q˘Ď r´a, as. By the classical Paley-Wiener theorem, ∆ s 2 f 0 extends to a function in P W a , that we denote by ∆ s 2 f . Let us focus on the surjectivity. Let h P P W a , then, by the previous lemma, there exists a sequence tϕ n u Ď th P P W a : h 0 P S 8 u such that }h´ϕ n } P Wa Ñ 0 as n Ñ 8. Set We observe that, since ϕ n P S 8 , we can write x ϕ n pξq|ξ|´sP m pizξq dξ since both the integrals converge absolutely. We are going to show that Φ n P P W s a , tΦ n u is a Cauchy sequence in P W s a , and that ∆ s 2 Φ n Ñ 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 P P W s a . Moreover, using [MPS20a, Corollary 3.4] and the fact that ϕ n P S 8 , we see that Hence, ∆ s 2 Φ n Ñ g in L 2 , and the surjectivity follows. In order to show that the inverse of ∆ s 2 has the expression (13), we observe that p h 0 pξq|ξ|´s P L 2 a p|ξ| 2s q, so that arguing as in the proof of Theorem 2, we see that F "`∆ s 2˘´1 h P P W s a . Now, if tg n u Ď C 8 c`t δ n ď |ξ| ď a´δ n u˘are such that δ n Ñ 0 and g n Ñ p h 0 pξq|ξ|´s in L 2 a p|ξ| 2s q, using [MPS20a, Corollary 3.4] again we have Thus, ∆ s 2 F " h and the surjectivity follows.

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.
Lemma 4.1. Let 1 ă p ă 8. Then, the space T " tf P E a : f 0 P S 8 u is dense in B p a . The proof of such lemma is somewhat elementary but not immediate and it is postponed to Section 7.
Theorem 4.2. Let s ą 0 such that s´1 p R N 0 . Then, the operator ∆ s 2 is a surjective isometry ∆ s 2 : B s,p a Ñ B p a and the inverse is as in (13) (with h P B p a ). Proof. We first notice that ∆ 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 9 W s,p . We now prove that ∆ and we claim that f ϕ P E a and pf ϕ q 0 P L p pRq. In fact, f ϕ is clearly entire and, for every ε ą 0, where the last integral converges since ϕ is continuous and compactly supported. Hence, f ϕ is of exponential type a. Now, let pf ϕ q 0 " f 0˚ϕ be the restriction of f ϕ to the real line. Since f 0 P E s,p there exists a sequence tϕ n u Ď S such that ϕ n Ñ f 0 in S 1 {P m , where m " ts´1{pu and t∆ s 2 ϕ n u is a Cauchy sequence in L p . We now argue as in (11). Since ϕ P S M , let Φ P S be such that ϕ " Φ pM q , so that I s ϕ " R ℓ I s´ℓ Φ pM´ℓq , where R denotes the Riesz transform. Then we have ż R |pϕ n˚ϕ qpxq| p dx " where 1 " 1 q´p s´ℓq, choosing ℓ " tsu in such a way that q ď 1. In particular, we get that tϕ n˚ϕ u is a Cauchy sequence in L p . Since ϕ n˚ϕ Ñ f 0˚ϕ in S 1 {P m and S m is dense in L p 1 , we see that Therefore, f ϕ is an exponential function of type a whose restriction to the real line is L pintegrable. Hence, f ϕ P B p a and supp z pf ϕ q 0 " suppp p f 0 p ϕq Ď r´a, as. From the arbitrariness of ϕ we conclude that supp p f 0 Ď r´a, as.
We now argue as in the proof of Lemma 3.5, to show that also supp { ∆ s 2 f 0 Ď r´a, as. Let Φ n P C 8 c ptδ n ď |ξ| ď auq, Φ n Ñ p f 0 in S 1 , and setting η n " F´1`Φ n˘w e have η n P S 8 . Now, for ψ P S, as n Ñ 8 we have xη n , ψy " xΦ n , p ψy Ñ x p f 0 , p ψy " xf 0 , ψy , where the pairings are in S 1 . Thus, η n Ñ f 0 in S 1 , which implies that ∆ Then, F n P E a , ∆ s 2 pF n q 0 " ph n q 0 and tF n u 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 tpF n q 0 u is a Cauchy sequence in E s,p . Hence, there exists a limit function r F P E s,p . We need to prove that r F is the restriction to the real line of a function in B s,p a . Since s ă 1 p , by Parseval's identity, we have F n pzq " ż R h n pxqF´1`|ξ|´se izξ χ r´a,as˘p xq dx , so that |F n pzq| ď }ph n q 0 } L p }F´1`|ξ|´se izξ χ r´a,as˘} L p 1 , where p, p 1 are conjugate indices. Observing that F´1`|ξ|´se izξ χ r´a,as˘" I s`F´1 pe izξ χ r´a,as q˘, from Theorem 2.1 we obtain }F´1`|ξ|´se izξ χ r´a,as˘} L p 1 ď C}F´1`e izξ χ r´a,as˘} L p 1 1`sp 1 , since p 1 1`sp 1 ą 1. However, for z fixed, F´1`e izξ χ r´a,as˘p tq " a 2π sincpapz`tqq belongs to B q a for any q P p1, 8q. Therefore, by the classical Plancherel-Pólya Inequality, we obtain }F´1`|ξ|´se izξ χ r´a,as˘} L p 1 ď Ce a|y| } sincpaξq} where y " Im z. In conclusion, |F n pzq| ď Ce a|y| }ph n q 0 } L p .
Since }ph n q 0 } L p " }h n } B p a , we just proved that the B p a -convergence of th n u implies the uniform convergence on compact subsets of C of tF n u to a function F of exponential type a. Necessarily, F | R " r F as we wished to show. Notice that we also have that that is, the inverse is given by equation (13). Suppose now that s ą 1 p , s´1 p R N 0 . Again, let h P B p a , th n u Ď T , h n Ñ h in B p a and set where m " ts´1{pu. Then, F n P E a and ∆ s 2 pF n q 0 " ph n q 0 by [MPS20a, Corollary 3.4]. Thus, tF n u is a Cauchy sequence in B s,p a , that is, tpF n q 0 u is a Cauchy sequence in E s,p , hence there exists a limit function r F P E s,p . We need to prove that r F is the restriction of some entire function of exponential type a.
Differentiating m`1 times, since s 1 :" m`1´s P p´1{p, 1{p 1 q the integrals below converge absolutely so that Then, if´1 p ă s 1 ă 0, the term on the right hand side in (15) equals ż R h n ptqR m`1 I s 1 sincpz´tq dt , and by Theorem 2.1 we obtain where 1 q " 1 p 1´s 1 , by the classical Plancherel-Pólya inequality. If 0 ď s 1 ă 1 p 1 , we repeat the same argument with s 1´1 in place of s 1 , observing that the the term on the right hand side in (15) equals ż R h n ptqR m I s 1´1`D sincpz´¨q˘ptq dt , and using the classical Bernstein inequality as well. Therefore, the convergence of th n u in B p a implies the uniform convergence on compact subsets of C of tF pm`1q n u 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 " 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 P B s,p a . From the identity, (13) with h P B p a , arguing as in (16), we obtain that |f pm`1q pzq| ď e a|y| }∆ s 2 f } B p a " e a|y| }f } B s,p a . Since P f ;m;0 " 0, it follows that for any compact K Ď C, sup zPK |f pzq| ď C K }f } B s,p a .
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 P T , and as in the proof of Theorem 4.2 define f pwq " p∆ is an orthonormal basis for P W s a . Proof. It is a well known fact that the family of functions tϕ n u nPZ , ϕ n pzq " 1 2 ? aπ ż á a e itpn π a´z q dt " a a{π sinc`apz´nπ{aqȋ s an orthonormal basis for P W a . The conclusion follows from Theorem 3.8.
We have the following consequences.
Corollary 5.2. For every f P P W s a we have the orthogonal expansion where the series converges in norm and uniformly on compact subsets of C. Moreover, Proof. By the classical theory of Hilbert spaces and Plancherel's formula, we get where the series convergences in P W s Proposition 5.5. The family te´i λnξ |ξ|´2 s u λnPΛ is a frame for L 2 a p|ξ| 2s q if and only if the family te´i λnξ |ξ|´su λnPΛ is a frame for L 2 a . Proof. Assume that te´i λnξ |ξ|´2 s u λnPΛ is a frame for L 2 a p|ξ| 2s q and let f be a function in P W a . Then, hence, te´i λnξ |ξ|´su λnPΛ is a frame for L 2 a p|ξ| 2s q. The reverse implication is similarly proved.
Therefore, the sampling problem for P W s a , 0 ă s ă 1 2 , is equivalent to study windowed frames for L 2 a . The following result, due to C.-K. Lai [Lai11] (see also [GL14]) implies that we cannot have real sampling sequences for P W s a . Hence, we cannot obtain an analogue of (18) with point evaluations of the function instead of point evaluations of its fractional Laplacian.
is a frame for L 2 a for some sequence of points Λ " tλ n u nPZ Ď R if and only if there exist positive constants m, M such that m ď gpξq ď M .
We conclude the section with one last comment about fractional Paley-Wiener spaces and de Brange spaces. These latter spaces were introduced by L. de Branges [dB68] and have been extensively studied in the last years. Among others, we recall the papers [OCS02, BBB15, BBB17, BBP17, BBH18, ABB19]. The space P W a is the model example of a de Branges space. A classical result, see [dB68], 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 P W s a , 0 ă s ă 1 2 . Therefore, the following result holds. Theorem 5.6. The fractional Paley-Wiener spaces P W s a , 0 ă s ă 1 2 , are not de Branges spaces.

Boundedness of the orthogonal projection
In the previous sections we proved that the spaces P W s a can be equivalently described as thus, it is clear that the spaces P W s a are closed subspaces of the Hilbert spaces E s,2 . Therefore, we can consider the Hilbert space projection operator P s : E s,2 Ñ P W s a . In the case 0 ă s ă 1 2 we get from Theorem 1 that the projection operator P s is explicitly given by the formula P s f pxq " 1 ? 2π ż R p f pξqχ r´a,as pξqe ixξ dξ .
Similarly, we explicitly deduce P s in the case s ą 1 2 from Theorem 2, P s f pxq " 1 2π ż R p f pξqχ r´a,as pξq`e ixξ´P m pixξq˘dξ .
Corollary 7.1. Let s ą 0, p P p1, 8q, s´1 p R N 0 , and set m " ts´1 p u. For s ą 1 p , set T m " f P E a : f 0 P S 8 , P f ;m;0 " 0 ( . Then, if 0 ă s ă 1 p the subspace T is dense in B s,p a , whereas if s ą 1 p the subspace T m is dense in B s,p a if s ą 1 p , s´1 p R N 0 . Proof. We only prove the case s ą 1 p , s´1 p R N 0 , the other case being easier. By Lemma 4.1 and Theorem 4.2 we have that p∆ s 2 q´1`T˘is dense in B s,p a . Thus, it suffices to show that this latter space is contained in T m . Let h P T and let f " ∆´s 2 h be given by (13). It is clear that P f ;m;0 " 0. Moreover, f pm`1q 0 " F´1`piξq m`1 |ξ|´s x h 0˘P S 8 and extends to a function in B s,p a , by Theorem 4. This easily implies that f 0 P S 8 and the conclusion follows.

Final remarks and open questions
We believe that the fractional spaces we introduced are worth investigating and, as we mentioned, they arise naturally in a several variable setting ( [MPS20b]).
We mention a few questions that remain open. First of all, it is certainly of interest to consider the cases s´1 p P N 0 . As we pointed out already, these cases correspond to the critical cases in the Sobolev embedding theorem. As shown by Bourdaud, when s´1 p R N 0 , the realization spaces E s,p of 9 W s,p are the unique realization spaces whose norms are homogeneous with respect the natural dilations. On the other hand, when s´1 p P N 0 there exists no realization space of 9 W s,p whose norm is homogeneous. In these case, it would be natural to define the realization space as the interpolating space between two spaces with s´1 p R N 0 . Thus, a natural definition may be B s,p a " f P E a : rf 0 s m P 9 W s,p and if m ě 1{p , P f 0 ;m;0 " 0 ( , where rf 0 s m denotes the equivalence class of f 0 in S 1 {P m . In any event, these spaces remain to be investigated. Naturally, another question that remains open is the boundedness of the orthogonal projection P : E s,p Ñ B s,p a in the cases s´1 p P N 0 . Such boundedness would allow one to explicitly describe the dual space of B s,p a , for the whole scale s ą 0 and p P p1, 8q. 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. [dB68,Rom14]. It would be interesting to determine whether the fractional Paley-Wiener spaces P W s a also arise to the solution of a canonical system defined in terms of the fractional derivative. In [BBH18] 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.