On Fourier frames

We solve the problem of Duffin and Schaeffer (1952) of characterizing those sequences of real frequencies which generate Fourier frames. Equivalently, we characterize the sampling sequences for the Paley-Wiener space. The key step is to connect the problem with de Branges' theory of Hilbert spaces of entire functions. We show that our description of sampling sequences permits us to obtain a classical inequality of H. Landau as a consequence of Pavlov's description of Riesz bases of complex exponentials and the John-Nirenberg theorem. Finally, we discuss how to transform our description into a working condition by relating it to an approximation problem for subharmonic functions. By this approach, we determine the critical growth rate of a nondecreasing function $\psi$ such that the sequence $\{\lamdda_k\}){k\in \Bbb Z}$ defined by $\lambda_k + \psi(\lambda_k)=k$ is a sampling.


Introduction
Following Duffin and Schaeffer [DS52], we say that a system of complex exponentials {e iλ k x }, with Λ = {λ k } k∈Z a sequence of real numbers, is a Fourier frame if there exist positive constants A and B such that for all f ∈ L 2 (−π, π). The purpose of this paper is to give a description of those sequences Λ that generate Fourier frames.
The main result of [DS52] is a sufficient density condition for {e iλ k x } to constitute a Fourier frame. Over the past two decades, the work of Duffin and Schaeffer has been remarkably influential, but mainly so because of its description of abstract frame expansions as an alternative to orthonormal bases; see, e.g., [Da92]. There has been little progress on the problem of characterizing the original Fourier frames of Duffin and Schaeffer.
Our work relies on several profound results. Specifically, it may be seen as an interplay between three themes: 1) de Branges' Hilbert spaces of entire functions. Our main theorem is based on de Branges' theory [dB68]. The Hilbert space of interest to us is the classical Paley-Wiener space, the prime example of a de Branges space. We denote this space by PW; it consists of all entire functions of exponential type at most π whose restrictions to R are square-integrable. We say that a sequence of real numbers Λ = {λ k } k∈Z is sampling for PW if there exist positive constants A and B such that holds for all f ∈ PW. By the Paley-Wiener theorem, Λ is sampling for PW if and only if {e iλ k x } is a Fourier frame. In other words, our task is to describe the sampling sequences for PW.
The Paley-Wiener space is a Hilbert space when equipped with its standard norm, the L 2 (R)-norm. The symbol PW will stand for this particular Hilbert space. However, if Λ is a sampling sequence, we may equip the Paley-Wiener space with an equivalent norm given by k |f (λ k )| 2 . In this way, we obtain a different Hilbert space, which is again a de Branges space. This simple observation has the amazing consequence that several basic results of de Branges' theory apply to our problem. In particular, two results of de Branges about norm identities may be combined to yield a result about norm equivalence (Theorem 1 below).
The required background on de Branges spaces is reviewed in Section 2, and then our main theorem -a necessary and sufficient condition for sampling -is stated and proved in Section 3.
2) Riesz bases of complex exponentials. Fourier frames are intimately connected with Riesz bases of complex exponentials, or equivalently, sampling sequences are intimately connected with complete interpolating sequences (see below for definition). This is of interest to us, because there exists a beautiful description of the latter kind of sequences, due to Pavlov [Pa79]; see [HNP81] for an extensive account and [LS97] for some recent progress on this topic.
We say that Λ is a complete interpolating sequence for PW if the interpolation problem f (λ k ) = a k has a unique solution f ∈ PW whenever {a k } is square-summable, or equivalently, if it is sampling but fails to be so on the removal of any one of the points λ k . Thus a complete interpolating sequence is a sampling sequence with no "redundant" points. The sequence of integers is of course the leading example of such a sequence.
Some additional terminology is required to state Pavlov's theorem. Let us assume that λ k ≤ λ k+1 for all k. A sequence Λ is separated if q is referred to as the separation constant of Λ. With a separated sequence Λ we associate a distribution function n Λ (t) defined such that for a < b and normalized such that n Λ (0) = 0. There is clearly a one-to-one correspondence between Λ and n Λ . It is plain that all complete interpolating sequences are separated. Let denote the Poisson extension of h to the upper half-plane y > 0, and let u →ũ stand for the usual conjugation operator. The following theorem, with the Helson-Szegö condition appearing in slight disguise, is a reformulation (see [HNP81,p. 286 To see the problem of relating this result to sampling sequences, we begin by stating two density conditions for sampling. The first is a fairly elementary result: Λ is sampling if there exist positive constants ε and C such that This is essentially what was obtained by Duffin and Schaeffer (see also Theorem 2.1 of [Se95]). A more profound result is the following inequality 1 of Landau [La67].
Theorem B (Landau). If Λ is a separated sampling sequence for PW, then when a < b, with constants A and B independent of a, b.
Landau's inequality is best possible, as shown by an example in Section 4. We shall see that Theorem A and our main theorem (Theorem 1) imply that Landau's inequality is a consequence of the John-Nirenberg theorem for BMO functions. This result will be proved in Section 4 below.
The two inequalities just recorded have the following consequence. Set which is the Beurling lower uniform density of Λ. Then Λ is sampling if D − (Λ) > 1 and fails to be sampling if D − (Λ) < 1. (See also [Ja91] for a somewhat different formulation.) It was proved in [Se95] that when D − (Λ) > 1, there exists a subsequence Λ ′ ⊂ Λ such that Λ ′ is a complete interpolating sequence, but an example was given of a sampling sequence Λ for which no subsequence is a complete interpolating sequence. The latter type of sampling sequences are the only ones which are essentially different from complete interpolating sequences. The third theme of this paper is what is needed to explore their subtle properties.
3) Approximation of subharmonic functions. We have in mind the following general type of problem. Given a subharmonic function U , find an entire function f such that log |f | is in some sense a good approximation to U . Our main theorem becomes a working condition only if we are able to solve certain problems of this kind. An exhaustive discussion of the solutions relevant for our sampling condition seems impracticable. Instead, we use an approach of Lyubarskii and Malinnikova [LM01] to elaborate an illustrative example. In Section 5, we determine the critical growth rate of a nondecreasing function ψ such that Λ = {λ k } k∈Z defined by λ k + ψ(λ k ) = k is sampling. Roughly speaking, we obtain that if ψ(x) = 0 for x < 0, ψ is sufficiently regular, and ψ(x) → ∞ when x → ∞, then Λ is sampling if and only if ψ(x) tends to ∞ at least as fast as log + x.
This result may seem surprising: A sequence is sampling if and only if it constitutes a sufficiently large deviation from Z. The central point here is that the sampling condition implies that we need to solve the above approximation problem for the logarithmic potential of the measure dψ. Then difficulties arise if the measure is too much "spread out".

Preliminaries on de Branges spaces
A Hilbert space H of entire functions is a de Branges space if the following conditions are met: (H1) Whenever f is in the space and has a nonreal zero ζ, the function g(z) = f (z)(z −ζ)/(z − ζ) is in the space and has the same norm as f .
It is clear that H(E) is a de Branges space. The following fundamental theorem of de Branges says that all de Branges spaces arise in this way [dB68, p. 57].
Theorem C. A Hilbert space H whose elements are entire functions, which satisfies (H1), (H2), and (H3), and which contains a nonzero element, is equal isometrically to some space H(E).
It follows from condition (H2) that for each nonreal ζ there exists a reproducing kernel K(ζ, z) for the space H(E). The kernel has the following representation, which also defines it for real ζ [dB68, p. 50].
Theorem D. For each ζ ∈ C the function considered as a function of z, belongs to H(E) and it is the reproducing kernel of H(E), i.e. .
We say that ϕ is a phase function of E. If there is a need to distinguish ϕ from phase functions of other functions in HB, then we set ϕ = ϕ E . We shall make use of the following remarkable extension of the Plancherel identity [dB68, p. 55].
Theorem E. Let H(E) be a de Branges space and ϕ a phase function associated with E. Suppose α is a real number and let Γ = {γ k } be the sequence of real numbers such that ϕ( In particular, The following theorem will also be crucial [dB60]: Theorem F. Let µ be a nonnegative measure on R and E some function belonging to HB. Then We write H(E) = H(F ) if H(E) and H(F ) coincide when considered as sets and the norm of H(F ) is equivalent to the norm of H(E). The functions E ∈ HB with the property that PW = H(E) have been characterized in [LS99].

Main theorem
If Λ is a sampling sequence, then there exists a separated subsequence Λ ′ which is also sampling (cf. Lemma 3.11 of [Se95]). Hence without loss of generality, we may restrict our attention to separated sequences Λ.
Theorem 1. A separated sequence Λ of real numbers is sampling for PW if and only if there exist two entire functions E, F ∈ HB such that The following notation will be used repeatedly below: g and g f .
Proof. Let us assume first that Λ is a sampling sequence. This means that the Paley-Wiener space equipped with the norm k |f (λ k )| 2 is a de Branges space. Therefore, Theorem C provides us with a function E ∈ HB such that H(E) = PW and Since the measure µ = k |E(λ k )| 2 δ λ k meets the hypothesis of Theorem F, there exists a bounded holomorphic function A in the upper half-plane with norm A ∞ ≤ 1 and a real number a such that Note that the right-hand side is a holomorphic function defined in the upper half-plane, but the left-hand side is a meromorphic function defined in the whole plane. We denote this meromorphic function by M . The following relationship holds when Im z > 0: The function M − 1 has poles at the points λ k as we can see from (3), and it vanishes whenever E * vanishes. Setting we may therefore write with F an entire function. We also have M * = −M and G * = G, and consequently M + 1 = EF/G. It follows that F * /F = −A in the upper half-plane and F has no zeros in Im z > 0. Since |A| is bounded by 1, it follows that F ∈ HB.
We will now see that G = (EF + E * F * )/2, which means that Λ is the zero sequence of EF + E * F * . We know that G = −M G + EF . If x ∈ R, then G(x) is real and M (x)G(x) is an imaginary number. Thus G is the real part of EF for real z, whence G(z) = (EF + E * F * )/2 for all z ∈ C.
We next prove the converse implication. The function E has no zeros on the real axis since PW = H(E). We may assume that F has no real zeros. If this were not the case, then F = S F , where F ∈ HB without real zeros and S is an entire function, with real zero set Z(S), and moreover S * = S. Therefore Λ = Λ 1 ∪ Z(S) and Λ 1 will be the zero set of E F + E * F * . If we prove that Λ 1 is a sampling sequence, then Λ is a also a sampling sequence.
Given α ∈ (0, π] we let Λ α be the sequence of points λ α,k such that ϕ EF (λ α,k ) = α + kπ, k ∈ Z. Observe that since Λ is the zero sequence of EF + E * F * , and E and F have no real zeros, we have Λ = Λ π/2 . For α = π/2 the sequence Λ α is interlaced with the sequence Λ. Hence since Λ is a separated sequence, Λ α can be expressed as the union of two separated sequences, each with separation constant not smaller than that of Λ. Thus the Plancherel-Pólya inequality [Yo80, implies that with C independent of α. Moreover, for all values of α except possibly one, Theorem E implies that for every g ∈ H(EF ) the following relation holds: .
For every f ∈ H(E) the function g = F f belongs to H(EF ) with f H(E) = F f H(EF ) . Therefore, for every f ∈ PW we have .
Since H(E) = PW, E does not vanish for x ∈ R and therefore by (2) where the constant C does not depend on α. This inequality may fail for one α ∈ [0, π). We may assume it fails for α = π/2, because otherwise we have proved that Λ is sampling. Take a sequence α n → π/2. Then λ αn,k → λ k for all k, and by (4) we may apply Lebesgue's dominated convergence theorem to conclude that the inequality holds also for α = π/2.
If Λ is a complete interpolating sequence, then there exists an E ∈ HB such that H(E) = PW and Λ constitutes the zero sequence of E + E * . This follows from Theorems C and E. Therefore, we may view the function F of Theorem 1 as accounting for the "redundancy" in Λ. There is a trivial case in which this is particularly transparent: Suppose Λ = Λ ′ ∪ (Λ \ Λ ′ ), where Λ ′ is a complete interpolating sequence. (Such a decomposition holds whenever D − (Λ) > 1; cf. Theorem 2.3 of [Se95].) Then the condition of Theorem 1 is met if we choose F to be the generating function of the "redundant" sequence Λ \ Λ ′ , i.e., if we set A somewhat different way of seeing F as measuring the "redundancy" in Λ is as follows. The notion of a complete interpolating sequence may be extended in a natural way: We say that Λ is a complete interpolating sequence for a de Branges space H if the interpolation problem f (λ k ) = a k has a unique solution f ∈ H whenever k |a k | 2 K(λ k , λ k ) < ∞.
Then Theorem 1 says that if Λ is sampling for PW, then Λ is a complete interpolating sequence for H(EF ), and H(E) (with H(E) = PW) is isometrically embedded into H(EF ) by the map f → F f . This interpretation of Theorem 1 has an interesting relation to the result of [Se95] which says that we cannot in general obtain from Λ a complete interpolating sequence for PW by making the sequence thinner: Instead, we can make the space bigger so that Λ becomes a complete interpolating sequence for the bigger space. We note in passing that Theorem 1 solves the following problem of norm equivalence for PW: Which nonnegative measures µ on R have the property that the L 2 (R, dµ)-norm yields a norm for PW equivalent to the standard L 2 (R)-norm? To apply Theorem 1 to this problem, we need a way of associating sampling sequences with such measures. Given a nonnegative measure µ and two positive numbers r, δ, define Λ µ (δ, r) = {kr : k ∈ Z and µ([kr, (k + 1)r)) ≥ δ}.
Then the Bernstein and the Plancherel-Pólya inequalities imply (see [Or98]): Proposition. The L 2 (R, µ)-norm yields an equivalent norm for PW if and only if the following holds: (i) There exists a positive constant C such that µ([x, x + 1)) ≤ C for all x ∈ R.
Problems about norm equivalence for spaces of entire functions of exponential type in one and several variables have been studied by many authors. See, e.g., [Li65] and [LLS92] for an extensive historical account.

Landau's inequality and the John-Nirenberg theorem
The following corollary says that a sampling sequence is "everywhere denser" than some complete interpolating sequence.
Corollary 1. If Λ is a separated sampling sequence for PW, then there exists a complete interpolating sequence Γ = {γ k } ∞ k=−∞ such that for every k ∈ Z there is at least one point λ ∈ Λ such that γ k ≤ λ < γ k+1 .
Proof. If Λ is a sampling sequence, then Λ consists of those points such that ϕ EF (λ) = π/2 + kπ for some k ∈ Z. On the other hand, ϕ E is an increasing function which grows more slowly than ϕ EF . Thus the sequence Γ α which consists of those points such that ϕ E (γ) = α + kπ for some k ∈ Z, has the claimed property. Since H(E) = PW, Theorem E implies that Γ α is a complete interpolating sequence for all values of α except possibly one, mod π. Since the choice of α is at our disposal, the proof is complete.
Let us return to Landau's inequality, which is stated as Theorem B above. To begin with, let us show by an example that the inequality is best possible. Set Λ = {k + log + |k|} k∈Z . This sequence is sampling, because the function is a sine-type function; i.e., |F (z)|e −π| Im z| ≃ 1 for | Im z| > 1. Thus a classical theorem of Levin [HNP81,p. 250] implies that Λ is a complete interpolating sequence. (Alternatively, note that this means v = 0 in Theorem A.) On the other hand, it is clear that Claim. Landau's inequality is a consequence of the John-Nirenberg theorem for BMO functions.
Proof. Suppose Λ is a sampling sequence, and let Γ be an associated complete interpolating sequence as described in Corollary 1. By Corollary 1 and Theorem A, where h ∈ BMO of a special form. (For this proof we will only need the fact that h ∈ BMO.) Set I = [a, b]. Then the triangle inequality in the form Then the John-Nirenberg theorem (cf. [JN61], [Ga81,p. 230 for some sufficiently big A, with C, c absolute positive constants. But h ′ (t) is bounded, in fact equal to −1 for t ∈ Λ, and so if the right-hand side is smaller than the separation distance of Λ, then |h(t) − h I | ≤ A log |I| + B for all t ∈ I. Plugging this into (6), we obtain Landau's inequality.
It should be stressed that we do not pretend to have given an easier proof of Landaus's inequality, as the John-Nirenberg theorem is admittedly a more sophisticated result. However, we find the link between the two results quite intriguing.

Sampling sequences and approximation of subharmonic functions
In Section 3, we explained how Theorem 1 could be interpreted in the trivial case in which there exists a subsequence Λ ′ ⊂ Λ which is a complete interpolating sequence. The purpose of this section is to demonstrate the applicability of our condition when D − (Λ) = 1 and no such subsequence Λ ′ exists.
To obtain positive results from Theorem 1, one needs information about those E ∈ HB for which H(E) = PW. As mentioned at the end of Section 2, such functions are described in [LS99]. However, we will not use this description, which involves a rather delicate statement about the location of the zeros of E. In fact, we will obtain precise results by using only the evident fact that H(E) = PW if |E(z)| ≃ e π Im z for Im z ≥ 0.
We will now demonstrate how our sampling condition is related to the problem of approximating U ψ by log |f |, with f an entire function.
Proof. If we could find e ∈ HB such that we would be done, because then Λ would constitute the zero sequence of ef + e * f * , and |e(z)| ≃ e π Im z for Im z ≥ 0. In general, however, it is impossible to find such an e. Instead, we will appeal to the following perturbation argument: If a separated sequence Γ = {γ k } is sampling, then Γ ′ = {γ k + δ k } is sampling whenever all γ k + δ k are distinct and δ k → 0 as |k| → ∞. (This follows from Lemma III in [DS52] and the elementary fact that Γ is sampling if δ k = 0 except for finitely many integers k.) Thus it is enough to construct an E ∈ HB such that when |x| → ∞ and |E(z)| ≃ e π Im z for Im z ≥ 0, because then the perturbation argument applies with the zero sequence of Ef + E * f * playing the role of Γ and Γ ′ = Λ. We may assume the function ω(x) = x+ψ(x)−ϕ f (x)/π satisfies ω ′ (x) ≃ 1. Partition the real line into a sequence of disjoint intervals I k = [x k , x k+1 ], k ∈ Z, with x 0 = 0, such that I k ω ′ (t)dt = 1 for all k, and choose γ k ∈ I k so that and Γ = {γ k }. Then by Lemma 3 in [OS99], Now choose two monomials P and Q of the same degree and with only real zeros such that B(z) = A(z − 1/2)P (z)/Q(z) is an entire function and the zeros of A and B are interlaced. Then according to a theorem of Meȋman (see [Le80,p. 314]), we have either A − iB ∈ HB or A + iB ∈ HB; we may set E = A − iB and assume E ∈ HB because P may be replaced by −P . It remains to show that E satisfies (8). Consider the sequence of functions A k (z) = A(z − x 2k ). A normal family argument shows that there exists a sequence c k ≃ 1 such that A k (x) − c k cos πx → 0 uniformly on compact subsets of the real line. Similarly, if we set B k (z) = B(z − x 2k ), we obtain that B k (x) − c k sin πx → 0 uniformly on compact subsets of the real line. Then it follows from the construction of E that To solve the approximation problem for U ψ , we shall adapt an approach from a recent paper by Lyubarskii and Malinnikova [LM01], which contains rather conclusive results about the possibility of approximating an arbitrary subharmonic function by log |f | with f an entire function.
To ease the exposition, we set ψ(x) = 0 for x ≤ 0. (It is not difficult to modify the construction below to get a corresponding result when both ψ(∞) = ∞ and ψ(−∞) = −∞.) Let {t n } ∞ n=0 be the sequence such that t 0 = 0 and ψ(t n ) = n, n = 1, 2, 3, ...; set d n = t n − t n−1 . We will say that ψ induces a logarithmically regular partition if d n ≃ d n+1 and We note that ψ(x) = √ x for x ≥ 0 corresponds to the example of [Se95] showing the existence of a sampling sequence containing no subsequence which is a complete interpolating sequence.
Case (i) is proved by means of Corollary 2, while case (ii) is proved by a direct argument. The point of case (ii) is to show that the growth condition ψ ′ (x) = 1/O(x) is critical. Thus Theorem 2 indicates that the condition of Corollary 2 is in a sense close to being necessary.
Proof. We consider first statement (i). According to Corollary 2, we need to show that the assumptions on ψ ensure the existence of a function f ∈ HB such that (7) holds when Im z ≥ 0.
The zeros of f are determined as follows. Define r n ∈ (t n−1 , t n ) by (9) log r n = tn t n−1 log t dψ(t), and then z n = r n e −icdn/rn with c > 0 so small that cd n /r n ≤ π/4 for all n. We choose f in such a way that it satisfies Clearly f belongs to HB. We set V = U ψ − log |f |; it satisfies the following relation: We see that it is enough to prove that the series converges uniformly on compact sets in C, and that V (z) = O(1) for Im z ≥ 0. The proof is a simplified version of an argument from [LM01]. We shall therefore be brief and refer to [LM01] for further details.
Given z ∈ C, we let n(z) be a positive integer n such that r n−1 < |z| ≤ r n . Then if Im z ≥ 0, it is plain that the smoothness of ψ ensures that n(z)+1 n=n(z)−1 j n (z) ≃ 1.
Next let n − (z) be the positive integer n such that r n−1 < |z|/2 ≤ r n , and n + (z) be the positive integer n such that r n−1 < 2|z| ≤ r n . Then for z ∈ C we see that ∞ n=n + (z)+1 j n (z) ≃ 1.
After observing that by (9) we may write we obtain in a similar way. Thus we have in particular established the uniform convergence on compact sets. We set (the set of "essential indices") and split the corresponding sum into two parts, now assuming that Im z ≥ 0.
To estimate V 1 , introduce the function which for each n ∈ N (z) is analytic in a domain containing those ω such that e ω ∈ [t n−1 , t n ]. Therefore, where t = e ω ∈ [t n−1 , t n ] and e ωn = r n . By (9), we obtain A direct estimate gives sup t∈[t n−1 ,tn] |Q n (z, t)| d 2 n |z − r n | 2 .
Using the assumption that ψ induces a logarithmically regular partition, we obtain that V 1 (z) ≃ 1.
To deal with V 2 , we write log |z − r n | − log |z − z n | = Re rn zn dζ ζ − z .
Using again the assumption that ψ induces a logarithmically regular partition, we have also proved that V 2 (z) ≃ 1, and are done with the proof of (10). We consider next statement (ii). Our plan is to construct a sequence of functions f n ∈ PW for which |f n (λ k )| 2 / f n 2 PW → 0. Let t n be the sequence such that ψ(t n ) = n, n = 1, 2, 3, ..., and suppose n is sufficiently big for the following construction to be feasible. We require ξ n ∈ (t n , t n+1 /2) to be such that ψ(ξ n ) = n + 1/2 + ǫ n , where ǫ n will be chosen below. Define a bounded, continuous function φ n such that φ n (t) = −t when |t| < 1/2, φ n (t) = ψ n (t) − n − 1/2 for t n < t < ξ n and φ n (t) = ψ(t) − n − 3/2 for 2ξ n < t < t n+1 , and otherwise φ n is linear. We choose ǫ n such that ξn tn φ n (x) x dx = 0.
The construction of f n is now identical to the construction of the function A in the proof of Corollary 2, with ω ′ replaced by φ ′ n + 1. Thus we obtain f n (z) = lim R→∞ |γ k |<R (1 − z/γ k ) with Γ = {γ k } a separated real sequence and such that |f n (z)|e −Un(z) ≃ min (1, dist(z, Γ)).
This f n is of exponential type π. By (12), f n ∈ PW, but also f n PW → ∞, in view of (11).
Observe that dist(λ k , Γ) → 0 uniformly for λ k ∈ Ω n , when n → ∞, and that γ k = k for all k < 0. Along with (13) and our estimates for f n , this implies that k |f (λ k )| 2 / f 2 PW → 0, so that Λ is not sampling.