Riesz bases of exponentials on multiband spectra

Let $S$ be the union of finitely many disjoint intervals on the real line. Suppose that there are two real numbers $\alpha, \beta$ such that the length of each interval belongs to $Z \alpha + Z \beta$. We use quasicrystals to construct a discrete set of real frequencies such that the corresponding system of exponentials is a Riesz basis in the space $L^2(S)$.

1. Introduction 1.1. Let S be the union of a finite number of bounded intervals on R. We denote by P W S (after Paley and Wiener) the space of all functions f ∈ L 2 (R) whose Fourier transform vanishes almost everywhere outside of S. A discrete set Λ ⊂ R is called a complete interpolation set for P W S if the restriction operator f → f | Λ is a bounded and invertible one from P W S onto ℓ 2 (Λ). In the context of communication theory this means that Λ provides a "stable and non-redundant" sampling of signals with spectrum in S.
It is well-known that the complete interpolation property of Λ is equivalent to the Riesz basis property of the corresponding exponential system If S is a single interval, then a complete description of the Riesz bases E(Λ) in L 2 (S) was given by B. S. Pavlov (1979). Much less is known, however, in the case when S is the union of more than one interval. In fact, it is unknown in general whether an exponential Riesz basis in L 2 (S) exists at all. This existence has been established in the following special cases: (i) S is a finite union of disjoint intervals with commensurable lengths [2,4].
(ii) S is the union of two general intervals [8].
For other results in the subject we refer to the survey paper [4].
In this note we extend the two results above and prove: Theorem 1. Let S be the union of finitely many disjoint intervals on R. Suppose that there are two real numbers α, β such that the length of each interval belongs to Zα + Zβ. Then there is Λ ⊂ R such that E(Λ) is a Riesz basis in L 2 (S).
Here Zα+ Zβ denotes the set of real numbers of the form nα+ mβ (n, m ∈ Z). The above mentioned results are thus obtained as special cases of Theorem 1.
1.2. In fact, we will prove the following more general result.
Theorem 2. Suppose that the indicator function of a set S ⊂ R can be expressed as a linear combination of indicator functions of intervals I 1 , . . . , I N whose lengths belong to Zα + Zβ, that is, where α, β ∈ R. Then there is Λ ⊂ R such that E(Λ) is a Riesz basis in L 2 (S).
Sets with the structure (1) form a wider class than unions of disjoint intervals with lengths in Zα + Zβ. For example, one may take an interval with "holes" obtained by the removal of disjoint sub-intervals, where the interval and its sub-intervals have their lengths in Zα + Zβ.
A result of similar type in the periodic setting was obtained in [3].

Quasicrystals. Duality.
Our approach is inspired by the papers [6,7] due to Matei and Meyer, who introduced the usage of so-called 'simple quasicrystals' in order to construct "universal" sets of sampling or interpolation for P W S spaces. Here we will use simple quasicrystals to construct complete interpolation sets for spectra S with the structure (1).
Following [6,7] we let Γ be a lattice in R 2 . Consider the projections p 1 (x, y) = x and p 2 (x, y) = y, and assume that the restrictions of p 1 and p 2 to Γ are injective. Let Γ * be the dual lattice, consisting of all vectors γ * ∈ R 2 such that γ, γ * ∈ Z, γ ∈ Γ.
Let S be the union of disjoint semi-closed intervals, and be a single semi-closed interval. Define In some sense, the quasicrystals Λ(Γ, I) and Λ * (Γ, S) are dual to each other. This duality was observed and used by Matei and Meyer in connection with sampling and interpolation. In the present context of exponential Riesz bases the duality can be formulated as follows. (ii) E(Λ * (Γ, S)) is a Riesz basis in L 2 (I).
The proof of Lemma 1 is along similar lines as in the paper [7] (Sections 6-7), but in our case there is an additional point concerned with the requirement that the intervals in (2) and (3) should be semi-closed. Indeed, the significance of this point is clarified once keeping in mind that the Riesz basis property of the exponential systems E(Λ(Γ, I)) and E(Λ * (Γ, S)) is not preserved upon either the addition or removal of any element.
For a proof of the duality lemma in the periodic setting see [3, Section 2].

Proof of Theorem 2
There is no loss of generality in assuming that the numbers α, β are linearly independent over the rationals. Moreover, by rescaling we may restrict ourselves to the case when α is an irrational number, and β = 1.
3.1. Define a lattice Γ = {(n(1 + α) − m, m − nα) : n, m ∈ Z}, and let I = [0, mes S) be an interval whose length coincides with the Lebesgue measure of S. We will prove that the exponential system E(Λ(Γ, I)) is a Riesz basis in L 2 (S). According to Lemma 1 it will be sufficient to show that the system E(Λ * (Γ, S)) is a Riesz basis in L 2 (I).
It is easy to check that the set Λ * (Γ, S) may be partitioned as follows, (where some of the sets Λ n may be empty). Let {s n } be a sequence of integers such that s n − s n−1 = #Λ n , and choose an enumeration {λ j , j ∈ Z} of the set Λ * (Γ, S) such that Λ n = {λ j : s n−1 j < s n } (n ∈ Z).
In order to prove that E(Λ * (Γ, S)) is a Riesz basis in L 2 (I) it will be sufficient, by a theorem of Avdonin [1], to check that the following three conditions hold: 3.2. We will next show that (b) holds. Consider the bounded, piecewise constant function This function is 1-periodic and hence may be viewed as a function on the circle group T = R/Z. The assumptions (1), (2) and (4) now provide the following representation for φ.
Given j there is n = n(j) such that λ j ∈ Λ n , or equivalently, such that s n−1 j < s n . Then It thus follows from (5), (7) and (8) that sup j |δ j | < ∞, which confirms condition (b) above.

It remains to establish (c).
In order to show that (6) holds we will first obtain a simple expression for the sum δ j where j goes through the interval s n−1 j < s n . Indeed, We evaluate each one of the sums S 1 (n), S 2 (n) separately. First we observe that by (5), Secondly, by a direct calculation and using (7) and (8) we find that We conclude that for an appropriately defined function τ : T → R (bounded and piecewise continuous) we have 3.4. Now we can finish the proof of (c) above. Given a ∈ Z and a positive (large) integer N there are n = n(a) and r = r(a, N ) such that s n−1 a < s n , s n+r−1 a + N < s n+r .
Since the sequence {δ j } is bounded and due to (9) we have uniformly with respect to n. Since (7) and (8) imply that N = r mes S + O(1), we get with the o(1) uniform with respect to a. This implies (c), and so Theorem 2 is proved.

Remarks
We have constructed Riesz bases of exponentials with real frequencies for multiband spectra subject to the diophantine condition (1). We would like though to comment on one result which is not covered by our theorems above. In the paper [8] existence of such Riesz bases was proved under certain nondiscrete conditions on the lengths of the gaps between the intervals. The restrictions obtained are rather severe; however, the result indicates that diophantine restrictions are not necessarily natural ones in the problem.
We also refer the reader to the paper [5] where the authors construct, for any finite union of intervals, a Riesz basis of exponentials with complex frequencies lying in a horizontal strip along the real axis.