Exponential bases on two dimensional trapezoids

We discuss existence and stability of Riesz bases of exponential type of L^2(T) for special domains T called trapezoids. We construct exponential bases on L^2(T) when T is a finite union of rectangles with the same height. We also generalize our main theorems in dimension d\ge 3.


Introduction
The study of Riesz bases and frames in Hilbert spaces is a fruitful topic of investigation since many decades. One of the central problems arising in many applications is the question of whether a family of exponential functions B = {e iλm.x } m∈Z with x = (x 1 , ..., x d ) and λ n ∈ C d , forms a Riesz basis of L 2 (D) or not. Here, D is a domain of R d , i.e., a bounded and measurable set of finite measure. If that is the case, we say that B is an exponential basis of L 2 (D).
When D is an interval of the real line, the problem is well studied since when Paley and Wiener explored the possibility of non harmonic Fourier series (see [PW]) but much less is known when D is a general domain of R n .
In this paper we construct Riesz bases and frames on a class of domains in R 2 that we call Trapezoid.
Let f : [a, b] → R be a bounded and measurable function. Assume also that f (y) > l > 0 for almost every y ∈ [a, b] (that is, with the possible exception of a set of measure zero). The trapezoid bounded by f is the set T = {(x, y) : −f (y) < x < f (y), a ≤ y ≤ b}. (1.1) The trapezoids bounded by step functions are special and significant. We say that a bounded and positive step function is regular if it is constant on intervals of equal length. We say that a trapezoid is a multi-rectangle if it is bounded by a regular step function. So, a multi-rectangle is a union of a finite family rectangles [−b j , b j ] × [y 0 + jh, y 0 + (j + 1)h), with h, b j > 0 and j = 1, ..., N . We also say that this multi-rectangle has N steps of height h > 0.
In Section 3 we use regular step functions to construct Riesz bases of L 2 of trapezoids, in the sense specified by the following is a Riesz basis for L 2 (T ).
A function f is piecewise continuous if it is continuous everywhere except at a finite number of points. The regular step functions are the "next best thing" after the constants, and Theorem 1.1 allows to construct a Riesz basis of L 2 (T ) in the form of {e i (λn(y),µm), (x,y) } n,m∈Z , with λ n (y) a regular step function. Theorem 1.1 is a corollary of the following Theorem 1.2. Let f (y) : [a, b] → R be bounded and measurable and such that f (y) ≥ l > 0 in [a, b]. Let T be the trapezoid bounded by f . Let {h n (y)} n∈N be a family of bounded and measurable functions such that for almost every y ∈ [a, b] and for every n ∈ N. Then, is a Riesz basis for L 2 (T ). The constant 1 4 cannot be replaced by any larger constant. This Theorem can be viewed as a Kadec stability theorem on trapezoids. We state Kadec's theorem and we compare it with Theorem 1.2 in Section 2.3.
In section 4 we show that exponential bases of L 2 of multi-rectangles can be obtained from exponential bases of L 2 of the union of finite families of disjoint and bounded intervals of R.
Exponential bases on the union of disjoint intervals can be found in special cases (see the discussion in Section 2.2). We use a theorem of N. Lev (see Theorem 2.5) to prove the following Theorem 1.3. Let f (y) = αy m + βy m ′ , where m and m ′ are non negative integers and α, β ∈ R are such that f (y) > 0 for every y ∈ [0, 1]. Let s n : [0, 1] → R: Let R n be the trapezoid bounded by s n . Then, L 2 (R n ) has an exponential basis for every n ≥ 1.
The plan of this paper is the following. In section 2 we state some preliminary definitions and results. In Section 3 we prove Theorems 1.1 and 1.2. In Section 4 we prove Theorem 1.3. We have stated some remarks and conjectures in Section 5.

Frames and Riesz bases
Here, || || and are the norm and the inner product in H. Unless otherwise specified, we will assume that the v j 's are normalized, that is, ||v j || = 1. A and B are called lower and upper frame bounds. We say that B is tight if A = B.
Definition. We say that B is a Bessel sequence if only the upper bound inequality holds in (2.6). If only the lower bound inequality holds in (2.6), B is called a Riesz-Fisher sequence.
Definition. We say that B is a Riesz basis if one of the following equivalent conditions is satisfied i) B is an exact frame (that is, it ceases to be a frame when any one of its elements is removed).
ii) B is complete and there exist constants A, B > 0 so that for every finite sequence of scalars {c 1 , c 2 , ... c n }, iii) B is the image of an orthonormal basis of H through a bounded invertible operator.
iv) There exists an orthonormal basis {w j } of H (called the dual basis of {v n }) such that every f ∈ H can be represented in a unique way as f = ∞ j=1 f, w j v j .
By Plancherel's theorem, an orthonormal Riesz basis is also a tight frame with bounds A = B = 1.
Definition. We say that B is a Riesz sequence if it satisfies ii) without the assumption of completeness. So, B is a Riesz basis of the Hilbert space Span(B), but may not be a a Riesz basis of H.
If H = L 2 (D), with D a domain in R d , a Riesz basis, (resp. frame) of L 2 (D) in the form of B = {e iλn.x }, with x = (x 1 , ..., x d ) and λ n ∈ C d , is called an exponential basis (resp. exponential frame) of L 2 (D).
Frames can be considered over-complete Riesz bases, and Riesz sequences can be viewed as under-complete Riesz bases, but it is not true that an exponential Riesz basis can always be extracted from an exponential frame of L 2 (D), or that an exponential Bessel sequence can always be completed to an exponential Riesz basis of L 2 (D). K. Seip proved in [S2] that there are frames {e iλnx } of L 2 (−π, π) from which it is not possible to extract a Riesz basis of L 2 (−π, π), and also Riesz sequences {e iλnx } that cannot be completed to a Riesz basis of L 2 (−π, π) with the addition of terms of the form of e iµnx .
The following proposition implies that it is very easy to find exponential frames of L 2 (D) for every domain D ⊂ R n .
Proposition 2.1. Let P be a domain in R n , and let F = {v n } n∈Z be a frame for L 2 (P ) with constants A and B. a) For every domain D ⊂ P , the sequence F | D of the restrictions of the v n 's to D forms a frame for L 2 (D) with the same frame bounds A and B. In particular, if F is an orthonormal basis of where by , P we mean the inner product on L 2 (P ). By definition off , and so we have proved that F | D is a frame for L 2 (D). If F is an orthonormal Riesz basis of The following proposition shows an easy construction of an exponential Bessel sequences on a multi-rectangle.
Proposition 2.2. Let R be a multi-rectangle with N steps of height h. Let 2bh be the measure of the largest step. Then, for every Proof. After a translation and a dilation, we can assume, without loss of generality, that is an orthonormal basis in L 2 ((−1, 1) × [j, j + 1)) for every integer j, and so, by Proposition 2.1, it is a tight frame of L 2 (R j ) with constants A = B = 1 for every By the elementary inequality ( All the inner products are in L 2 (R), but since the f j have support in R j , we can consider them in L 2 (R j ). By the frame inequality,

Exponential bases of L 2 of the union of segments
Proposition 2.1 shows that it is easy to find exponential frames of L 2 (D), but it is often very difficult, and sometimes impossible, to extract exponential Riesz bases from them. Here, we discuss some special and significant cases for which the answer is known. a) for every 0 < a < b. From this particular frame, we can always extract a Riesz basis of L 2 (−a, a), as specified by the following Lemma 2.3 is equivalent to a Lemma proved by K. Seip in [S2]. In the same paper the Author shows that it is always possible to construct an exponential basis of L 2 of the union of two intervals, as specified by the following Theorem 2.4. Let I 1 = (a 1 , b 1 ) and I 2 = (a 2 , b 2 ) be intervals of the real line, with −∞ < a 1 < b 1 ≤ a 2 < b 2 < ∞. There exists a sequence of integers {n k } k∈Z so that the set e 2πin k b 2 −a 1 k∈Z is an exponential basis of L 2 (I 1 ∪ I 2 ).
The main theorem in [S2] is about the construction of a Riesz basis of L 2 of the union of a finite family of disjoint intervals of finite measure. We call such sets multi-segment. In [S2] the author construct a Riesz basis {e 2πiλn N } λn∈Z on a multi-segment of diameter N without making any assumption on the length of the intervals, but with strong assumptions on the gaps between them.
N. Lev proved in [L] that an exponential basis on a multi-segment exist whenever the length of the intervals are in a special Diophantine form.
Theorem 2.5. Let I be the union of finitely many disjoint intervals on R. Suppose that there are two real numbers α and β such that the length of each interval belongs to αZ + βZ. Then L 2 (I) has an exponential basis.
To the best of our knowledge, the problem of finding an exponential basis on L 2 of a general multi-segment is not solved yet.

Stability of Riesz bases
Riesz bases are stable, in the sense that a small perturbation of a Riesz basis produces a Riesz basis. Let us recall Paley-Wiener stability theorem, and the celebrate Kadec's stability theorem for exponential bases of L 2 (−a, a). The proof of both theorems can be found e.g. in [Y]. Kadec's theorem was originally proved in [Ka].
for some constant 0 < λ < 1 and all choices of scalars c 1 , ... c n . Then {y n } n∈N is a Riesz basis for H.
Note that if {x n } n∈N is an orthonormal basis of H, and j c 2 j = 1, then the right hand side of (2.7) equals to λ. A powerful generalization of Kadec's theorem is proved in [A]. An extension of Kadec's theorem to exponential Riesz bases of L 2 of parallelepipeds in R n is in [SZ].

3
Bases of exponentials for L 2 of trapezoids.
In this section we prove Theorem 1.2 and its corollary, Theorem 1.1. We start with an easy construction of a Riesz basis of L 2 (T ). It is easy to verify that L is an invertible isometry. Since C = (2(b − a)) − 1 2 e πinx+ 2πiky b−a n,k∈Z is an orthonormal basis of L 2 (R), the set is an orthonormal basis of L 2 (T ).
We change the order of the summation and use triangular inequality. n,k∈Z We use a change of variables and Plancherel theorem to estimate A. and since δ n → 1 − sin x x is increasing when x ∈ [0, π] and we have assumed that f (y)δ n < L, we obtain We argue in a similar way to show that are the partial fraction expansions of the function 1 L − cot L and tan L respectively. Hence, we have proved that and since |L| < π 4 , the right hand side is < 1, as required.
Observe that since we have assumed |f (y)| ≤ 1, (3.9) follows if we prove that By our assumptions on f , also 1/f (y) is positive and bounded in (0, 1). If f is continuous, then 1/f (y) is continuous too, and so it is uniformly continuous in [0, 1]. If we let ǫ = 1 4n , there exists δ = δ(n) > 0 so that 1 f (y 1 ) − 1 f (y) < 1 4n if |y 1 − y| < δ. So, if we fix δ < δ, we partition [0, 1] with N = N (n) intervals of length δ, and we let y j = j/N , we can see at once We can define a step function s n (y) which is ≡ f (j/N ) if j N < y < j+1 N for every 0 ≤ j ≤ N −1. Clearly, the s n 's satisfy (3.9). Since sup y∈(0,1) we have proved that the s n 's converge uniformly to f , as required. ✷

Bases of L 2 of multi-rectangles
In this section we construct Riesz bases of special multi-rectangles. We recall that a multirectangle is a trapezoid bounded by a regular step function. First of all, we show that the construction of an exponential basis on L 2 of a multi-rectangle can be reduced to the construction of an exponential basis on L 2 of a multi-segment of R, i.e., a union of a finite family of disjoint intervals of finite measure. Let a 1 , ..., a N > 0 and h > 0; let R = ∪ N j=1 (−a j , a j ) × (jh + y 0 , (j + 1)h + y 0 ) be a multi-rectangle. After a translation and a dilation, we can assume y 0 = 0 and h = 1. We can also translate R by v = (a 1 , 0), and consider insteadR = ∪ N j=1 (−a j + a 1 , a j + a 1 ) × (j, j + 1). So, the first rectangle inR is R 1 = (0, 2a 1 ) × (0, 1). After a dilation, we can assume that max j≤N {a j + a 1 } = 1. We let I = ∪ N j=1 (−a j + a 1 + 2j, a j + a 1 + 2j). The intervals of I do not intersect because for every j ≥ 0, a j + a 1 + 2j ≤ −a j+1 + a 1 + 2(j + 1) ⇐⇒ a j + a 1 ≤ −a j+1 + a 1 + 2 ⇐⇒ (a j + a 1 ) + (a j+1 + a 1 ) ≤ 2a 1 + 2 and since a j + a 1 ≤ 1 for every j, the last inequality is verified.
Theorem 4.1. Let R and I be defined as above. Suppose that L 2 (I) has an exponential basis. Then L 2 (R) has an exponential basis as well.
Proof. Clearly, R has an exponential basis if and only ifR has an exponential basis. We let for simplicity a j − a 1 = a ′ j and a j + a 1 = a ′′ j , so thatR = ∪ N j=1 (a ′ j , a ′′ j ) × (j, j + 1) and I = ∪ N j=1 (a ′ j + 2j, a ′′ j + 2j). Assume a ′′ N = 1 (the proof of the theorem can be easily modified if this is not the case). With this position, the diameter of I is 2N + 1. Let T : L 2 (R) → L 2 (I × (0, 1)) it is easy to verify that T is a linear and invertible isometry, and so, by Lemma 3.1, if B is a basis of L 2 (I × (0, 1)) then T −1 (B) is a basis of L 2 (R).
Let N ′ = 2N + 1, and let {e 2πiλn x N ′ } n∈Z be a basis of L 2 (I); for every n ∈ Z, we chose a sequence {µ n(m) } m∈Z in such a way that (4.11) Clearly, (4.11) is verified if j(2λ n − µ n(m) ) is an integer multiple of N ′ for every j; we can chose µ n(m) = {2λ n } + N ′ m, where {2λ n } is reminder of the division of 2λ n by N ′ , and observe that, for every n ∈ Z, the sequence {e Proof. Let {v ′ n (x)} n∈Z (resp. {w ′ n(m) (y)} m∈Z ) be the dual basis of {v n (x)} in L 2 (D) (resp. the dual basis of {w n(m) (y)} in L 2 (E)). Let f (x, y) ∈ L 2 (D×E). Then, f n (y) = f, v ′ n D ∈ L 2 (E), and so it can be written in a unique way as ∞ m=−∞ f n , w ′ n(m) E w n(m) (y). By Fubini's theorem, f n , w ′ n(m) E = f, v ′ n D , w ′ n(m) E = f, v ′ n w ′ n(m) D×E , and so, as required. ✷ Our Theorem 4.1 reduced the problem of finding an exponential basis on a multi-rectangle to the problem of finding an exponential basis on a multi-segment I. The proof of Theorem 1.3 is now an easy consequence of Theorem 4.1 and Theorem 2.5.
By Theorem 2.5, a basis of the multi-segment formed by the disjoint union of the translates of the (−f (k/n), f (k/n)) exists, and by Theorem 4.1 the multi-rectangle bounded by s n has an exponential basis as well. ✷

Remarks and open problems
We have constructed exponential bases of L 2 (R), where R is a special multi-rectangle, but can we find exponential bases when R is a union of a generic family of disjoint rectangles? In our Theorem 4.1 we strongly use the fact that the rectangles in the R are all of the form of (−b j , b j ) × (j, j + 1), and we have tied up the construction of an exponential basis of L 2 (R) to the construction of an exponential basis of L 2 of a multi-segment. Our construction does not seem to work well for general multi-rectangles. Complex analysis methods have been often used in these problems. We cite for example the recent paper by J. Marzo [M], where the Author proves the existence of a Riesz basis of exponentials on a finite union of congruent cubes of R n by finding complete interpolating sequences in a suitable Paley-Wiener space. It seems to us that the proof in [M] cannot be easily generalized when the cubes are replaced by parallelepiped, but it is worthwhile remarking that the distances between the cubes have no importance for the proof.
We are also wondering if the construction of exponential Riesz bases on a family of multirectangles R n that approximate a trapezoid T , in the sense that the measure of the symmetric difference of R n and T goes to zero when n → ∞ could lead to the construction of an exponential basis of L 2 (T ). We feel that this should be possible, and we plan to pursue this investigation in a subsequent paper.