Exponential bases on rectangles of R d

Transcription

1 Exponential bases on rectangles of R d Laura De Carli Abstract. We produce explicit exponential bases on finite unions of disjoint rectangles of R d with rational vertices. 1. Introduction Exponential bases are important to represent square integrable functions on domains of R d. Let D R d be bounded and measurable; an exponential basis in L 2 (D) is a Riesz basis in the form of E(Λ) = {e 2πi λ, x } λ Λ, where Λ R d, and, is the standard inner product in R d (see section 2 for definitions and properties of Riesz bases). If E(Λ) is an exponential basis, every function in L 2 (D) has a unique and stable representation in terms of the functions in E(Λ) with coefficients that are easy to calculate. Unfortunately, our understanding of exponential bases is still very incomplete. There are very few examples of domains in which it is known how to construct exponential bases, and no example of domain for which exponential bases can be shown not to exist (see [20], [12] and the references cited there). The existence of exponential bases on finite unions of disjoint intervals of the real line and disjoint rectangles in R d was recently proved by G. Kozma and S. Nitzan in [13] and [14]. See also [24] and [6]. Explicit exponential bases are not produced in these papers. In this paper we construct exponential bases on multi-rectangles with vertices in Q d, i.e, on finite unions of disjoint rectangles [a 1, b 1 )... [a d, b d ), with a j < b j Q. After perhaps a translation and dilation of coordinates (see Corollary 5.1) we restrict our attention to multi-rectangles with vertices in ( Z)d, the lattice of points in R d with half-integer coordinates. These multi-rectangles can be written as disjoint unions of translated of the unit cube Q 0 = [ 1 2, 1 2 )d. We let (1.1) Q = Q 1... Q N, Q p = M p +Q 0, with M p Z d, Mp M q if p q Mathematics Subject Classification. 42C15, 42C30. 1

2 2 LAURA DE CARLI We find exponential bases of L 2 (Q) in the form of (1.2) B = N {e 2πi n+ δ j, x } n Z d where δ p = (δ p,1,..., δ p,d ) R d. The idea of combining exponential bases on individual rectangles to form a basis on their disjoint union is not new. See e.g. the introduction of [13] and [14] and [23, Sect. 4]. Our main result is the following Theorem 1.1. B is a Riesz basis of L 2 (Q) if and only if the matrix Γ = {e 2πi δ j, M p } 1 j,p N is nonsingular. The optimal frame constants of B are the maximum and minimum eigenvalues of Γ Γ, where Γ is the conjugate transpose of Γ. The optimal frame constants of B can be explicitly evaluated when N = 2 (see Corollary 5.4); the case N > 2 is discussed in Section 6. Exponential bases or frames on finite unions of unit intervals of the real line have been studied by several authors. See [2], [13] and [23, Sect. 4] and the references cited in these papers. The main theorem and Remark 4 in [2] imply that E = N {e2πi(n+δ j)x } n Z is a Riesz basis on E = [m 1, m 1 + 1)... [m N, m N +1), with m 1 <... < m N Z, if the matrix z{e 2πiδ jm p } 1 j,p N is nonsingular. In general, it is not true that an exponential frame contains a Riesz basis or that an exponential Riesz sequences can be completed to a Riesz basis (see [29]). But on L 2 (Q), sets of exponentials (1.2) enjoy the following remarkable properties: Theorem 1.2. Let B as in (1.2). The following are equivalent B is a Riesz sequence on L 2 (Q) B is a frame B is a Riesz basis. The following is a special and significant case of Theorem 1.1. Theorem 1.3. Let δ R d be fixed, and let (1.3) S(δ) = N {e 2πi n+(j 1) δ, x } n Z d. S(δ) is a Riesz basis of L 2 (Q) if and only if, for every 1 p q N, (1.4) M p M q, δ Z. The frame constants of S(δ) are the maximum and minimum eigenvalue of the matrix B = { β p,q } 1 p,q N, where sin(πn δ, M q M p ) (1.5) βp,q = sin(π δ, M q if p q, M p ) N if p = q

3 EXPONENTIAL BASES ON RECTANGLES IN R d 3 Theorem 1.3 shows that the set of δ R d for which S(δ) is not an exponential basis on L 2 (Q) is union of inverse images of Z under the linear applications δ δ, Mq M p, with 1 p q N, and so it has measure zero in R d. The proof of Theorem 1.1 relies on the semigroup properties and precise norm estimates of a remarkable family of linear operators on l 2 (Z d ) that we will discuss in section 3. When d = 1, these operators have been introduced and investigated in [5]; versions of these operators appear also in [2]. The rest of the paper is organized as follows. In Section 4 we prove our main results. We have collected a number of corollaries and examples in Section 5. In Section 6 we estimate the frame constants of B and S(δ). 2. Preliminaries 2.1. Bases and frames. We have used [18] for standard linear algebra results, and the excellent textbook [9] for definitions and properties of bases and frames in Hilbert spaces. See also [4] and [31]. A sequence of vectors V = {v j } j N in a separable Hilbert space H is a frame if there exist constants A, B > 0 such that for every w H, (2.1) A w 2 w, v j 2 B w 2. Here,, and =, are the inner product and the norm in H. The sequence V is a tight frame if A = B, it is a Parseval frame if A = B = 1, and a Riesz sequence if the following inequality is satisfied for all finite sequences {a j } j J C. (2.2) A 2 a j 2 a j v j B a j 2. j J j J j J A Riesz basis is a frame and a Riesz sequence, i.e., a set of vectors that satisfies (2.1) and (2.2). If H = L 2 (D), with D R d of finite Lebesgue measure D, Riesz bases (or frames) made of exponential functions are especially relevant in the applications. An exponential basis of L 2 (D) is a Riesz basis in the form of E(Λ) = {e 2πi λ, x } λ Λ, where Λ is a discrete set of R d and x D. Because frames are not necessarily linearly independent, they are often more easily constructible than bases. For example, when D Q 0 = [ 1 2, 1 2 )d we can easily verify that E(Z d ) is a Parseval frame on D. See also [6, Prop. 2.1]. The construction of exponential frames on unbounded sets of finite measure is a difficult problem that has been recently solved in [26]. Many different exponential bases can be produced on L 2 (D) when D is a rectangle in R d. For example, the set {e 2πi(λn 1 x λ nd x d ) } (n1,...,n d ) Z d is

4 4 LAURA DE CARLI an exponential basis on Q 0 = [ 1 2, 1 2 )d if and only if the sets {e 2πiλn j x j } nj Z are exponential bases on [ 1 2, 1 2 ) (see [30, Lemma 2.1]). It is proved in [22] that E(Λ) is an orthonormal basis on L 2 (Q 0 ) if and only if the sets {Q 0 + λ} λ Λ tiles R d, i.e: λ Λ (Q 0 + λ) = R d and (Q 0 + λ j ) (Q 0 + λ i ) = 0 whenever λ i λ j. When D is arbitrary, L 2 (D) may not have orthogonal exponential bases; we say that D is spectral if an orthogonal exponential basis exists in L 2 (D). The connection between tiling and spectral properties of domains in R d is deep and fascinating and has spur intense investigation since when B. Fuglede formulated his famous tiling spectral conjecture in [7]. See also [11] and the references cited there. Non-orthogonal exponential bases in L 2 ( 1 2, 1 2 ) were first investigated by Paley and Wiener [27] and Levinson[21] and extensively studied by several other authors. A complete characterization of exponential bases on L 2 ( 1 2, 1 2 ) was given by Pavlov in [28]. To the best of our knowledge, no complete characterization of exponential bases on L 2 (Q 0 ) exists in the literature. Riesz bases are stable, in the sense that a small perturbation of a Riesz basis produces a Riesz basis. The celebrated Kadec stability theorem states that any set {e 2πi(n+ɛn)x } n Z where ɛ n < 1 4 is Riesz basis of L2 ( 1 2, 1 2 ). The constant 1 4 cannot be replaced by any larger constant. See [10] or [31]. The following multi-dimensional version of Kadec theorem is in [30]. In the theorem below we have let v = sup 1 j d v j. Theorem 2.1. Let Λ = {λ n = (λ n,1,..., λ n,d ) R d } n Z d in R d for which be a sequence λ n n L < 1 4 whenever n Z d. Then, E(Λ) is an exponential basis of L 2 (Q 0 ). The constant 1 4 cannot be replaced by any larger constant Three useful lemmas. Lemma 2.2. Let V be a Riesz basis in H; the optimal constants A and B in the inequalities (2.1) and (2.2) are the same. Proof. Follows from [4, Proposition 3.5.5]. The following lemma is in [14]. Lemma 2.3. E(Λ) E(Z d ) is a frame on a domain D Q 0 if and only if E(Z d Λ) is a Riesz sequence on Q 0 D. The following is Proposition in [4] Lemma 2.4. A sequence of unit vectors V H is a Parseval frame if and only if it is an orthonormal Riesz basis.

5 EXPONENTIAL BASES ON RECTANGLES IN R d 5 From Lemma 2.4 follows that an exponential basis of L 2 (D) is orthognal if and only if it is a tight frame of L 2 (D) with frame constant D. 3. Families of isometries in l 2 (Z d ) The proof of Theorem 1.1 in dimension d = 1 lead to the investigation of a one-parameter family of operators {T t } t R defined in l 2 = l 2 (Z) as follows: sin(πt) a n if t Z (3.1) (T t ( a)) m = π m n + t n Z ( 1) t a m+t if t Z. When t is not an integer (and in particular when t ( 1, 1)), these operators can be viewed as discrete versions of the Hilbert transform in L 2 (R) (see [5] and also [17]). The main result in [5] is the following: Theorem 3.1. The family {T t } t R defined in (3.1) is a strongly continuous Abelian group of isometries in l 2 ; the discrete Hilbert transform (H( a)) m = 1 a n π m n is its infinitesimal generator. n Z n m That is, we proved that T s T t = T s+t ; that for every a l 2, the application t T t ( a) is continuous in R; that T t ( a) l 2 = a l 2; and finally that, for every a l 2 T, lim t( a) a t 0 t = H( a), where the limit is in l 2. In this paper we define multi-variable versions of the operators T t. We let l 2 (Z d ) = { a = {a n } n Z d : a 2 l 2 (Z d ) = n Z d a n 2 < }. We denote with e j the vector in R d whose components are all = 0, with the exception of the j th component which is = 1. Let j d and s R; we define the operator T s ej : l 2 (Z d ) l 2 (Z d ) as follows: sin(πs) a (n1,..., n j 1, m, n j+1,...n d ) if s Z (3.2) (T s ej ( a)) n = π n j m + s m Z ( 1) s a (n1,..., n j +s,...n d ) if s Z. For a given t = (t 1,..., t d ) R d, we let T t = T t 1 e 1... T td e d. For example, when t (0, 1) d we can see at once that (T t ( a)) m = d sin(πt j) π d n Z d a n d (m j n j +t j ). Theorem 3.1 and the definition of T t yield the following Theorem 3.2. For every d, t R d and every a l 2 (Z d ), (3.3) T d T t ( a) = T t T d ( a) = T t+ d ( a); furthermore (3.4) T 1 ( a) = T t t ( a),

6 6 LAURA DE CARLI and We also prove the following Theorem 3.3. a) Let T t T d ( a) l 2 (Z d ) = a l 2 (Z d ). be the adjoint of T t. Then, T t ( a) = T t ( a). b) For every t R d and every a, b l 2 (Z d ) Proof. Observe that T t T t ( a), T t ( b) = a, b. = (T t1 e 1... T td e d ) = T t d e d... T t 1 e 1. By Lemma 4.4 in [5], the expression above equals T td e d... T t1 e 1 = T t, and a) follows. The second part of the theorem follows from a) and (3.4). 4. Proofs Let Q = Q 1... Q N be as in (1.1). By (2.1) and (2.2), the set B = N p=1 {e2πi x( n+ δ p) } n Z d is a Riesz basis in L 2 (Q) if and only if there exists A, B > 0 for which the following hold: N (4.1) A f 2 L 2 (Q) n Z d f, e 2πi n+ δj, x 2 L 2 (Q) B f 2 L 2 (Q) whenever f L 2 (Q), and (4.2) A a j, n 2 a j, n e 2πi n+ δj, x n Z d n Z d for every set of coefficients { a j, n } j N n Z d set of n Z d. 2 L 2 (Q) B n Z d a n j 2 C which are 0 for all but a finite To prove Theorem 1.1 we need the following important Lemma 4.1. Let Q M = Q 0 + M, with M R d. Let a, b l 2 (Z d ) and t, s R d. Let T t be as in Section 3; we have (4.3) a n e 2πi n+ s, x, b m e 2πi m+ t, x L 2 (Q M ) n Z d m Z d = e 2πi s t, M T s ( a), T t ( b) l 2 (Z d ).

7 EXPONENTIAL BASES ON RECTANGLES IN R d 7 Proof. Assume d = 1 since the proof for d > 1 is similar. We have n Z a n e 2πi (n+s)x, = n,m Z b m e 2πi(m+t)x L 2 (M 1 2,M+ 1 2 ) m Z a n bm ˆ M+ 1 2 = e 2πi(s t)m M 1 2 n,m Z 2πi(s t)m sin(π(s t)) = e π e 2πi (n m+s t)x dx a n bm sin(π(n m + s t)) π(n m + s t) n,m Z = e 2πi(s t)m T s t ( a), b l 2 (Z). Theorem 3.3 and the identity (3.3) yield (4.3). a n bm n m + s t 4.1. Proof of Theorem 1.1. We prove (4.1). Let f L 2 (Q). For every j N, we obtain f, e 2πi n+ δj, x L 2 (Q) 2 2 = f, e 2πi n+ δ j, x L n Z d n Z 2 (Q p) d p=1 = f, e 2πi n+ δ j, (4.4) x L 2 (Q p) f, e 2πi n+ δ j, x L 2 (Q q). n Z d p,q=1 Observe that ˆ f, e 2πi n+ δ j, x L 2 (Q p) = f(x)e 2πi n+ δj, x dx Q 0 + M ˆ p = f(y + M p )e 2πi n+ δj, y+ Mp dy Q 0 ˆ = e 2πi δ j, M p τ Mp f(y)e 2πi n+ δj, y dy Q 0 (4.5) = e 2πi δ j, M p τ Mp f, e 2πi n+ δ j, x L 2 (Q 0 ) where τ v f(x) = f(x + v). Since the set {e 2πi n+δj,x } n Z d is an orthonormal basis on L 2 (Q 0 ), by Parseval identity and (4.5) the sum in (4.4) equals e 2πi δ j, M q M p τ Mp f, e 2πi n+ δ j, x L 2 (Q 0 ) τ Mq f, e 2πi n+ δ j, x L 2 (Q 0 ) n Z d p,q=1 (4.6) = e 2πi δ j, M q M p τ Mp f, τ Mq f L 2 (Q 0 ) p,q=1

8 8 LAURA DE CARLI and from (4.4) and (4.6) follow that (4.7) where we have let (4.8) β p,q = n Z d f, e 2πi n+ δj, x L 2 (Q) 2 = p,q=1 e 2πi δ j, M q M p, 1 p, q N. β p,q τ Mp f, τ Mq f L 2 (Q 0 ) Let B be the N N matrix whose elements are the β p,q. We can let γ j,p = e 2πi δ j, M p and write β p,q = N γ j,pγ j,q ; thus, B = Γ Γ where Γ = {γ j,p } 1 j,p N is as in the statement of Theorem 1.1. The maximum (minimum) of the Hermitian form B v, v on S N 1 C, the unit sphere of C N, equal the maximum (minimum) eigenvalue of B. Let Λ = max x S N 1 Bx, x, and λ = min x S N 1 Bx, x. We have p,q=1 ˆ β p,q τ Mq f, τ Mp f L 2 (Q 0 ) = ˆ Λ Q 0 p=1 Q 0 p,q=1 β p,q f(x + M p )f(x + M q )dx f(x + M ˆ p ) 2 dx = Λ f(x) 2 dx = Λ f 2 L 2 (Q). Q p Similarly, we prove that N p,q=1 β p,q τ Mq f, τ Mp f L 2 (Q 0 ) λ f 2 L 2 (Q). From (4.7) and the inequalities above follows (4.1), with A = λ and B = Λ. We show that λ and Λ are optimal. Let w = (w 1,..., w d ) be an eigenvector of B with eigenvalue Λ; let g(x) = N q=1 w qχ Q0 M q (x), where χ D (x) is the characteristic function of D. The identity (4.7) yields ˆ g, e 2πi n+ δj, x L 2 (Q) 2 = β p,q τ Mp g(x)τ Mq g(x)dx n Z d Q 0 p,q=1 = β p,q ω k ω k χ Q0 M ˆQ h (x + M p )χ Q0 M k (x + M q )dx 0 = p,q=1 h,k=1 p,q=1 β p,q w p w q = Λ p=1 w p 2 = Λ g 2 L 2 (Q). p=1 The proof is similar for the lower bound. This argument shows also that B is a frame in L 2 (Q) if and only if det B 0. We now prove (4.2). Let a 1,..., a N l 2 (Z d ) be such that a j, n = 0 for all but a finite set of n Z d. Let S j = n Z d a j, n e 2πi n+ δj, x. The central sum in

9 EXPONENTIAL BASES ON RECTANGLES IN R d 9 (4.2) equals S S N 2 L 2 (Q). By Lemma 4.1 S S N 2 L 2 (Q) = S i, S j L 2 (Q) (4.9) where we have let (4.10) α i,j = 1 i,j N = 1 i,j N p=1 S i, S j L 2 (Q p). = α i,j T δi ( a i ), T δj ( a j ) L 2 (Z d ) p=1 e 2πi δ i δ j, Let A be the N N matrix whose elements are the α i,j. It is easy to verify that A = ΓΓ, where Γ is as in the statement of Theorem 1.1 and Γ is the conjugate transpose of Γ. The matrices A = ΓΓ and B = Γ Γ have the same eigenvalues (see Lemma 4.2 below) and for every v C N, (4.11) λ v 2 Mp α i,j v i v j Λ v 2 i, where Λ and λ are as in the first part of the proof. Let T δk ( a k ) = b k. In view of (4.9), N (4.12) S S N 2 L 2 (Q) = α i,j b i, b j l 2 (Z d ) = Fix n Z d ; by (4.11), λ( b 1, n b N, n 2 ) i, n Z d i, α i,j b i, n b j, n. α i,j b i, n b j, n Λ( b 1, n b N, n 2 ) i, and when we sum with respect to n we obtain (4.13) λ b j 2 n Z d i, Recalling that b j = T δj ( a j ) and that the T δj α i,j b i, n b j, n Λ b j 2. are invertible isometries, we obtain b j l 2 (Z d ) = a j l 2 (Z d ). From (4.12) and (4.13) follows that λ N a j 2 S S N 2 L 2 (Q) Λ a j 2. and (4.2) is proved with B = Λ and A = λ. By Lemma 2.2, these are the optimal constants in (4.2). The proof of Theorem 1.1 is concluded.

10 10 LAURA DE CARLI Remark. The proof of Theorem 1.1 shows that B is a frame det B 0 det Γ 0 det A 0 B is a Riesz sequence This observation proves Theorem 1.2. For the sake of completeness, we prove the following easy Lemma 4.2. Let M be a square matrix. The matrices MM and M M have the same eigenvalues. Proof. Let λ be an eigenvalue of MM with eigenvector v. We have MM v = λ v, and if we multiply this identity to the right by M, we obtain (M M)M v = λm v. Thus, λ is an eigenvalue of M M with eigenvector M v Proof of Theorem 1.3. Let S(δ) be as in (1.3). By Theorem 1.1, S(δ) is a Riesz basis of L 2 (Q) if and only if the matrix Γ whose entries ( are γ j,p = e 2πi M p, (j 1) δ is nonsingular. But γ j,p = e 2πi M p, δ ) j 1, and Γ is a Vandermonde matrix, i.e., a matrix with the terms of a geometric progression in each row. Its determinant is det Γ = ( ) e 2πi M p, δ e 2πi M q, δ. Therefore, S(δ) is a Riesz basis if and only if e 2πi M p, δ e 2πi M q, δ 0 whenever p q, or if and only if M p M q, δ Z. By Theorem 1.1 and (4.8), the frame constants of S(δ) are the minimum and maximum eigenvalue of Γ Γ = B = {β p,q } 1 p,q N, where β p,q = e 2πi(j 1) δ, M q M p = 1 e2πin δ, M q M p 1 e 2πi δ, M q M p =e πi(n 1) δ, M q M p sin(πn δ, Mq M p ) sin(π δ, Mq M p ). If B is the matrix whose elements are β p,q = sin(πn δ, M q M p ) sin(π δ, M q M when p q p ) and β p,q = N when p = q, and v R d, we have B v, v = B w, w, where w = (e πi(n 1) δ, M 1 v 1,..., e πi(n 1) δ, M N v N ). Since w = v, we can easily verify that the maximum and minimum of the Hermitian forms v B v, v and w B w, w on S N 1 C are the same; consequently, the maximum and minimum eigenvalue of B and B are the same. This concludes the proof of the theorem. Remark. It is interesting to observe that

11 EXPONENTIAL BASES ON RECTANGLES IN R d 11 det Γ 2 = det B = e 2πi M p, δ e 2πi M q, δ 2 = 2 N(N 1) 2 (1 cos(2π M p M q, δ )) = 2 N(N 1) sin 2 (π M p M q, δ ). 5. Corollaries and examples In this section we prove a number of corollaries of Theorems 1.1 and 1.3. We use the following notation: if v = (v 1,..., v d ) R d is a given vector and A is a nonempty subset of R d, we let D v x = D (v1,...v d ) x = (v 1 x 1,..., v d x d ), and D v (A) = {D v x : x A}. Let n, k Z, with k > 0; we denote with [n] k (or simply with [n] when there is no risk of confusion), the remainder of the division of n by k Multi-rectangles in Q d and a stability theorem. Let R be a multi-rectangle with vertices in Q d. Let l 1,..., l d be the smallest positive integers for which R = D (l1,...,l d ) (R) has vertices in Z d. Let L = N l j. The multi-rectangle R is a union of N = L R disjoint unit cubes with vertices in Z d. If B as in (1.2) is an exponential basis of L 2 ( R), a simple scaling in (4.1) and (4.2) proves the following Corollary 5.1. B = N {e2πi n+ δ j, x } n Z d is a Riesz basis of L 2 ( R) with constant A and B if and only if B = N {e 2πi D l ( n+ δ j ), x } n Z d is a Riesz basis for L 2 (R) with constants A L and B L. We use Corollary 5.1 to prove the following interesting Corollary 5.2. Let N 2, and let ɛ 0,..., ɛ N 1 R. The set B = {e 2πi(m+ɛ [m] N )x } m Z is a Riesz basis of L 2 (0, 1) if and only if, for every 0 i j N 1, (5.1) We prove first the following i j + ɛ i ɛ j N Z. Lemma 5.3. Let I R be an interval of length N 1. The set U = N 1 j=0 {e2πi(n+δj)x } n Z is a Riesz basis on L 2 (I) if and only if δ i δ j Z. The optimal frame constants of U are the maximum{ and minimum eigenvalues sin(πn(δi δ j )) of the matrix à = { α i,j } 1 i,j N, where α i,j = sin(π(δ i δ j )) if i j, N if i = j.

12 12 LAURA DE CARLI Proof. Without loss of generality, we can let I = [ 1 2, N 1 2 ) = N 1 p=0 [p 1 2, p+ 1 2 ); by Theorem 1.1, U = N 1 j=0 {e2πi(n+δj)x } n Z is a Riesz basis of L 2 (I) if and only if the matrix Γ = {e 2πipδ j } 0 j,p N 1 is nonsingular. But e 2πipδ j = ( e 2πiδ j) p, and so Γ is a Vandermonde matrix; thus, det Γ = (e 2πiδ i e 2πiδ j ) 0 δ i δ j Z. 0 i<j=n 1 By Theorem 1.1, the frame constant of U are the maximum and minimum eigenvalue of the matrix A = {α i,j } 1 i,j N, with α i,j = N 1 p=0 e2πi(δ i δ j )p (see (4.10)). We can easily verify that α i,j = e πi(n 1)(δ i δ j ) sin(πn(δ i δ j ) sin(π(δ i δ j ) when i j, and α i,j = N when i = j; if we argue as in the second part of the proof of Theorem 1.3, we can conclude that the frame constants of U are the maximum and minimum eigenvalue of the matrix à defined above. for sim- Proof of Corollary 5.2. We let [m] N = [m] and s k = k+ɛ k N plicity of notation. By Corollary 5.1, B = {e 2πi(m+ɛ [m])x } m Z = N 1 j=0 {e2πi(nn+j+ɛ j)x } n Z = N 1 j=0 {e2πin(n+s j)x } n Z is a Riesz basis of L 2 (0, 1) if and only if the set B = N 1 j=0 {e2πi(n+sj)x } n Z is a Riesz basis of L 2 (0, N); by Lemma 5.3, this is equivalent to s i s j Z whenever i j, which is (5.1). Example 1. Let s (0, 1), and let U = {e 2πi(m+µm)x } m Z where µ m = s when m is even and µ m = s when m is odd. With the notation of Corollary 5.2, we can let N = 2, ɛ 0 = s and ɛ 1 = s; by (5.1), U is a Riesz basis of L 2 (0, 1) if and only if 2s 1 2 Z, or 0 < s < 1 2. It is interesting to compare Example 1 with a famous example by Ingham. In [15, p. 378] it is proved that U is not a Riesz basis of L 2 (0, 1) when µ m = 1 4 if m > 0, µ m = 1 4 if m < 0 and µ 0 = 0. Ingham s example shows that the constant 1 4 in Kadec s theorem cannot be replaced by any larger constant. See also [31] Two cubes in R d. Let Q 1 = τ M1 (Q 0 ), Q 2 = τ M2 (Q 0 ) be disjoint unit cubes with vertices in ( Z)d. We prove the following Corollary 5.4. a) The set B = {e 2πi n+ δ 1, x } n Z d {e 2πi n+ δ 2, x } n Z d is a Riesz basis of L 2 (Q 1 Q 2 ) if and only if M 1 M 2, δ 1 δ 2 Z. The optimal frame constants of B are A = 2(1 cos(π M 1 M 2, δ 1 δ 2 ) ), B = 2(1+ cos(π M 1 M 2, δ 1 δ 2 ) ).

13 EXPONENTIAL BASES ON RECTANGLES IN R d 13 b) B is an orthogonal Riesz basis if and only if 2 M 1 M 2, δ 1 δ 2 Z and M 1 M 2, δ 1 δ 2 Z. Proof. After a translation, we can let Q 1 = Q 0, Q 2 = τ M (Q 0 ), with M = M 2 M 1. By Theorem 1.1, B is a Riesz basis if and only if the matrix ( ) 2, 1 + e A = 2πi M, δ 1 δ e 2πi M, δ 1 δ 2, 2 is nonsingular. The eigenvalues of A are the solution of the characteristic polynomial, det(a si) = (2 s) e 2πi M, δ 1 δ 2 2 = 0 where I is the identity. We can easily verify that det(a si) = s 2 4s + 4 sin 2 (π M, δ 1 δ 2 ) = 0 s = 2(1 ± cos(π M δ 1 δ 2 ) ). Thus, λ = 2(1 cos(π M δ 1 δ 2 ) ) and Λ = 2(1 + cos(π M δ 1 δ 2 ) ) are the optimal frame constants of B. When cos(π M δ 1 δ 2 ) = 0, i.e. when M, δ 1 δ 2 is an odd multiple of 1 2, B is a tight frame with constants λ = Λ = 2. Since the functions in B have norm = 2, Lemma 2.4 implies that B is orthogonal. Remark. The union of two disjoint unit cubes with vertices in Z d tiles R d by translation, and by Corollary 5.4, it is a spectral domain in R d. It is proved in [16] that the union of two disjoint intervals of nonzero length is a spectral domain if and only it tiles the real line by translation. To the best of our knowledge, the analog of the main theorem in [16] has not been proved (or disproved) for couples of rectangles in R d Spectral domains in R d. In this section we show examples of multi-rectangles that are spectral domains. We let Q = Q 1... Q N R d as in (1.1) and S( δ) = N 1 {e2πi n+(j 1) δ, x } n Z d be as in (1.3). We prove the following Corollary 5.5. S( δ) is an orthogonal basis of L 2 (Q) if and only if, for every p q, (5.2) M p M q, δ Z and N M p M q, δ Z. Proof. Assume that (5.2) holds. The first condition implies that S( σ) is a Riesz basis; the second condition that the matrix B in the statement Theorem 1.3 is diagonal. Thus, S( σ) is a tight frame with frame constant N = Q, and by Lemma 2.4, it is an orthogonal basis of L 2 (Q). Conversely, assume that S(δ) is orthogonal. The maximum and minimum eigenvalues B equal N, the frame constant of S(δ). Since C N has a

14 14 LAURA DE CARLI basis of eigenvectors of B, we can infer that B v = N v for every v C N and that B is diagonal. Recalling the elements of B are as in (1.5), (5.2) follows. Example 2. Let Q be as in (1.1), with N d. Assume that Q N = Q 0 and that the M j are linearly independent. We show that Q is a spectral domain. Let M be the matrix whose rows are M 1,..., MN 1. By assumption, M has rank N 1. We can find σ R d that satisfies M j, σ = j, j = 1,..., N 1. N By Corollary 5.5, S( σ) is an orthogonal basis of L 2 (Q) Extracting Riesz bases from frames. Let T be the side of the smallest cube that contains Q, i.e., T = sup p,q N M p M q. After perhaps a translation, we can assume that Q [0, T ) d. In [14] it is proved that a basis of L 2 (Q) can be extracted from E((T 1 Z) d ), an orthogonal basis of [0, T ) d Note that E((T 1 Z) d ) is an exponential frame of L 2 (Q), and E((T 1 Z) d ) = {e 2πi T (n 1x n d x d ) } (n1,...n d ) Z d = T j 1,...,j d =0{e 2πi((m 1+ j 1 T )x (m 1 + j 1 T )x d ) } (m1,...m d ) Z d T j=0{e 2πi((m 1+ j T )x (m 1 + j T )x d) } (m1,...m d ) Z d = S( T 1 ) where we have let a 1 = ( 1 a,..., 1 a ). Our Theorem 1.3 allows to prove a related result. Corollary 5.6. There exists an integer L T such that S( L 1 ) is a Riesz basis of L 2 (Q) Proof. Let m p,q = M p M q, 1, with 1 p q N. If the m p,q are not divisible by T, we can chose L = T. Otherwise, we can let L be the smallest positive integer for which no m p,q is divisible by L. By Theorem 1.3, S( L 1 ) is a Riesz basis of L 2 (Q). Note that if m p,q 0 for every p q, S( t 1 ) is a Riesz basis of L 2 (Q) for every irrational t. Corollary 5.7. Let T be as in Corollary 5.6 and let L T be an integer. Then, S( L 1 ) is a Riesz basis of L 2 (Q) if and only if E( L 1 Z d ) S( L 1 ) is a Riesz basis on L 2 ([0, L) d Q). Proof. Follows from Lemma 2.3 and Theorem 1.2.

15 EXPONENTIAL BASES ON RECTANGLES IN R d 15 Example 3. We construct an exponential basis of L 2 (Q) that is also a Riesz basis of a rectangle of measure N = Q. Let e 1 = (1, 0,..., 0). Observe that S( 1 N e 1) = N 1 j=0 {e2πi (n 1+ j N )x 1+...n d x d } n Z d is an exponential basis on R = [0, N) [0, 1)... [0, 1). If M p M q, e 1 is not divisible by N when p q, by Theorem 1.3 we can conclude that S( 1 N e 1) is an exponential basis on L 2 (Q). If that is not the case, we can chose η R d, with η 1 4, for which M p M q, e 1 + η is divisible by N only when p = q. By Theorem 1.3 S( δ) is a Riesz basis of L 2 (Q) when δ = 1 N ( e 1 + η); by Theorem 2.1, S( δ) is also a Riesz basis of L 2 (R). 6. Estimating the frame constants Let Q be as in (1.1); let B as in (1.2) be a a Riesz basis on L 2 (Q) with optimal frame constants 0 < λ Λ. By Theorem 1.1 Λ and λ are the maximum and minimum eigenvalues of the matrices A = {α i,j } 1 i,j N and B = {β p,q } 1 p,q N, with (6.1) α i,j = e 2πi δ i δ j, Mp, β p,q = p=1 e 2πi M p M q, δ j as defined in (4.10) and (4.8). When B = S(δ) is as in (1.3), λ and Λ are the maximum and minimum eigenvalues of the matrix B = { β p,q } 1 p,q N defined in (1.5). Gershgorin theorem provides a powerful tool for estimating the eigenvalues of complex matrices. It states that each eigenvalue of a square matrix M = {m i,j } 1 i,j n is in at least one of the disks D j = {z C : z m j,j R j }, and in at least one of the disks D j = {z C : z m j,j C j }, where R j (resp. C j ) are the sum of the off-diagonal elements of the j th row (column) of M, i.e. n n (6.2) R j = m j,i, C j = m i,j. i=1 i j See [8], and also [25, pg. 146] and [3]. Observe that if m j,j > max{r j, C j } for every j, (i.e., if M is diagonally dominant), then M is nonsingular. The following refinement of Gershgorin theorem is in [1]. Theorem 6.1. Let M be an Hermitian matrix with eigenvalues λ 1,..., λ n. Let R j = C j be as in (6.2). We have i=1 i j max {m j,j R j } max λ j max {m j,j + R j } 1 j n 1 j n 1 j n (6.3) min 1 j n {m j,j R j } min 1 j n λ j min 1 j n {m j,j + R j }.

16 16 LAURA DE CARLI We can use Theorem 6.1 to estimate the optimal frame constants of B and S(δ). Theorem 6.2. Let B be a Riesz basis of L 2 (Q). Let r i = 1 4 sin 2 (π δ i δ j, Mp M q ) 1 j N j i N 2 (6.4) ρ p = q N q p N 2 p,q=1 1 2 sin 2 (π δ i δ j, Mp M q ). i, i<j The optimal frame constants of B satisfy (6.5) N(1 min 1 i N 1 p N b) If B = S(δ), we have {r i, ρ p } λ Λ N(1 + max {r i, ρ p }) 1 i N 1 p N (6.6) N(1 min s p) λ Λ N(1 + max s p) 1 p N 1 p N where s p = sin(πn δ, M q M p ) N sin(π δ, M q M p ). 1 q N q p Proof. a) The optimal frame constants of B are the maximum and minimum eigenvalues of the matrices A and B. In view of (6.1), it is easy to verify that 1 1 j N j i α i,j = 1 j N j i = 1 j N j i = 1 j N j i = Nr i. N + 2 N + 2 N 2 4 p,q=1 1 2 cos(2π δ i δ j, Mp M q ) (1 2 sin 2 (π δ i δ j, Mp M q ) p,q=1 p,q=1 sin 2 (π δ i δ j, Mp M q ) We can prove that 1 q N β p,q = Nρ p in a similar manner. q p Since A and B have the same eigenvalues, we can apply Theorem 6.1 with m j,j = N and R i = Nr i or R i = Nρ i, and (6.5) follows.,

17 EXPONENTIAL BASES ON RECTANGLES IN R d 17 b) When B = S(δ) we can apply Theorem 6.1 to the matrix B; by (1.5), (6.6) follows. Corollary 6.3. a) Let B, λ and Λ be as in in Theorem 6.2. If ( ( ) ) (6.7) min sin 2 (π δ i δ j, Mp M N 1 a 2 q ) 1 1 p,q N 2(N 1) N 1 for some 0 < a < 1, we have b) Let B = S(δ), with min 1 p,q N an λ Λ (2 a)n. sin(π δ, M q M p ) N 1 N(1 a). Then, Na λ Λ N (2 + a). Proof. a) Let s = min 1 p,q N sin(π δ i δ j, Mp M q ). By (6.4) and ( ) (6.7), r j (N 1) 1 2(N 1) 1 N s2 2 1 a. A similar inequality holds for ρ p. By Theorem 6.2, part a) of the corollary is proved. For part b), we let s = of Theorem 6.2 b), s p 1 q N q p min sin(π δ, M q M p ). With the notation 1 p,q N 1 N sin(π δ, M q M p ) N 1 Ns. By (6.6), the proof of the corollary is concluded References [1] Anderson, N.; Best, G. A Gerschgorin-Rayleigh inequality for the eigenvalues of Hermitian matrices. Linear and Multilinear Algebra 6 (1978/79), no. 3, [2] Bezuglaya, L.; Katsnelson, Y. The sampling theorem for functions with limited multiband spectrum. Z. Anal. Anwendungen 12 (1993), no. 3, [3] Brualdi, R.; Mellendorf, S. Regions in the complex plane containing the eigenvalues of a matrix. Amer. Math. Monthly 101 (1994), no. 10, [4] Christiansen, O. An Introduction to Frames and Riesz Bases, Birkhäuser, [5] De Carli, L.; Shaikh Samad, S. One-parameter groups of operators and discrete Hilbert transforms, arxiv: (2015) [6] L. De Carli and A. Kumar, Exponential bases on two dimensional trapezoids, Proc. Amer. Math. Soc. 143 (2015), no. 7, [7] Fuglede, B. Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974), [8] Gerschgorin, S. Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk 6, , [9] Heil, C. A basis theory primer, Appl. Num. Harm. Analysis, Birkhäuser (2011). [10] Kadec, M.I. The exact value of the Paley-Wiener constant, Sov. Math. doklady, 5, , (1964). [11] Kolountzakis, M. The study of translational tiling with Fourier Analysis. Fourier Analysis and Convexity, Birkhäuser, 2004.

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