Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group

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Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group S. Arati, R. Radha ∗ Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India Received 16 April 2018; received in revised form 20 August 2018; accepted 13 September 2018 Communicated by Henk Broer

Abstract Let G be a second countable locally compact abelian group. The aim of this paper is to characterize the left translates on the Heisenberg group H(G) to be frames and Riesz bases in terms of the group Fourier transform. c 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Frames; Heisenberg group; Locally compact abelian group; Riesz basis; Shift invariant spaces; Twisted translates

1. Introduction The study of shift invariant spaces has been of utmost interest and, over the last two decades, it has been researched upon largely along two perspectives, one being from the application point of view where these spaces are applied to the problems of sampling, interpolation and approximation theory and the other, from the theoretical point of view involving various characterizations of frames and Riesz bases associated with these spaces. In fact, the system of translates has been studied widely, starting from the real line to various group settings. We are interested in studying the frames and Riesz bases on the shift invariant space associated with the abstract Heisenberg group introduced by Weil. ∗ Corresponding author.

E-mail addresses: [email protected] (S. Arati), [email protected] (R. Radha). https://doi.org/10.1016/j.indag.2018.09.001 c 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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In [2], Bownik studied the system of translates for a shift invariant space with countably many generators and characterized frame sequences and Riesz sequences in terms of the range function. These results were later generalized to a locally compact abelian group by Cabrelli and Paternostro in [4] and independently by Kamyabi Gol and Raisi Tousi in [10]. Later, many of these results were extended to nonabelian group setting. Radha and Kumar studied shift invariant spaces for a compact nonabelian group in [15]. In [1], Barbieri et al. considered the polarized Heisenberg group and studied the characterization of frame sequences and Riesz sequences. In [6], Currey et al. generalized some results of [2] to shift invariant spaces associated with a class of nilpotent Lie groups. Recently, the shift invariant spaces associated with twisted translates in L 2 (Cn ) were studied by Radha and Adhikari in [14]. They also studied the problem of characterizing frames and Riesz sequences associated with shift invariant spaces having countably many mutually orthogonal generators on the Heisenberg group in [13]. In 2015, Bownik considered translation-invariant spaces which allow translation by real numbers instead of only integers as is the case of shift invariant spaces and characterized these spaces in terms of range function in [3]. In 1964, A. Weil [17] introduced the Heisenberg group H(G) for a locally compact abelian group G. In particular, he considered G to be an ad`ele group or the additive group of vector space over a local field in the context of quadratic forms. On the group H(G), Radha and Shravan Kumar recently studied Weyl multipliers for L p spaces in [16]. In this paper, our interest is to study characterizations of frames and Riesz bases for a system of translates on H(G). In order to study shift-invariant spaces on H(G), we first consider the ˆ We shall explain the reason behind this by taking twisted shift invariant spaces on G × G. n G = R . In many problems on the Heisenberg group Hn , whose underlying manifold is Rn × Rn × R, an important technique is to take the partial Fourier transform in the t variable to reduce the study to the case of R2n . In particular, for f, g ∈ L 1 (Hn ), the convolution of f and g on Hn is defined to be ∫ ( f ∗ g)(z, t) = f ((z, t)(w, s)−1 )g(w, s)dwds. Hn

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This group convolution on Hn can be reduced to R2n as a non-standard convolution, known as twisted convolution. For f, g ∈ L 1 (R2n ), the twisted convolution of f and g is defined to be ∫ ( f × g)(z) = f (z − w)g(w)eπi I m(z·w) dw. R2n

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If we define f ♮ (z, t) = e−2πit f (z), then one can show that f ♮ ∗ g ♮ = ( f × g)♮ . Further, for f, g ∈ L 1 (Hn ), one has ˆ f ∗ g(λ) = ˆ f (λ)ˆ g (λ), λ ∈ R∗ , as in the case of Euclidean Fourier transform. This leads to W ( f × g) = W ( f )W (g), where ∫ W( f ) = f (x, y)π (x, y)d xdy R2n

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is the Weyl transform of f ∈ L 1 (R2n ). Thus in order to study the left translations on ˆ and obtain the the Heisenberg group H(G), we consider the twisted translations on G × G characterization theorems in terms of the kernel of the Weyl transform. Later we look into the characterization theorems for the left translations on the Heisenberg group H(G) in terms of the group Fourier transform. The following definitions are used in the course of the paper. Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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Let H be a separable Hilbert space.

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Definition 1.1. A sequence { f k : k ∈ Z} of elements in H is a frame for H if there exist constants A, B > 0 such that ∑ A∥ f ∥2 ≤ |⟨ f, f k ⟩|2 ≤ B∥ f ∥2 , ∀ f ∈ H.

2 3 4

k∈Z

The numbers A and B are called frame bounds. If the right hand side inequality holds, then { f k : k ∈ Z} is said to be a Bessel sequence with bound B. If the frame bounds satisfy A = B = 1 then the frame is called a Parseval frame. A sequence { f k : k ∈ Z} in H is said to be a frame sequence if it is a frame for span{ f k : k ∈ Z}. Definition 1.2. Let { f k : k ∈ Z} be a frame for H. The operator ∑ S : H → H, S f = ⟨ f, f k ⟩ f k

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9

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k∈Z

is called the frame operator. It is bounded, linear and invertible. The canonical dual frame of { f k : k ∈ Z} is the frame {S −1 f k : k ∈ Z} for H. Definition 1.3. A Riesz basis for H is a family of the form {U ek : k ∈ Z}, where {ek : k ∈ Z} is an orthonormal basis for H and U : H → H is a bounded invertible operator. Alternatively, a sequence { f k : k ∈ Z} is a Riesz basis for H if { f k : k ∈ Z} is complete in H, and there exist constants A, B > 0 such that ∑ ∑ ∑ A |ck |2 ≤ ∥ ck f k ∥2 ≤ B |ck |2 k

k

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k

holds for every finite scalar sequence {ck }. A sequence { f k : k ∈ Z} in H is a Riesz sequence if it is a Riesz basis for span{ f k : k ∈ Z}. For a study on frames, we refer to [5] and [9]. Definition 1.4 (See [3]). Let G be a second countable locally compact abelian group and Γ be its subgroup. Then the collection {Tγ ϕ : γ ∈ Γ } for ϕ ∈ L 2 (G) is said to be a continuous Riesz sequence if for any continuous function a on Γ with compact support, there exist bounds 0 < A ≤ B < ∞ such that ∫ 2   2 2  A∥a∥ L 2 (Γ ) ≤  a(γ )Tγ ϕdµΓ (γ )  ≤ B∥a∥ L 2 (Γ ) .

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Γ

Definition 1.5. A closed subspace V ⊂ L 2 (R) is said to be shift invariant if for every f ∈ V we also have Tk f ∈ V for any k ∈ Z, where Tu f (x) = f (x − u) is the translation by u ∈ R. In particular, for ϕ ∈ L 2 (R), the shift invariant space V (ϕ) = span{Tk ϕ : k ∈ Z} is called the principal shift invariant space. We wish to remark here that many of the results pertaining to twisted translations on the ˆ generalize those on R2n and so we do not mention them in detail. However, abstract group G × G such is not the case with the left translations on the Heisenberg group. These results interestingly turn out to be totally non-trivial and form the main essence of our paper. 2. Fourier and Weyl transforms in connection with H(G) Let G be a second countable locally compact abelian group which admits a countable uniform lattice. In general, a locally compact abelian group need not contain a uniform lattice. However, Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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in the case of G being compactly generated, the existence of uniform lattices in G is assured (Remark 2 in [11]). Further, we assume that for a non-zero integer j, the map x ↦→ j x is an automorphism on G. This assumption is made only to obtain an explicit Plancherel formula for ˆ denote the dual group of G. Let H(G) = G × G ˆ × T denote A. Weil’s H(G) as in (2.6). Let G Heisenberg group where the group operation is defined by (x, γ , θ ) · (x ′ , γ ′ , θ ′ ) = (x + x ′ , γ + γ ′ , θ θ ′ γ ′ (x)). It is well known from Stone–von Neumann theorem that every infinite dimensional irreducible unitary representation of H(G) is unitarily equivalent to ρ j , j ∈ Z \ {0}, where ρ j (x, γ , θ )ψ(y) = θ j (γ (y)) j ψ(x + y), ψ ∈ L 2 (G). We refer to Folland [7] for further details. For f ∈ L 1 (H(G)), the Fourier transform fˆ of f is given by the operator fˆ( j), j ∈ Z \ {0} on L 2 (G) defined as ∫ fˆ( j)ψ = f (x, γ , θ )ρ j (x, γ , θ )ψdµG (x)dµGˆ (γ )dµT (θ ), ψ ∈ L 2 (G), ˆ G×G×T

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ˆ and T. The above where dµG (x)dµGˆ (γ )dµT (θ) is a product of the Haar measures on G, G integral is a Bochner integral taking values in the Hilbert space L 2 (G). Further, the operator fˆ( j) is a bounded operator on L 2 (G) satisfying ∥ fˆ( j)∥B(L 2 (G)) ≤ ∥ f ∥ L 1 (H(G)) . For j ∈ Z, let f j given by ∫ j f (x, γ , θ )θ j dµT (θ ) (2.1) f (x, γ ) = T

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ˆ denote the inverse Fourier transform of f ∈ L 1 (H(G)) in the θ variable. Then f j ∈ L 1 (G × G). j ˆ Using f , f can be written as ∫ ∫ fˆ( j)ψ = f j (x, γ )ρ j (x, γ , 1)ψdµGˆ (γ )dµG (x), ψ ∈ L 2 (G). (2.2) G

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ˆ G

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ˆ the operator W j (g) on L 2 (G) is defined as For g ∈ L (G × G), ∫ W j (g)ψ = g(x, γ )ρ j (x, γ , 1)ψdµG (x)dµGˆ (γ ), for ψ ∈ L 2 (G). ˆ G×G

Then it follows from (2.2) that fˆ( j) = W j ( f j ). The bounded operator W j (g) can further be expressed as ∫ W j (g)ψ(y) = K gj (u, y)ψ(u)dµG (u), G ∫ where K gj (u, y) = g(u − y, γ )(γ (y)) j dµGˆ (γ ). (2.3) ˆ G

j

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Thus W j (g) is an integral operator on L 2 (G) with kernel K g : G × G → C. In particular when j = 1, W j (g) is denoted by W (g) which is called the Weyl transform of g and the associated kernel is denoted by K g . As in the case of the Euclidean Fourier transform, the definitions of ˆ respectively through the fˆ and W j can be extended to functions in L 2 (H(G)) and L 2 (G × G) ˆ we have density argument. For g ∈ L 2 (G × G), ∫ K gj (u, y) = g(u − y, γ )γ (− j y)dµGˆ (γ ) = F2 g(u − y, − j y), ˆ G

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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where F2 denotes the Fourier transform with respect to the second variable and so 2 ∥K gj ∥ L 2 (G×G)

−1

= (C j,G )

1

∥g∥2L 2 (G×G) ˆ,

2

where C j,G is a constant dependent on j and G such that C j,G dµG ( j −1 x) = dµG (x), for j j ∈ Z \ {0} and C0,G is taken to be zero. Thus K g ∈ L 2 (G × G) and W j (g) is a Hilbert–Schmidt ˆ we have operator on L 2 (G). For g ∈ L 2 (G × G), ∥W j (g)∥2B (L 2 (G)) 2

=

2 ∥K gj ∥ L 2 (G×G)

= (C j,G )

−1

∥g∥2L 2 (G×G) ˆ.

⟨W j (g), W j (h)⟩B2 (L 2 (G)) =

∥

2

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j K h ⟩ L 2 (G×G)

= (C j,G ) ⟨g, h⟩ L 2 (G×G) ˆ. −1

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Further, the Fourier transform fˆ of f ∈ L 2 (G) satisfies 2 fˆ( j)∥B2 (L 2 (G))

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ˆ On polarization, we have for g, h ∈ L 2 (G × G), ⟨K gj ,

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j

= ∥W j ( f j )∥B2 (L 2 (G)) = ∥K f j ∥

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2 L 2 (G×G)

(2.4)

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2

= (C j,G )−1 ∥ f j ∥ L 2 (G×G) ˆ. ˆ where f 0 (x, γ ) is as in (2.1). Define h on Let f 0 (x, γ , θ ) = f 0 (x, γ ), ∀ θ ∈ T, (x, γ ) ∈ G × G, 0 2 0 H(G) by h = f − f . Then h ∈ L (H(G)) since f ∈ L 2 (H(G)) and it can be easily seen that { j f (x, γ ) , j ̸= 0 j h (x, γ ) = 0 , j = 0.

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Now we shall show that there is an isometry f − f 0 ↦→ fˆ from L 2 (H(G)) into l 2 (Z \ {0}, B2 (L 2 (G)); C j,G ), where l 2 (Z \ {0}, B2 (L 2 (G)); C j,G ) denotes the space of all sequences indexed in Z \ {0}, taking values in B2 (L 2 (G)) and square summable with weight C j,G . In fact ∑ ∑ 2 2 2 ∥ f j ∥ L 2 (G×G) ∥ fˆ( j)∥B2 (L 2 (G)) C j,G = ∥ fˆ∥l 2 (Z\{0},B2 (L 2 (G));C j,G ) = ˆ j̸=0

j̸=0

=

∑∫ ∫ j̸=0

G

2

ˆ G

=

∫ ∫ ∑

=

∫ ∫ ∑

G

G

ˆ G j̸=0

ˆ G j∈Z

| f j (x, γ )| dµGˆ (γ )dµG (x) 2

| f j (x, γ )| dµGˆ (γ )dµG (x) 2

|h j (x, γ )| dµGˆ (γ )dµG (x).

Thus

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∥

2 fˆ∥l 2 (Z\{0},B2 (L 2 (G));C j,G )

=∥f −

2 f 0 ∥ L 2 (H(G)) .

(2.5)

ˆ × T, in order to arrive at the Remark 2.1. Since the group under consideration is H(G) = G × G Plancherel formula for the group Fourier transform, one needs to consider the one-dimensional ˆ in addition to representations of H(G) namely, the characters of the abelian group G × G, the infinite dimensional irreducible unitary representations. Hence in this case, the Plancherel formula is given as follows. For f ∈ L 2 (H(G)), ∫ ∫ 2 2 |(ˆ f 0 )(χ , y)| dµG (y)dµGˆ (χ). (2.6) ∥ f ∥2L 2 (H(G)) = ∥ ˆ f ∥l 2 (Z\{0},B2 (L 2 (G));C j,G ) + ˆ G

G

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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However, in the case of functions f − f 0 , for f ∈ L 2 (H(G)) which has mean value zero in the central variable, the Plancherel formula reduces to (2.5). For Plancherel formula in a more general setting, we refer to [12]. Let L be a countable uniform lattice in G. In other words, L is a countable discrete subgroup of G such that G/L is compact. Then the annihilator L ⊥ of L given by ˆ : γ (x) = 1, ∀ x ∈ L} L ⊥ = {γ ∈ G ˆ In fact, L ⊥ is a uniform lattice in G. ˆ Let H be a is a countable closed discrete subgroup of G. closed subgroup of T. Then H is compact and T/H is also compact. Consider for ϕ ∈ L 2 (H(G)), the collection {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } of left translates on H(G), where L (l,m,t) f (x, γ , θ ) = f ((l, m, t)−1 (x, γ , θ )), (x, γ , θ ) ∈ H(G). In this paper we aim at characterizing this collection to be a Parseval frame, frame and Riesz sequences. In order to obtain such characterizations, we make use of the concept of twisted ˆ and g ∈ L 2 (G × G), ˆ the j-twisted translate, (T t ′ ′ ) j g translates. For j ∈ Z, (x ′ , γ ′ ) ∈ G × G (x ,γ ) is given by ˆ (T(xt ′ ,γ ′ ) ) j g(x, γ ) = g(x − x ′ , γ − γ ′ )(γ (x ′ )) j , (x, γ ) ∈ G × G.

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When j = 1, we simply denote it by T(xt ′ ,γ ′ ) g and call it the twisted translate of g. In the following t section we first obtain characterizations for the collection {T(l,m) ϕ : (l, m) ∈ L × L ⊥ } for ˆ Since it is well-known that the quotient groups G × G/L ˆ × L ⊥ and G/L × G/L ˆ ⊥ ϕ ∈ L 2 (G × G). ⊥ ˆ are isomorphic, L × L turns out to be a discrete and co-compact subgroup of G × G. Later in Section 4, we shall obtain the required characterizations for the system of left translates on H(G).

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3. System of twisted translates

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t We now list here the properties of j-twisted translation (T(l,m) ) j , for (l, m) ∈ L × L ⊥ .

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t t t (1) The adjoint ((T(l,m) ) j )∗ of (T(l,m) ) j is (T(−l,−m) )j.

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(2) (T(lt 1 ,m 1 ) ) j (T(lt 2 ,m 2 ) ) j = (T(lt 2 +l1 ,m 2 +m 1 ) ) j .

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The proofs of (1) and (2) are straightforward. ˆ the kernel of W j ((T t ) j g) satisfies (3) For g ∈ L 2 (G × G), (l,m) j

K (T t

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j (l,m) ) g

(u, y) = (m(y)) j K gj (u, l + y), (u, y) ∈ G × G.

Proof. By using (2.3), we have ∫ j t K (T t ) j g (u, y) = (T(l,m) ) j g(u − y, γ )(γ (y)) j dµGˆ (γ ) (l,m) ˆ G ∫ = g(u − y − l, γ − m)(γ (l)) j (γ (y)) j dµGˆ (γ ) ˆ G ∫ = g(u − (l + y), χ)((χ + m)(l)) j ((χ + m)(y)) j dµGˆ (χ) ˆ G ∫ = g(u − (l + y), χ)(χ (l)) j (χ(y)) j (m(y)) j dµGˆ (χ) ˆ G

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= (m(y)) j =

∫

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g(u − (l + y), χ)(χ (l + y)) j dµGˆ (χ)

ˆ G j (m(y)) K gj (u, l

+ y).

□

ˆ we define the weight function wϕ on G/L by For ϕ ∈ L 2 (G × G), ∑∫ wϕ (u + L) = |K ϕ (x, u + η)|2 dµG (x), u + L ∈ G/L . η∈L

1

2

G

Clearly wϕ ∈ L 1 (G/L). We extend this definition to a section of G/L and then to G as well. It is known that there exists a section of G/L which is Borel measurable and we shall denote it by S L . The cross-section map κ : G/L → S L , [u] ↦→ [u] ∩ S L is bijective and can be used to carry over the algebraic and topological structure of G/L to S L . Thus S L becomes a locally compact abelian group and has Haar measure m given by m(E) = µG/L (κ −1 (E)), where E is a Borel subset of S L . Since the restriction of µG to S L is a multiple of m, µG is taken to be the Haar measure on S L . For u ∈ S L , we may define wϕ as ∑∫ −1 wϕ (u) = wϕ (κ (u)) = |K ϕ (x, u + η)|2 dµG (x). η∈L

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10

G

For y ∈ G, there is a unique expression of the form y = u + η0 , where u ∈ S L and η0 ∈ L. So, we may define ∑∫ wϕ (y) = wϕ (u) = |K ϕ (x, u + η)|2 dµG (x). η∈L

3

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13

G

Now, we are in a position to state the results in connection with the twisted translates.

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ˆ Suppose {T t ϕ : (l, m) ∈ L × L ⊥ } is orthonormal in Theorem 3.1. Let ϕ ∈ L 2 (G × G). (l,m) 2 ˆ L (G × G). Then wϕ (x) = 1 a.e. x ∈ G.

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Proof. Consider t ⟨T(lt 1 ,m 1 ) ϕ, T(lt 2 ,m 2 ) ϕ⟩ L 2 (G×G) ˆ = ⟨ϕ, T(l2 −l1 ,m 2 −m 1 ) ϕ⟩ L 2 (G×G) ˆ

= ⟨K ϕ , K T t ϕ ⟩ L 2 (G×G) ∫ ∫ (l2 −l1 ,m 2 −m 1 ) = K ϕ (x, y)(m 2 − m 1 )(y)K ϕ (x, l2 − l1 + y)dµG (x)dµG (y) G ) ∫G (∫ = K ϕ (x, y)K ϕ (x, l2 − l1 + y)dµG (x) (m 1 − m 2 )(y)dµG (y), G

G

by properties (1) and (2). Now by Weil’s formula, we have ⟨T(lt 1 ,m 1 ) ϕ, T(lt 2 ,m 2 ) ϕ⟩ L 2 (G×G) ˆ ⎛ ⎞ ∫ ∑∫ ⎝ = K ϕ (x, y + η)K ϕ (x, l2 − l1 + y + η)dµG (x)⎠ G/L

η∈L

17

(3.1)

18

G

× (m 1 − m 2 )(y)dµG/L (y + L) since (m 1 − m 2 )(η) = 1. In particular, if we take l1 = l2 = l in (3.1), then ∫ δm 1 m 2 = wϕ (y + L)(m 1 − m 2 )(y + L)dµG/L (y + L) = w ˆϕ (m 2 − m 1 ). G/L

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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ˆ In other words, for γ ∈ G/L, { 1 , if γ = e, w ˆϕ (γ ) = 0 , if γ ̸= e, ˆ given by e(u + L) = 1, ∀ u + L ∈ G/L. Further, for where e is the identity element in G/L ˆ γ ∈ G/L and the constant function 1 ∈ L 1 (G/L), we have { 1 , if γ = e, ˆ 1(γ ) = 0 , if γ ̸= e, ˆ is an orthonormal basis for L 2 (G/L) when G/L is using the well-known result that G/L ˆ from which it follows that wϕ = 1 in compact. (See [8]). Thus w ˆϕ (γ ) = ˆ 1(γ ), ∀ γ ∈ G/L, L 1 (G/L). So wϕ (u + L) = 1 for a.e. u + L ∈ G/L. This implies that wϕ (u) = 1 for a.e. u ∈ S L . Finally, wϕ (x) = 1 for a.e. x ∈ G. □ ˆ Suppose ϕ satisfies Theorem 3.2. Let ϕ ∈ L 2 (G × G). ∫ ∑ K ϕ (x, u + η)K ϕ (x, l + u + η)dµG (x) = 0 a.e. u ∈ S L , ∀ l ∈ L \ {0}

12

(3.2)

G

η∈L

t ˆ and wϕ (x) = 1 a.e. x ∈ G. Then {T(l,m) ϕ : (l, m) ∈ L × L ⊥ } is orthonormal in L 2 (G × G).

Proof. From (3.1), we have for l1 = l2 ⟨T(lt 1 ,m 1 ) ϕ, T(lt 2 ,m 2 ) ϕ⟩ L 2 (G×G) ˆ ⎛ ⎞ ∫ ∫ ∑ ⎝ |K ϕ (x, y + η)|2 dµG (x)⎠ (m 1 − m 2 )(y)dµG/L (y + L) = G/L

∫

η∈L

G

wϕ (y + L)(m 1 − m 2 )(y + L)dµG/L (y + L).

= G/L

Since wϕ = 1 a.e. on G, we have wϕ = 1 a.e. on S L and so wϕ = 1 a.e. on G/L. So ⟨T(lt 1 ,m 1 ) ϕ, T(lt 2 ,m 2 ) ϕ⟩ L 2 (G×G) ˆ ∫ = 1(y + L)(m 1 − m 2 )(y + L)dµG/L (y + L) G/L

=ˆ 1(m 2 − m 1 ) = δm 1 m 2 . We also have from (3.1) that for l1 ̸= l2 , ⟨T(lt 1 ,m 1 ) ϕ, T(lt 2 ,m 2 ) ϕ⟩ L 2 (G×G) ˆ ⎛ ⎞ ∫ ∫ ∑ ⎝ K ϕ (x, y + η)K ϕ (x, l2 − l1 + y + η)dµG (x)⎠ = SL

η∈L

G

× (m 1 − m 2 )(y)dµG (y) = 0, 13

14

15

by (3.2). Thus ⟨T(lt 1 ,m 1 ) ϕ, T(lt 2 ,m 2 ) ϕ⟩ L 2 (G×G) ˆ = δl1 l2 δm 1 m 2 thereby proving our assertion. □ Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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9

Remark 3.3. The condition given in (3.2) reduces to the ‘condition C’ in [14] when G = Rn . ˆ If {T t ϕ : (l, m) ∈ L × L ⊥ } is orthonormal in L 2 (G × G), ˆ Theorem 3.4. Let ϕ ∈ L 2 (G × G). (l,m) then ϕ satisfies (3.2). Proof. Let l ∈ L \ {0}. Consider the function F on G/L given by ∑∫ K ϕ (x, u + η)K ϕ (x, l + u + η)dµG (x), u + L ∈ G/L . F(u + L) = η∈L

1

2 3

4

5

G

ˆ Then using Cauchy Schwarz inequality, it clearly follows that F ∈ L 1 (G/L). Now for γ ∈ G/L, ∫ ˆ )= F(u + L)γ (u + L)dµG/L (u + L) F(γ G/L ∫ ∑∫ K ϕ (x, u + η)K ϕ (x, l + u + η)dµG (x)γ (u)dµG/L (u + L) = G/L η∈L

G

∫ ∫ K ϕ (x, u)K ϕ (x, l + u)γ (u)dµG (x)dµG (u)

= ∫G ∫G

K ϕ (x, u)K T t

= G

(l,γ ) ϕ

G

= ⟨K ϕ , K T t

⟩

2 (l,γ ) ϕ L (G×G)

(x, u)dµG (x)dµG (u)

t = ⟨ϕ, T(l,γ ˆ = 0. ) ϕ⟩ L 2 (G×G)

This shows that F(u + L) = 0 a.e. u + L ∈ G/L which in turn implies that F(u) = 0 a.e. u ∈ S L thereby proving our assertion. □ The proofs of the following results can be written along similar lines and as in the case of the twisted translates on R2n [14]. Let σ (S L ) denote the space of trigonometric polynomials on S L . Let c00 (L , σ (S L )) represent the space of sequences indexed by L and consisting of only finitely many non-zero terms, wherein each non-zero term is a trigonometric polynomial on S L . ˆ is said to be a twisted shift invariant space Definition 3.5. A closed subspace V ⊂ L 2 (G × G) t if for every f ∈ V we also have T(l,m) f ∈ V for any (l, m) ∈ L × L ⊥ . In particular, for ˆ the space V t (ϕ) = span{T t ϕ : (l, m) ∈ L × L ⊥ } is called the principal ϕ ∈ L 2 (G × G), (l,m) twisted shift invariant space. t

Let A (ϕ) denote

t span{T(l,m) ϕ

⊥

: (l, m) ∈ L × L }.

∑ ˆ satisfy (3.2). For f ∈ At (ϕ) given by f = cl,m T t ϕ, the Theorem 3.6. Let ϕ ∈ L 2 (G × G) (l,m) ∑ sequence P defined by P(u) = {Pl (u)}l∈L with Pl (u) = m cl,m m(u), u ∈ S L is in c00 (L , σ (S L )). The map f ↦→ P defined initially between At (ϕ) and c00 (L , σ (S L )) can be extended to an isometric isomorphism of V t (ϕ) onto L 2 (S L , l 2 (L); wϕ ). ˆ satisfying (3.2), f ∈ V t (ϕ) if and only if K f (x, y) = Remark 3.7. For ϕ ∈ L 2 (G × G) ∑ 2 2 l Pl (y)K ϕ (x, l + y) a.e. (x, y) ∈ G × G for some P = {Pl }l∈L ∈ L (S L , l (L); wϕ ) which is further extended to G by Pl (x) = Pl (u + η) = Pl (u), x ∈ G. ˆ satisfy (3.2). Then {T t ϕ : (l, m) ∈ L × L ⊥ } is a Parseval Theorem 3.8. Let ϕ ∈ L 2 (G × G) (l,m) t frame for V (ϕ) if and only if wϕ (u) = 1 a.e. u ∈ Ω , where Ω = {u ∈ S L : wϕ (u) ̸= 0}. Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

6 7

8 9 10 11 12

13 14 15 16

17

18 19 20 21

22 23 24

25 26

INDAG: 607

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2

3

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As a consequence of Theorem 3.8, we shall prove the following ˆ satisfy (3.2). Let ψ ∈ L 2 (G × G) ˆ be given by Corollary 3.9. Let ϕ ∈ L 2 (G × G) { − 21 y ∈ Ωϕ x ∈ G, K ψ (x, y) = K ϕ (x, y)wϕ (y) , 0, y ̸∈ Ωϕ , t where Ωϕ = {y ∈ G : wϕ (y) ̸= 0}. Then {T(l,m) ψ : (l, m) ∈ L × L ⊥ } is a Parseval frame for t V (ϕ).

Proof. Consider ∥K ψ ∥2L 2 (G×G) =

∫

∑∫ G/L η∈L

∫ =

∑∫

S L η∈L

∫ S L ∩Ωϕ

∫ S L ∩Ωϕ

8 9 10

11

wϕ (u)

∑∫ η∈L

|K ϕ (x, u + η)|2 dµG (x)dµG (u) G

wϕ (u) wϕ (u)dµG (u) ≤ m(S L ) < ∞ −1

=

7

|K ψ (x, u + η)|2 dµG (x)dµG (u) G −1

=

6

|K ψ (x, y + η)|2 dµG (x)dµG/L (y + L) G

ˆ as S L is compact. Further, the map f ↦→ W ( f ) is an isometric isomorphism between L 2 (G × G) 2 2 ˆ and B2 (L (G × G)). Thus the integral operator on L (G) associated with this kernel is a Hilbert– ˆ Schmidt operator on L 2 (G) which turns out to be the Weyl transform of a function in L 2 (G × G), 2 ˆ denoted by ψ. Hence ψ is indeed in L (G × G). We now observe from the above calculations that for u ∈ S L , { ∑∫ 1, u ∈ S L ∩ Ωϕ 2 wψ (u) = |K ψ (x, u + η)| dµG (x) = 0, u ∈ S L ∩ (Ωϕ )c G η∈L

12 13 14 15 16 17 18

which shows that wψ (u) = χΩϕ (u) and Ωψ = Ωϕ . Let l ∈ L \ {0}. If u ∈ S L ∩ (Ωϕ )c , then for η ∈ L, u + η ̸∈ Ωϕ and K ψ (x, u + η) = 0, ∀ x ∈ G. Clearly in this case (3.2) holds. On the other hand if u ∈ S L ∩ Ωϕ , then for every η ∈ L, u + η ∈ Ωϕ and further ϕ satisfying (3.2) implies t that ψ also satisfies the same. It now follows from Theorem 3.8 that {T(l,m) ψ : (l, m) ∈ L × L ⊥ } is a Parseval frame for V t (ψ). Using Remark 3.7, one can show that ψ ∈ V t (ϕ) as is done t t in [14]. By making use of Property (2), this ∑ in turn implies that V (ψ) ⊂ V (ϕ). On the t other hand { ϕ ∈ V (ψ) since K ϕ (x, y) = l Q l (y)K ψ (x, l + y) a.e. (x, y) ∈ G × G, where 1 wϕ (x) 2 , 0,

and Q given by Q(x) = {Q l (x)}l∈L is in L 2 (S L , l 2 (L); wψ ). This leads

19

Q l (x) =

20

to V t (ϕ) ⊂ V t (ψ) and hence V t (ψ) = V t (ϕ) which proves the corollary. □

21 22

23 24

25 26

l=0 l ̸= 0

ˆ We shall now present a decomposition theorem for a twisted shift invariant space in L 2 (G× G) for which we need the following definition. ˆ If the system {T t ψ : (l, m) ∈ L × L ⊥ } is a Parseval Definition 3.10. Let ϕ, ψ ∈ L 2 (G × G). (l,m) t frame for V (ϕ), then the function ψ is called a Parseval frame generator of V t (ϕ). ˆ then there exists a family Theorem 3.11. If V is a twisted shift invariant space in L 2 (G × G), ⨁ 2 t ˆ of functions {ϕα }α∈I in L (G × G) (where I is an index set) such that V = α∈I V (ϕα ). In Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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11

ˆ addition, if all the ϕα satisfy (3.2), then there exists a family of functions {ψα }α∈I in L 2 (G × G) such that each ψα is∑a Parseval frame generator of V t (ϕα ). Moreover, in this case, if f ∈ V , 2 2 2 then ∥ f ∥2L 2 (G×G) ˆ = α∈I ∥Pα ∥ L 2 (S ,l 2 (L);w ) , where Pα ∈ L (S L , l (L); wϕα ). L

ϕα

Proof. By Zorn’s lemma there exists a maximal ⨁ collection {V t (ϕα )}α∈I of mutually orthogonal twisted shift invariant subspaces of V . Suppose α∈I V t (ϕα ) is not all of V , then there exists a non-zero ψ ∈ V such that it is orthogonal to each V t (ϕα ). This in turn leads to V t (ψ) being t orthogonal (ϕα ) for each α ∈ I , thereby contradicting the maximality of {V t (ϕα )}α∈I . Thus ⨁ to V t V = α∈I V (ϕα ). The rest of the proof follows by using Corollary 3.9 and Theorem 3.6 as in [14]. □ ˆ satisfy (3.2). Then {T t ϕ : (l, m) ∈ L × L ⊥ } is a Theorem 3.12. Let ϕ ∈ L 2 (G × G) (l,m) t frame for V (ϕ) with frame bounds A,B if and only if A ≤ wϕ (u) ≤ B a.e. u ∈ Ω , where Ω = {u ∈ S L : wϕ (u) ̸= 0}. ˆ Suppose {T t ϕ : (l, m) ∈ L ×L ⊥ } is a frame for L 2 (G× G) ˆ Theorem 3.13. Let ϕ ∈ L 2 (G× G). (l,m) with frame operator S. Then the canonical dual frame also has the same structure and is given t by {T(l,m) S −1 ϕ : (l, m) ∈ L × L ⊥ }. The proofs of the above theorems follow as in [14]. For the characterization of twisted translates as a Riesz sequence, we make use of the following isometric isomorphism and proceed as in [14]. ⊥ Theorem 3.14. For c = {cl,m }(l,m)∈L×L ⊥ ∈ c00 (L × L ), let P be the sequence defined by ∑ P(u) = {Pl (u)}l∈L with Pl (u) = m cl,m m(u), u ∈ S L . Then the map c ↦→ P is an isometry between c00 (L × L ⊥ ) and c00 (L , σ (S L )) which can be extended to an isometric isomorphism of l 2 (L × l ⊥ ) onto L 2 (S L , l 2 (L)).

ˆ satisfy (3.2). Then {T t ϕ : (l, m) ∈ L × L ⊥ } is a Riesz Theorem 3.15. Let ϕ ∈ L 2 (G × G) (l,m) t basis for V (ϕ) with Riesz bounds A,B if and only if A ≤ wϕ (u) ≤ B a.e. u ∈ S L . 4. System of left translates on the Heisenberg group

2 3

4 5 6 7 8 9

10 11 12

13 14 15 16 17 18

19 20 21 22

23 24

25

Now we consider the system of left translates {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } on H(G), for ϕ ∈ L 2 (H(G)). We recall that for (l, m, t) ∈ L × L ⊥ × H , f ∈ L 2 (H(G)) and (x, γ , θ ) ∈ H(G), L (l,m,t) f (x, γ , θ ) = f ((l, m, t)−1 (x, γ , θ )) = f (x − l, γ − m, θt −1 γ (−l)).

26 27 28 29

The left translates L (l,m,t) f satisfy the following properties.

30

j ˆ (1) Lˆ (l,m,t) f ( j) = t L (l,m,1) f ( j). t )j f j. (2) (L (l,m,t) f ) j = t j (T(l,m)

31 32

In the sequel, we let SZ/H ⊥ to denote the section corresponding to the quotient Z/H ⊥ . The following theorem characterizes the system of left translates to be a Parseval frame sequence. Theorem 4.1. Suppose ϕ ∈ L 2 (H(G)) satisfies ⊥ ⟨ˆ ϕ ( j), Lˆ (l,m,1) ϕ( j)⟩B2 (L 2 (G)) C j,G = 0, ∀ j ∈ Z, (l, m) ∈ L × L \ {(e, γ0 )},

1

33 34 35

36

(4.1)

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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where e and γ0 are the identity elements of L and L ⊥ respectively. Then {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } is a Parseval frame for its closed linear span, denoted by V(ϕ − ϕ 0 ), if and only if ωϕ ( j) = 1 for all j ∈ SZ/H ⊥ \ N , where ∑ ωϕ ( j) = ∥ˆ ϕ ( j + k)∥2B (L 2 (G)) C j+k,G (4.2) 2

k∈H ⊥ 5

and N = { j ∈ SZ/H ⊥ : ωϕ ( j) = 0}. In order to prove the necessary condition of the theorem we need the following

6

7

Proposition 4.2. The mapping T : L 2 (H(G)) → l 2 (SZ/H ⊥ , l 2 (H ⊥ , B2 (L 2 (G))))

8

9

defined by 1 T f ( j) = { ˆ f ( j + k)(C j+k,G ) 2 }k∈H ⊥ , for f ∈ L 2 (H(G)), j ∈ SZ/H ⊥ ,

10

= ∥ f − f 0 ∥ L 2 (H(G)) , where SZ/H ⊥ is the Borel section of

11

satisfies ∥T f ∥l 2 (S

12

Z/H ⊥ .

13

Proof. The proof is straightforward by using Weil’s formula and the isometry in (2.5). □

Z/H ⊥

,l 2 (H ⊥ ,B2 (L 2 (G))))

Using the map T , (4.2) can be rewritten as

14

ωϕ ( j) = ∥T ϕ( j)∥l22 (H ⊥ ,B

15

2 2 (L (G)))

.

Proof of Theorem 4.1. Let A(ϕ − ϕ 0 ) denote the span of the collection {L (l,m,t) (ϕ − 0 ⊥ 0 ϕ ∑) : (l, m, t) ∈ L × L × 0H }. Then for f ∈ A(ϕ − ϕ ), we may write f = ′ ′ ′ ′ ′ ′ a finite set. By linearity, f = ∑(l ′ ,m ′ ,t ′ )∈F cl ,m ,t L (l ,m ,t ) (ϕ − 0ϕ ), where F denotes cl ′ ,m ′ ,t ′ (L (l ′ ,m ′ ,t ′ ) ϕ − L (l ′ ,m ′ ,t ′ ) ϕ ). In fact, L (l,m,t) ϕ 0 = (L (l,m,t) ϕ)0 on H(G) for, ∫ L (l,m,t) ϕ(x, γ , θ )dµT (θ) (L (l,m,t) ϕ)0 (x, γ , θ ) = (L (l,m,t) ϕ)0 (x, γ ) = T ∫ = ϕ((l, m, t)−1 (x, γ , θ ))dµT (θ ) ∫T ∫ −1 = ϕ(x − l, γ − m, θt γ (l))dµT (θ ) = ϕ(x − l, γ − m, θ ′ )dµT (θ ′ ) T

T

= ϕ 0 (x − l, γ − m) = ϕ 0 ((l, m, t)−1 (x, γ , θ )) = L (l,m,t) ϕ 0 (x, γ , θ ). 16

17

So f =

(∑

) (∑ )0 cl ′ ,m ′ ,t ′ L (l ′ ,m ′ ,t ′ ) ϕ − cl ′ ,m ′ ,t ′ L (l ′ ,m ′ ,t ′ ) ϕ .

(4.3)

Then by the isometry between L 2 (H(G)) and l 2 (Z \ {0}, B2 (L 2 (G)); C j,G ), we have 2 ∑ ∑  ′ ,m ′ ,t ′ ) ϕ( j) ∥ f ∥2L 2 (H(G)) = cl ′ ,m ′ ,t ′ L (lˆ C j,G  2 j̸=0

B2 (L (G))

2 ∑ ∑  ′ ,m ′ ,1) ϕ( j) = cl ′ ,m ′ ,t ′ (t ′ ) j L (lˆ  j̸=0

B2 (L 2 (G))

C j,G .

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

INDAG: 607

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S. Arati, R. Radha / Indagationes Mathematicae xx (xxxx) xxx–xxx ′ j ) and P( j) = {Pl ′ ,m ′ ( j)}(l ′ ,m ′ )∈L×L ⊥ , we have  2  ∑ ∑   ˆ ′ ′ ′ ′ C j,G Pl ,m ( j) L (l ,m ,1) ϕ( j) =    j̸=0 l ′ ,m ′ 2

Defining Pl ′ ,m ′ ( j) = ∥ f ∥2L 2 (H(G))

∑

t ′ cl ′ ,m ′ ,t ′ (t

B2 (L (G))

2

∑∑

=

′ ,m ′ ,1) ϕ( j)∥ ∥Pl ′ ,m ′ ( j) L (lˆ B2 (L 2 (G)) C j,G

j̸=0 l ′ ,m ′

+

∑

∑

′ ,m ′ ,1) ϕ( j), Pl,m ( j) Lˆ ⟨Pl ′ ,m ′ ( j) L (lˆ (l,m,1) ϕ( j)⟩B2 (L 2 (G)) C j,G

j̸=0 (l ′ ,m ′ )̸=(l,m)

=

∑∑

+

∑

2

′ ,m ′ ,1) ϕ( j)∥ |Pl ′ ,m ′ ( j)|2 ∥ L (lˆ B2 (L 2 (G)) C j,G

j̸=0 l ′ ,m ′

∑

′ ,m ′ ,1) ϕ( j), Lˆ Pl ′ ,m ′ ( j)Pl,m ( j)⟨ L (lˆ (l,m,1) ϕ( j)⟩B2 (L 2 (G)) C j,G

(4.4)

j̸=0 (l ′ ,m ′ )̸=(l,m)

= I1 + I2 . But by (2.4) and properties of left translates and twisted translates, j ′ ,m ′ ,1) ϕ( j), Lˆ ⟨ L (lˆ (l,m,1) ϕ( j)⟩B2 (L 2 (G)) = ⟨K (L

j

(l ′ ,m ′ ,1) ϕ)

j

, K (L

(l,m,1) ϕ)

⟩

j L 2 (G×G)

j

j

= ⟨K (T t ) j ϕ j , K (T t ) j ϕ j ⟩ L 2 (G×G) ′ ′ (l,m) ∫ (l ,m ) j j K (T t ) j ϕ j (u, y)K (T t ) j ϕ j (u, y)dµG (u)dµG (y) = ′ ′ (l,m) (l ,m ) ∫G×G j j (m ′ (y)) j K ϕ j (u, l ′ + y)(m(y)) j K ϕ j (u, l + y)dµG (u)dµG (y) = G×G ∫ j j = K ϕ j (u, v)((m − m ′ )(v)) j K ϕ j (u, l − l ′ + v)dµG (u)dµG (v) ∫G×G j j K ϕ j (u, v)K (T t = (u, v)dµG (u)dµG (v) )jϕ j =

G×G j ⟨K ϕ j ,

(l−l ′ ,m−m ′ )

j K (L ′ j ⟩ L 2 (G×G) (l−l ,m−m ′ ,1) ϕ)

′ ,m−m ′ ,1) ϕ( j)⟩B (L 2 (G)) . = ⟨ˆ ϕ ( j), L (l−lˆ 2

(4.5)

Now, using (4.5) and (4.1), it is clear that I2 = 0. On the other hand, from (4.5), we also get 2 ′ ,m ′ ,1) ϕ( j)∥ ∥ L (lˆ ϕ ( j)∥2B (L 2 (G)) which in turn gives B2 (L 2 (G)) = ∥ˆ 2 ∑∑ 2 I1 = |Pl ′ ,m ′ ( j)| ∥ˆ ϕ ( j)∥2B (L 2 (G)) C j,G 2

j̸=0 l ′ ,m ′

=

∑

∥P( j)∥l22 (L×L ⊥ ) ∥ˆ ϕ ( j)∥2B

2 2 (L (G))

C j,G .

j̸=0

Thus by Weil’s formula, ∥ f ∥2L 2 (H(G)) =

∑

∑

∥P( j + k)∥l22 (L×L ⊥ ) ∥ˆ ϕ ( j + k)∥2B

2 2 (L (G))

C j+k,G

j+H ⊥ ∈Z/H ⊥ k∈H ⊥

=

∑ j∈SZ/H ⊥

∥P( j)∥l22 (L×L ⊥ )

∑

∥ˆ ϕ ( j + k)∥2B

2 2 (L (G))

C j+k,G ,

k∈H ⊥

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

INDAG: 607

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where SZ/H ⊥ is the Borel section of Z/H ⊥ . Further by (4.2) ∑ ∥P( j)∥l22 (L×L ⊥ ) ωϕ ( j). ∥ f ∥2L 2 (H(G)) =

(4.6)

j∈SZ/H ⊥

Having computed the norm of f in L 2 (H(G)), we shall next compute ∫ 2 0 H |⟨ f, L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) | dµT (t). Consider

∑

l∈L

∑

m∈L ⊥

⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) = ⟨ f, L (l,m,t) ϕ − (L (l,m,t) ϕ)0 ⟩ L 2 (H(G)) ⟨∑ ⟩ ˆ ˆ = cl ′ ,m ′ ,t ′ L (l ′ ,m ′ ,t ′ ) ϕ, L (l,m,t) ϕ l 2 (Z\{0},B2 (L 2 (G));C j,G )

using (4.3) and (2.5). Further, ⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) ⟨ ⟩ ∑ ∑ j ′ ,m ′ ,1) ϕ( j), t Lˆ = Pl ′ ,m ′ ( j) L (lˆ (l,m,1) ϕ( j) l ′ ,m ′

j̸=0

=

∑∑

=

∑

+

∑

C j,G

B2 (L 2 (G))

⟨ ⟩ ′ ,m ′ ,1) ϕ( j), Lˆ Pl ′ ,m ′ ( j) L (lˆ ϕ( j) (l,m,1)

B2 (L 2 (G))

j̸=0 l ′ ,m ′

C j,G t − j

2

−j Pl,m ( j)∥ Lˆ (l,m,1) ϕ( j)∥B2 (L 2 (G)) C j,G t

j̸=0

⟨ ⟩ ′ ,m ′ ,1) ϕ( j), Lˆ Pl ′ ,m ′ ( j) L (lˆ (l,m,1) ϕ( j)

∑

B2 (L 2 (G))

j̸=0 (l ′ ,m ′ )̸=(l,m)

C j,G t − j .

Again, by (4.5) and (4.1), the second term on the right hand side is zero and the first term can be rewritten as follows. ∑ ⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) = Pl,m ( j)∥ˆ ϕ ( j)∥2B (L 2 (G)) C j,G t − j 2

j̸=0

∑

=

∑

Pl,m ( j + k)∥ˆ ϕ ( j + k)∥2B

2 2 (L (G))

C j+k,G t −( j+k)

j+H ⊥ ∈Z/H ⊥ k∈H ⊥

=

∑

4

∥ˆ ϕ ( j + k)∥2B

2 2 (L (G))

C j+k,G t − j .

k∈H ⊥

j∈SZ/H ⊥ 3

∑

Pl,m ( j)

∑ Defining F( j) = Pl,m ( j) k∈H ⊥ ∥ˆ ϕ ( j + k)∥2B (L 2 (G)) C j+k,G , for j ∈ SZ/H ⊥ , we have 2 ∑ 0 ˆ ⟨ f, L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) = F( j)t − j = F(t) j∈SZ/H ⊥

and so ∑ ∑ ∫ l∈L m∈L ⊥

2

|⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) H

=

∑ ∑ ∫

=

∑ ∑

l∈L m∈L ⊥

2

ˆ | F(t)| dµT (t) H

∑

|F( j)|2

l∈L m∈L ⊥ j∈SZ/H ⊥ Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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⎞2

⎛ ∑

=

∥P( j)∥l22 (L×L ⊥ ) ⎝

∑

∑

∥ˆ ϕ ( j + k)∥2B

2 2 (L (G))

C j+k,G ⎠

k∈H ⊥

j∈SZ/H ⊥

=

15

( )2 ∥P( j)∥l22 (L×L ⊥ ) ωϕ ( j) .

(4.7)

j∈SZ/H ⊥

Now suppose ωϕ ( j) = 1 for all j ∈ SZ/H ⊥ \ N . Then from (4.6) and (4.7), it follows that for all f ∈ A(ϕ − ϕ 0 ), ∑ ∑ ∫ 2 ∥ f ∥2L 2 (H(G)) = |⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t). (4.8) l∈L m∈L ⊥

1 2

3

H

We shall now show that this equality holds for all f ∈ V(ϕ − ϕ 0 ) as well. Let f ∈ V(ϕ − ϕ 0 ). Then there exists a sequence { f n } in A(ϕ − ϕ 0 ) such that ∥ f n − ⊥ f ∥ L 2 (H(G)) → 0 as n → ∞. Since ⋃∞L × L × H ⊥is σ -compact, there exist compact subsets K i such that K 1 ⊂ K 2 ⊂ · · · and i=1 K i = L × L × H . Now ⏐ ⏐ ⏐⟨ f n , L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) − ⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) ⏐ ≤ ∥ f n − f ∥ L 2 (H(G)) ∥ϕ − ϕ 0 ∥ L 2 (H(G)) ⏐2 ⏐ which tends to zero as n → ∞. So ⏐⟨ f n , L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) ⏐ tends to ⏐ ⏐ ⏐⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) ⏐2 as n → ∞. Also ⏐ ⏐ ⏐⟨ f n , L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) ⏐2 ≤ A∥L (l,m,t) (ϕ − ϕ 0 )∥2 2 L (H(G)) , ∀ n, 2

where A = sup ∥ f n ∥ . Then by Lebesgue’s dominated convergence theorem, for each K i , we have ∫ ⏐ ⏐ ⏐⟨ f n , L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) ⏐2 dµG (l)dµGˆ (m)dµT (t)

4

5 6

7

8 9

10

Ki

converges to ∫ ⏐ ⏐ ⏐⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) ⏐2 dµG (l)dµGˆ (m)dµT (t)

11

12

Ki

as n → ∞. Further by (4.8), ∫ ⏐ ⏐ ⏐⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) ⏐2 dµG (l)dµGˆ (m)dµT (t) ≤ ∥ f ∥2 2 . L (H(G))

13

14

Ki

Taking the limit as i → ∞, we get ∑ ∑ ∫ 2 |⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) ≤ ∥ f ∥2L 2 (H(G)) . l∈L m∈L ⊥

15

(4.9)

H

For the reverse inequality, for a given ϵ > 0 we may choose f 1 ∈ A(ϕ − ϕ 0 ) such that ∥ f − f 1 ∥ L 2 (H(G)) < ϵ. Then ∥ f ∥ ≤ ∥ f − f 1 ∥ + ∥ f 1 ∥ < ∥ f 1 ∥ + ϵ and ∥ f ∥ − 2ϵ < ∥ f1∥ − ∥ f − f1∥ ⎛ ⎞1 2 ∑ ∑ ∫ 2 =⎝ |⟨ f 1 , L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t)⎠ − ∥ f − f 1 ∥. l∈L m∈L ⊥

H

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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Using (4.9), we get ⎞1 ⎛ 2 ∑ ∑ ∫ 2 0 ⎠ ⎝ |⟨ f 1 , L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) | dµT (t) ∥ f ∥ − 2ϵ < H

l∈L m∈L ⊥

⎞1 ⎛ 2 ∑ ∑ ∫ 2 0 ⎠ ⎝ |⟨ f − f 1 , L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) | dµT (t) . − H

l∈L m∈L ⊥ 1

2

Defining for f ∈ V(ϕ − ϕ 0 ), a function G f : L × L ⊥ × H → C by G f (l, m, t) = ⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) ,

(4.10)

we have G f ∈ L 2 (L × L ⊥ × H ) by (4.9) and ∥ f ∥ L 2 (H(G)) − 2ϵ < ∥G f1 ∥ L 2 (L×L ⊥ ×H ) − ∥G f − G f1 ∥ L 2 (L×L ⊥ ×H ) ≤ ∥G f ∥ L 2 (L×L ⊥ ×H ) ⎛ ⎞1 2 ∑ ∑ ∫ 2 0 ⎝ ⎠ = |⟨ f, L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) | dµT (t) . l∈L m∈L ⊥ 3

4

Being true for every ϵ > 0, we have ∑ ∑ ∫ 2 |⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) ∥ f ∥2L 2 (H(G)) ≤ H

l∈L m∈L ⊥ 5

6

and thus by (4.9) ∥

f ∥2L 2 (H(G))

=

∑ ∑ ∫ l∈L m∈L ⊥

7 8

H

2

|⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) H

for all f ∈ V(ϕ − ϕ 0 ) which proves that {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } is a Parseval frame for V(ϕ − ϕ 0 ). Conversely, suppose this collection is a Parseval frame for V(ϕ − ϕ 0 ) and ωϕ ( j) = 1 does not hold for all j ∈ SZ/H ⊥ \ N . Then without loss of generality we may assume that there exist j0 ∈ SZ/H ⊥ \ N and ϵ > 0 such that ωϕ ( j0 ) < 1 − ϵ. Define g ∈ V(ϕ) such that T g( j) = δ j j0 T ϕ( j), j ∈ SZ/H ⊥ . Let {gn } be a sequence in A(ϕ) such that ∥gn − g∥ L 2 (H(G)) → 0 as n → ∞. Now 2

∥(gn )0 − g 0 ∥ L 2 (H(G)) ∫ 2 = |(gn )0 (x, γ ) − g 0 (x, γ )| dµG (x)dµGˆ (γ )dµT (θ ) ˆ ∫G×G×T 2 = |(gn − g)0 (x, γ )| dµG (x)dµGˆ (γ ) ≤ ∥gn − g∥2L 2 (H(G)) ˆ G×G

which shows that g 0 is the limit of (gn )0 in L 2 (H(G)). Let h = g − g 0 . Then h ∈ V(ϕ − ϕ 0 ) for h = limn→∞ (gn − (gn )0 ) and gn − (gn )0 ∈ A(ϕ − ϕ 0 ). By Proposition 4.2, we have ∥h∥2L 2 (H(G)) = ∥T g∥l22 (S

Z/H ⊥

,l 2 (H ⊥ ,B2 (L 2 (G))))

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

INDAG: 607

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=

∑

∥T g( j)∥l22 (H ⊥ ,B

2 2 (L (G)))

= ∥T ϕ( j0 )∥l22 (H ⊥ ,B

2 2 (L (G)))

j∈SZ/H ⊥

= ωϕ ( j0 )

(4.11)

and

1

⟨h, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) = ⟨T g, T (L (l,m,t) ϕ)⟩l 2 (S

Z/H ⊥

,l 2 (H ⊥ ,B2 (L 2 (G)))) .

2

But T (L (l,m,t) ϕ)( j) = t j T (L (l,m,1) ϕ)( j). So ∑ ∑ ∫ 2 |⟨h, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) H

l∈L m∈L ⊥

=

∑ ∑ ∫ l∈L m∈L ⊥

=

∑ ∑ ∫

=

∑ ∑

l∈L m∈L ⊥

2

∑

|

H j∈S Z/H ⊥

|

∑

H

∑

⟨T g( j), T (L (l,m,t) ϕ)( j)⟩l 2 (H ⊥ ,B2 (L 2 (G))) | dµT (t) 2

t − j ⟨T g( j), T (L (l,m,1) ϕ)( j)⟩l 2 (H ⊥ ,B2 (L 2 (G))) | dµT (t) |⟨T g( j), T (L (l,m,1) ϕ)( j)⟩l 2 (H ⊥ ,B2 (L 2 (G))) |2

l∈L m∈L ⊥ j∈SZ/H ⊥

by Plancherel theorem. By the definition of T g, we obtain ∑ ∑ ∫ 2 |⟨h, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) H

l∈L m∈L ⊥

=

∑ ∑

|⟨T ϕ( j0 ), T (L (l,m,1) ϕ)( j0 )⟩l 2 (H ⊥ ,B2 (L 2 (G))) |2

l∈L m∈L ⊥

=

∑ ∑

|

∑

C j0 +k,G ⟨ˆ ϕ ( j0 + k), Lˆ (l,m,1) ϕ( j0 + k)⟩B2 (L 2 (G)) |

2

l∈L m∈L ⊥ k∈H ⊥

⎞2

⎛ =⎝

∑

C j0 +k,G ∥ˆ ϕ ( j0 + k)∥2B

2 2 (L (G))

⎠

k∈H ⊥

∑

+

|

∑

2

C j0 +k,G ⟨ˆ ϕ ( j0 + k), Lˆ (l,m,1) ϕ( j0 + k)⟩B2 (L 2 (G)) | .

(l,m)̸=(e,γ0 ) k∈H ⊥

The second term on the right hand side is zero by (4.1). So ∑ ∑ ∫ 2 |⟨h, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) = (ωϕ ( j0 ))2 . l∈L m∈L ⊥

4

H

By our assumption on j0 and (4.11) , we have ∑ ∑ ∫ 2 |⟨h, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) < (1 − ϵ)∥h∥2L 2 (H(G)) l∈L m∈L ⊥

3

5

6

H

which contradicts the fact that {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } is a Parseval frame for V(ϕ − ϕ 0 ). □ Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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Theorem 4.3. Suppose ϕ ∈ L 2 (H(G)) satisfies (4.1). Then the collection {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } is a frame for V(ϕ − ϕ 0 ) with frame bounds A, B > 0 if and only if A ≤ ωϕ ( j) ≤ B for all j ∈ SZ/H ⊥ \ N , where ωϕ ( j) and N are as in Theorem 4.1. Proof. Suppose there exist A, B > 0 such that A ≤ ωϕ ( j) ≤ B for all j ∈ SZ/H ⊥ \ N . Then by (4.6) and, ∑ ∑ ∫ 2 |⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) A∥ f ∥2L 2 (H(G)) ≤ ≤

4 5 6 7 8

9

l∈L m∈L ⊥ H B∥ f ∥2L 2 (H(G)) ,

for all f ∈ A(ϕ − ϕ 0 ).

0 0 Now, as in the proof of Theorem ⏐2 { f n } in A(ϕ−ϕ ) and compact ∫ ⏐4.1, for f ∈ V(ϕ−ϕ0 ), there exist ⏐ ⏐ sets K i such that for each i, K i ⟨ f n , L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) dµG (l)dµGˆ (m)dµT (t) converges ⏐2 ∫ ⏐ 0 ⏐ ⏐ as n → ∞ and to ˆ (m)dµT (t) K i ⟨ f, L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) dµG (l)dµG ⏐ ⏐ ∫ 2 ⏐⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) ⏐ dµG (l)dµGˆ (m)dµT (t) ≤ B∥ f ∥2 2 . Taking the limit as Ki L (H(G)) i → ∞, we get ∑ ∑ ∫ 2 |⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) ≤ B∥ f ∥2L 2 (H(G)) . l∈L m∈L ⊥

H

For the lower frame bound inequality, we may choose f 1 ∈ A(ϕ − ϕ 0 ) such that ∥ f − f 1 ∥ L 2 (H(G)) < ϵ, for any ϵ > 0. Then ∥ f ∥ < ∥ f 1 ∥ + ϵ. Further √ √ √ √ √ √ √ A∥ f ∥ − ( B + A)ϵ < A∥ f 1 ∥ − Bϵ < A∥ f 1 ∥ − B∥ f − f 1 ∥ ⎛ ⎞1 2 ∑ ∑ ∫ 2 0 ≤⎝ |⟨ f 1 , L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) | dµT (t)⎠ l∈L m∈L ⊥

H

√ −

⎛ ≤⎝

2

∑ ∑ ∫ l∈L m∈L ⊥

B∥ f − f 1 ∥ ⎞1

2

|⟨ f 1 , L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) | dµT (t)⎠ 0

H

⎛ ⎞1 2 ∑ ∑ ∫ 2 0 ⎝ ⎠ − |⟨ f − f 1 , L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) | dµT (t) . l∈L m∈L ⊥

H

As in the proof of Theorem 4.1, using the definition of G f in (4.10), we obtain √ √ √ A∥ f ∥ − ( B + A)ϵ ⎛ ⎞1 2 ∑ ∑ ∫ 2 0 ⎝ ⎠ < |⟨ f, L (l,m,t) (ϕ − ϕ )⟩ L 2 (H(G)) | dµT (t) l∈L m∈L ⊥ 10

11

H

and hence 2

A∥ f ∥ ≤

∑ ∑ ∫ l∈L m∈L ⊥

2

|⟨ f, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t), H

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

INDAG: 607

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thereby proving that {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } is a frame for V(ϕ − ϕ 0 ) with frame bounds A and B. Conversely, suppose this system of translates is a frame sequence with bounds A, B > 0 and there exist j0 ∈ SZ/H ⊥ \ N and ϵ > 0 such that ωϕ ( j0 ) < A − ϵ. As in the proof of Theorem 4.1, there exists h ∈ V(ϕ − ϕ 0 ) such that ∥h∥2L 2 (H(G)) = ωϕ ( j0 )

1 2 3 4 5

6

and

7

∑ ∑ ∫

2

|⟨h, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) = (ωϕ ( j0 ))2 .

8

H

l∈L m∈L ⊥

Since j0 ̸∈ N , we get ∑ ∑ ∫ 2 |⟨h, L (l,m,t) (ϕ − ϕ 0 )⟩ L 2 (H(G)) | dµT (t) < (A − ϵ)ωϕ ( j0 ) H

l∈L m∈L ⊥

< A∥h∥2L 2 (H(G)) which is a contradiction to our assumption. We arrive at a similar contradiction if ωϕ ( j0 ) > B +ϵ, thereby proving the theorem. □ Remark 4.4. When G = Rn , the Heisenberg group H(G) can also be written as Rn × Rn × R with the group operation given by (x, y, t)(x ′ , y ′ , t ′ ) = (x + x ′ , y + y ′ , t + t ′ + x · y ′ ).

9 10

11 12

13

In this case Theorem 4.3 can be written as follows. “Suppose ϕ ∈ L (H ) satisfies for (k, l) ∈ Z2n \ {(0, 0)}, ∑ n ⟨ˆ ϕ (λ + r ), Lˆ (k,l,0) ϕ(λ + r )⟩B2 (L 2 (Rn )) |λ + r | = 0 a.e. λ ∈ T. 2

n

14 15

16

r ∈Z

Then {L (k,l,m) ϕ : (k, l, m) ∈ Zn × Zn × Z} is a frame for its closed linear span with frame bounds A, B > 0 if and only if A ≤ ωϕ (λ) ≤ B a.e. λ ∈ {ξ ∈ T : ωϕ (ξ ) > 0}, where ∑ ωϕ (λ) = ∥ˆ ϕ (λ + r )∥2B (L 2 (Rn )) |λ + r |n .′′

17 18

19

2

r ∈Z

(Theorem 1.5 in [1] in this connection.)

20

Definition 4.5. We say that the system of left translates {L (l,m,t) (ϕ−ϕ 0 ) : (l, m, t) ∈ L ×L ⊥ × H } is a Riesz basis for V(ϕ − ϕ 0 ) if there exist A, B > 0 such that ∑ ∑ ∫ A |cl,m,t |2 dµT (t) l∈L m∈L ⊥

≤∥

H

∑ ∑ ∫ l∈L m∈L ⊥

≤B

cl,m,t L (l,m,t) (ϕ − ϕ 0 )dµT (t)∥2 H

∑ ∑ ∫ l∈L m∈L ⊥

|cl,m,t |2 dµT (t), H

where {cl,m,t } is any sequence which is non-zero for finitely many l and m and for a fixed (l, m) it is continuous on H as a function of t. Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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INDAG: 607

20 1 2 3 4 5

6 7

8 9 10

11

12

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0 ⊥ 0 Remark 4.6. Suppose { {L (l,m,t) (ϕ − ϕ ) : (l, m, t) ∈ L × L × H } is a Riesz basis for V(ϕ − ϕ ). 0 , (l, m) ̸= (e, γ0 ) Then taking cl,m,t = a(t) , where a is a continuous function on H , it turns out that , (l, m) = (e, γ0 ) 0 {L (e,γ0 ,t) (ϕ − ϕ ) : t ∈ H } is a continuous Riesz sequence. It now follows that H is discrete since there are no continuous Riesz sequences otherwise as is mentioned in [3]. So we also have that ˆ = Z/H ⊥ is compact. In other words SZ/H ⊥ is a finite subset of Z. H

We now deduce a characterization for {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } to be a Riesz sequence under the additional necessary assumption that H is discrete. Theorem 4.7. Suppose ϕ ∈ L 2 (H(G)) satisfies (4.1). Then the collection {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } is a Riesz basis for V(ϕ − ϕ 0 ) with Riesz bounds A, B > 0 if and only if A B ≤ ωϕ ( j) ≤ , for all j ∈ SZ/H ⊥ , (4.12) (m(SZ/H ⊥ ))2 (m(SZ/H ⊥ ))2 where ωϕ ( j) is as in (4.2). Proof. Suppose there exist A, B > 0 satisfying (4.12). Let {cl,m,t } be a sequence with finitely many non-zero terms. We define Pl,m ( j) and P( j) as in the proof of Theorem 4.1 and obtain (4.6), namely ∑ ∑ ∑ ∑ ∥ cl,m,t L (l,m,t) (ϕ − ϕ 0 )∥2 = ∥P( j)∥l22 (L×L ⊥ ) ωϕ ( j). l∈L m∈L ⊥ t∈H

17

18

j∈SZ/H ⊥

Now for t ∈ H , consider ∑ ∑ ˆ P Pl,m ( j)t − j = cl,m,t ′ l,m (t) = t′

j∈SZ/H ⊥

∑

(t ′ ) j t − j

j∈SZ/H ⊥

using the definition of Pl,m ( j). By the orthogonality of H as elements in l 2 (SZ/H ⊥ ), we get ˆ P l,m (t) = cl,m,t m(SZ/H ⊥ ) and so ∑ ∑ ∑ ∑ ∑ ∑ 1 2 ˆ |P |cl,m,t |2 = l,m (t)| 2 (m(S ⊥ )) Z/H l∈L m∈L ⊥ t∈H l∈L m∈L ⊥ t∈H ∑ ∑ ∑ 1 |Pl,m ( j)|2 = (m(SZ/H ⊥ ))2 l∈L ⊥ j∈S m∈L

19

20

by Plancherel theorem. Thus, we obtain ∑ ∑ ∑ 1 |cl,m,t |2 = (m(SZ/H ⊥ ))2 ⊥ t∈H l∈L m∈L

21 22

∑

Z/H ⊥

∥P( j)∥l22 (L×L ⊥ ) .

j∈SZ/H ⊥

Clearly it follows from the assumption of (4.12) that {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } is a Riesz basis for its closed linear span with Riesz bounds A and B. Conversely, we assume that {L (l,m,t) (ϕ − ϕ 0 ) : (l, m, t) ∈ L × L ⊥ × H } is a Riesz sequence with bounds A and B. Let j0 ∈ SZ/H ⊥ . Define R( j) = δ j j0 for j ∈ SZ/H ⊥ . Then R ∈ l 2 (SZ/H ⊥ ) and with SZ/H ⊥ being compact there exists a sequence {Q k } of trigonometric polynomials on SZ/H ⊥ such that ∥Q k − R∥l 2 (S ⊥ ) → 0 as k → ∞. For each k ∈ N, we define the sequence Z/H

(k) (k) (k) c(k) = {cl,m,t } by taking cl,m,t = 0 for (l, m) ̸= (e, γ0 ) and ce,γ to be the coefficients of the 0 ,t Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

INDAG: 607

S. Arati, R. Radha / Indagationes Mathematicae xx (xxxx) xxx–xxx

21

trigonometric polynomial Q k . Then each c(k) is a sequence with finitely many non-zero terms and so ∑ ∑ ∑ (k) 2 ∑ ∑ ∑ (k) A |cl,m,t | ≤ ∥ cl,m,t L (l,m,t) (ϕ − ϕ 0 )∥2 l∈L m∈L ⊥ t∈H

l∈L m∈L ⊥ t∈H

≤B

∑ ∑ ∑

2

(k) |cl,m,t |.

l∈L m∈L ⊥ t∈H

Considering

(k) ( j) Pl,m

and P (k) ( j) analogous to Pl,m ( j) and P( j) respectively, we obtain ∑ 2 ∥P (k) ( j)∥l 2 (L×L ⊥ )

A (m(SZ/H ⊥ ))2 j∈S Z/H ⊥ ∑ 2 ≤ ∥P (k) ( j)∥l 2 (L×L ⊥ ) ωϕ ( j) j∈SZ/H ⊥

≤

B

∑

(m(SZ/H

2 ⊥ ))

2

∥P (k) ( j)∥l 2 (L×L ⊥ ) .

j∈SZ/H ⊥

By the definition of P (k) , this turns out to be ∑ ∑ A |Q k ( j)|2 ≤ 2 (m(SZ/H ⊥ )) j∈S j∈S Z/H ⊥

|Q k ( j)|2 ωϕ ( j)

Z/H ⊥

≤

B (m(SZ/H ⊥ ))2

Taking the limit as k → ∞, we obtain ∑ ∑ A 2 |R( j)| ≤ (m(SZ/H ⊥ ))2 j∈S j∈S Z/H ⊥

∑

|Q k ( j)|2 .

j∈SZ/H ⊥

|R( j)|2 ωϕ ( j)

Z/H ⊥

≤

B (m(SZ/H

∑ 2 ⊥ ))

|R( j)|2

j∈SZ/H ⊥

which in turn by the definition of R shows that (4.12) holds for j0 and hence for all j ∈ SZ/H ⊥ , thereby proving our assertion. □ Acknowledgment We thank the referee for scrupulously reading the manuscript, providing valuable suggestions and letting us know some important references, which added real value to our manuscript. We are also grateful to the referee, particularly for clarifying our misconception regarding the existence of uniform lattices. References [1] D. Barbieri, E. Hernández, A. Mayeli, Bracket map for the Heisenberg group and the characterization of cyclic subspaces, Appl. Comput. Harmon. Anal. 37(2) (2014) 218–234. [2] M. Bownik, The structure of shift-invariant subspaces of L 2 (Rn ), J. Funct. Anal. 177 (2) (2000) 282–309. [3] M. Bownik, K.A. Ross, The structure of translation-invariant spaces on locally compact abelian groups, J. Fourier Anal. Appl. 21 (4) (2015) 849–884.

Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.

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Please cite this article in press as: S. Arati, R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indagationes Mathematicae (2018), https://doi.org/10.1016/j.indag.2018.09.001.