Frames of exponentials and sub-multitiles in LCA groups
Davide Barbieri, Carlos Cabrelli, Eugenio Hernández, Peter Luthy, Ursula Molter, Carolina Mosquera
aa r X i v : . [ m a t h . C A ] O c t FRAMES OF EXPONENTIALS AND SUB–MULTITILES IN LCAGROUPS.
D. BARBIERI, C. CABRELLI, E. HERN ´ANDEZ, P. LUTHY, U. MOLTER,AND C. MOSQUERA
Abstract.
In this note we investigate the existence of frames of exponentials for L (Ω) in the setting of LCA groups. Our main result shows that sub–multitilingproperties of Ω ⊂ b G with respect to a uniform lattice Γ of b G guarantee the existenceof a frame of exponentials with frequencies in a finite number of translates of theannihilator of Γ. We also prove the converse of this result and provide conditionsfor the existence of these frames. These conditions extend recent results on Rieszbases of exponentials and multitilings to frames. Introduction and main result
We begin by stating several known results. • Let Ω be a measurable subset of R d with positive, finite measure, let Λ bea complete lattice of R d (i.e. Λ = A Z d for some d × d invertible matrix A with real entries), and denote by Γ the annihilator of Λ . Recall that Γ = { γ ∈ R d : e πi h λ,γ i = 1 , ∀ λ ∈ Λ } . In 1974, B. Fuglede ([5], Section 6) proved that { e πi h λ, • i : λ ∈ Λ } is an orthogonal basis for L (Ω) if and only if (Ω , Γ) is a tiling pair for R d , that is X γ ∈ Γ χ Ω ( x + γ ) = 1, a. e. x ∈ R d . • The result of B. Fuglede just stated also holds in the setting of locally compactabelian (LCA) groups. Let G be a second countable LCA group, and letΛ be a uniform lattice in G (i.e. Λ is a discrete and co-compact subgroupof G ). Denote by b G the dual group of G. For a character ω ∈ b G we usethe notation e g ( ω ) = ω ( g ) , for g ∈ G. Let Γ be the annihilator of Λ . (i.e.Γ = { γ ∈ b G : e λ ( γ ) = 1 for all λ ∈ Λ } ). The dual group b G of G is alsoa second countable LCA group, and Γ is also a uniform lattice. Let Ω be ameasurable subset of b G with positive and finite measure. In 1987, S. Pedersen([10], Theorem 3.6) proved that { e λ : λ ∈ Λ } is an orthogonal basis for L (Ω)if and only if (Ω , Γ) is a tiling pair for b G , that is P γ ∈ Γ χ Ω ( ω + γ ) = 1, a. e. ω ∈ b G. • Recent results in this area focused on multitiling pairs . Let Ω be a bounded,measurable subset of R d , and let Γ be a lattice of R d . If there exists a positive Date : August 7, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Frames of exponentials, submultitiles, LCA groups. integer ℓ such that X γ ∈ Γ χ Ω ( x + γ ) = ℓ , a.e x ∈ R d , we will say that (Ω , Γ) is a multitiling pair , or an ℓ -tiling pair for R d . Fora lattice Λ ⊂ R d and a , . . . , a ℓ ∈ R d , let E Λ ( a , . . . , a ℓ ) := { e πi h a j + λ , • i : j = 1 , . . . , ℓ ; λ ∈ Λ } . S. Gresptad and N. Lev ([6], Theorem 1) proved in 2014 that if Γ is theannihilator of Λ, Ω is a bounded, measurable subset of R d whose boundary hasmeasure zero, and (Ω , Γ) is an ℓ -tiling pair for R d , there exist a , . . . , a ℓ ∈ R d such that E Λ ( a , . . . , a ℓ ) is a Riesz basis for L (Ω) . The proof of this resultin [6] uses Meyer’s quasicrystals. In 2015 M. Kolountzakis ([9], Theorem 1)found a simpler and shorter proof without the assumption on the boundary ofΩ.For the reader’s convenience we recall that a countable collection of elementsΦ = { φ j : j ∈ J } of a Hilbert space H is a Riesz basis for H if it is the imageof an orthonormal basis of H under a bounded, invertible operator T ∈ L ( H ) . Riesz bases provide stable representations of elements of H . • This result has been extended to second countable LCA groups by E. Agora,J. Antezana, and C. Cabrelli ([1], Theorem 4.1). Moreover, they prove theconverse ([1], Theorem 4.4): with the same notation as in the second item ofthis section, given a relatively compact subset Ω of b G , if L (Ω) admits a Rieszbasis of the form E Λ ( a , . . . , a ℓ ) := { e a j + λ : j = 1 , , . . . , ℓ ; λ ∈ Λ } for some a , . . . , a ℓ ∈ G, then (Ω , Γ) is an ℓ -tiling pair for b G. The purpose of this note is to investigate the situation when (Ω , Γ) is a sub-multitiling pair for b G. Let Ω be a measurable set in b G with positive and finite Haarmeasure. For Γ a lattice in b G and ω ∈ b G define F Ω , Γ ( ω ) := X γ ∈ Γ χ Ω ( ω + γ ) . If there exists a positive integer ℓ such thatess sup ω ∈ b G F Ω , Γ ( ω ) = ℓ , (1.1)we will say that (Ω , Γ) is a sub-multitiling pair or an ℓ - subtiling pair .Denote by Q Γ a fundamental domain of the lattice Γ in b G, i.e. it is a Borel mea-surable section of the quotient group b G/ Γ . (Its existence is guaranteed by Theorem 1in [4]). Since F Ω , Γ ( ω ) is a Γ-periodic function, it is enough to compute the ess sup in(1.1) over a fundamental domain Q Γ . Observe that (Ω , Γ) is an ℓ -tiling pair for b G if F Ω , Γ ( ω ) = ℓ for a. e. ω ∈ Q Γ . Another structure that allows for stable representations, besides orthonormal andRiesz bases, is that of a frame . A collection of elements Φ = { φ j : j ∈ J } of a Hilbertspace H is a frame for H if it is the image of an orthonormal basis of H under a RAMES OF EXPONENTIALS AND SUB–MULTITILES 3 bounded, surjective operator T ∈ L ( H ) or, equivalently, if there exist 0 < A ≤ B < ∞ such that A k f k ≤ X j ∈ J |h f, φ j i| ≤ B k f k , for all f ∈ H . (See [11], Chapter 4, Section 7.) The numbers A and B are called frame bounds ofΦ . In this note we prove the following relationship between frames of exponentials inLCA groups and ℓ -subtiling pairs. Theorem 1.1.
Let G be a second countable LCA group and let Λ be a uniform latticeof G . Let b G be the dual group of G , and let Γ be the annihilator of Λ . Let Ω ⊂ b G bea measurable set of positive, finite measure, and let ℓ be a positive integer. (1) If for some a , . . . , a ℓ ∈ G the collection E Λ ( a , . . . , a ℓ ) is a frame of L (Ω) , then (Ω , Γ) must be an m -subtiling pair of b G for some positive integer m ≤ ℓ. (2) If Ω ⊆ b G is a measurable, bounded set and (Ω , Γ) is an ℓ -subtiling pair of b G ,then there exist a , . . . , a ℓ ∈ G such that E Λ ( a , . . . , a ℓ ) is a frame of L (Ω) . Remark 1.2.
Recall that any locally compact and second countable group is metriz-able, and its metric can be chosen to be invariant under the group action (see [8] ,Theorem 8.3). Thus, it makes sense to talk about bounded sets in the group b G. The proof of Theorem 1.1 will be given in Section 2. In Section 3 we give otherconditions for a set of exponentials of the form E Λ ( a , . . . , a ℓ ) to be a frame of L (Ω)and provide expressions to compute the frame bounds. Acknowledgements . The research of D. Barbieri and E. Hern´andez is supportedby Grants MTM2013-40945-P and MTM2016-76566-P (Ministerio de Econom´ıa yCompetitividad, Spain).The research of C. Cabrelli, U. Molter and C. Mosquerais partially supported by Grants PICT 2014-1480 (ANPCyT), PIP 11220150100355(CONICET) Argentina, and UBACyT 20020130100422BA. P. Luthy was supportedby Grant MTM2013-40945-P while this research started at UAM.2.
Proof of Theorem 1.1
We start with a result that will be used in the proof of part (2) of Theorem 1.1
Proposition 2.1. If Ω is a measurable, bounded set in b G and Γ is a uniform latticein b G such that (Ω , Γ) is an ℓ -subtiling pair for b G , there exists a bounded measurableset ∆ ⊂ b G such that Ω ⊂ ∆ and (∆ , Γ) is an ℓ -tiling pair for b G. Proof.
Let Q Γ be a fundamental domain of Γ in b G. Modifying Ω in a set of mea-sure zero, we can assume that sup ω ∈ Q Γ F Ω , Γ ( ω ) = ℓ. Define e Γ = { γ ∈ Γ : ω + γ ∈ Ω for some ω ∈ Q Γ } . Since Ω is bounded, the set e Γ is finite and, by the definition of ℓ -subtiling pair, has at least ℓ different elements.Set Q k = { ω ∈ Q Γ : F Ω , Γ ( ω ) = k } for k = 0 , , ..., ℓ. Clearly Q Γ = ℓ [ k =0 Q k , D. BARBIERI, C. CABRELLI, E. HERN ´ANDEZ, P. LUTHY, U. MOLTER, AND C. MOSQUERA and the union is disjoint.Now, for k = 1 , . . . , ℓ, let B k = { B ⊂ e Γ : B = k } . For B ∈ B k set Q k ( B ) = { ω ∈ Q k : ω + γ ∈ Ω , for all γ ∈ B } . Since Ω is measurable, Q k is measurable and since Q k ( B ) = T γ ∈ B ((Ω − γ ) ∩ Q k ) , then Q k ( B ) is also measurable. Observe that the collection B k is finite since e Γ isfinite. Also if B and B ′ are different sets in B k then Q k ( B ) ∩ Q k ( B ′ ) = ∅ . Indeed, if ω ∈ Q k ( B ) ∩ Q k ( B ′ ), ω + γ ∈ Ω for all γ ∈ B and ω + γ ′ ∈ Ω for all γ ′ ∈ B ′ . Since B = B ′ , there exists γ ∈ B ′ \ B . Then, since ω ∈ Q k ,k = X γ ∈ Γ χ Ω ( ω + γ ) ≥ X γ ∈ B χ Ω ( ω + γ ) + χ Ω ( ω + γ ) = k + 1 , which is a contradiction. Observe that Q k = S B ∈B k Q k ( B ), k = 1 , . . . , ℓ , and theunion is disjoint. Therefore,Ω = ℓ [ k =1 [ B ∈B k [ γ ∈ B Q k ( B ) + γ , (2.1)and the union is disjoint.For k = 1 , . . . , ℓ and B ∈ B k , we extend B ⊆ e Γ to e B by inserting ℓ − k distinctelements from e Γ \ B into B . Let e B be a set of ℓ different elements from e Γ . We recallhere that e Γ ≥ ℓ since sup F Ω , Γ = ℓ .Finally we define:∆ = (cid:16) [ γ ∈ e B Q + γ (cid:17) ∪ (cid:16) ℓ [ k =1 [ B ∈B k [ γ ∈ e B Q k ( B ) + γ (cid:17) . The set ∆ is measurable since it is a finite union of measurable sets. From (2.1) itis clear that Ω ⊂ ∆ . Moreover, if ω ∈ Q k ( B ) , for some B ∈ B k , ω + γ ∈ Ω only when γ ∈ B. Hence, if ω ∈ Q k ( B ) , ω + e γ ∈ ∆ only when e γ ∈ e B. Since e B has precisely ℓ elements, if ω ∈ Q k ( B ), X γ ∈ Γ χ ∆ ( ω + γ ) = X e γ ∈ e B χ ∆ ( ω + e γ ) = ℓ . Also, if ω ∈ Q X γ ∈ Γ χ ∆ ( ω + γ ) = X e γ ∈ e B χ ∆ ( ω + e γ ) = ℓ . Taking into account that Q Γ = S ℓk =0 Q k = Q ∪ (cid:16) S ℓk =1 S B ∈B k Q k ( B ) (cid:17) is a disjointunion, we conclude that for ω ∈ Q Γ , P γ ∈ Γ χ ∆ ( ω + γ ) = ℓ , proving that (∆ , Γ) is an ℓ -tiling pair for b G. (cid:3) Remark 2.2.
The ℓ -tile found in Proposition 2.1 is not necessarily unique. It dependson the choice of the sets e B and e B . For the proof of part (2) of Theorem 1.1 we will use the fiberization mapping T : L ( G ) −→ L ( Q Γ , ℓ (Γ)) given by T f ( ω ) = { b f ( ω + γ ) } γ ∈ Γ ∈ ℓ (Γ) , ω ∈ Q Γ . (2.2) RAMES OF EXPONENTIALS AND SUB–MULTITILES 5
The mapping T is an isometry and satisfies T ( t λ f )( ω ) = e − λ ( ω ) T f ( ω ) , λ ∈ Λ , f ∈ L ( G ) , (2.3)(see Proposition 3.3 and Remark 3.12 in [3]), where t λ denotes the translation by λ that is t λ f ( g ) = f ( g − λ ) . The next result is Theorem 4.1 of [3] adapted to our situation. For ϕ , . . . , ϕ ℓ ∈ L ( G ) denote by S Λ ( ϕ , . . . , ϕ ℓ ) := span { t λ ϕ j : λ ∈ Λ , j = 1 , . . . , ℓ } the Λ-invariant space generated by { ϕ , . . . , ϕ ℓ } . The measurable range function as-sociated to S Λ ( ϕ , . . . , ϕ ℓ ) is J ( ω ) = span {T ϕ ( ω ) , . . . , T ϕ ℓ ( ω ) } ⊂ ℓ (Γ) , ω ∈ Q Γ . (2.4) Proposition 2.3.
Let ϕ , . . . , ϕ ℓ ∈ L ( G ) and let J ( ω ) be the measurable range func-tion associated to S Λ ( ϕ , . . . , ϕ ℓ ) as in (2.4) . Let < A ≤ B < ∞ . The followingstatements are equivalent:(i) The set { t λ ϕ j : λ ∈ Λ , j = 1 , . . . , ℓ } is a frame for S Λ ( ϕ , . . . , ϕ ℓ ) with framebounds A and B .(ii) For almost every ω ∈ Q Γ the set {T ϕ ( ω ) , . . . , T ϕ ℓ ( ω ) } ⊂ ℓ (Γ) is a frame for J ( ω ) with frame bounds A | Q Γ | − and B | Q Γ | − . Proof.
Let f ∈ S Λ ( ϕ , . . . , ϕ ℓ ). Use that the fiberization mapping given in (2.2) is anisometry satisfying (2.3) to write X λ ∈ Λ ℓ X j =1 |h t λ ϕ j , f i L ( G ) | = X λ ∈ Λ ℓ X j =1 |hT ( t λ ϕ j ) , T f i L ( Q Γ ,ℓ (Γ)) | = ℓ X j =1 X λ ∈ Λ (cid:12)(cid:12)(cid:12) Z Q Γ e − λ ( ω ) hT ( ϕ j )( ω ) , T f ( ω ) i ℓ (Γ) dω (cid:12)(cid:12)(cid:12) . Since { √ | Q Γ | e λ ( ω ) : λ ∈ Λ } is an orthonormal basis of L ( Q Γ ) it follows that X λ ∈ Λ ℓ X j =1 |h t λ ϕ j , f i L ( G ) | = | Q Γ | ℓ X j =1 Z Q Γ |hT ϕ j ( ω ) , T f ( ω ) i ℓ (Γ) | dω . From here, the proof continues as in the proof of Theorem 4.1 in [3]. Details are leftto the reader. (cid:3)
Remark 2.4.
Notice that the factor | Q Γ | − that appears in ( ii ) of Proposition 2.3does not appear in Theorem 4.1 of [3] . This is due to the fact that in [3] the measureof Q Γ is normalized (see the beginning of Section 3 in [3] ). Although this fact is notimportant to prove (2) of Theorem 1.1, it will be crucial in Section 3 to obtain optimalframe bounds of sets of exponentials. Proof of Theorem 1.1 (1) Assume that E Λ ( a , . . . , a ℓ ) is a frame for L (Ω) . We define ϕ ∈ L ( G ) by b ϕ := χ Ω , and ϕ j := t − a j ϕ, j = 1 , . . . , ℓ, where t a j denotes the translation by a j , that is t a j ϕ ( g ) = ϕ ( g − a j ) . D. BARBIERI, C. CABRELLI, E. HERN ´ANDEZ, P. LUTHY, U. MOLTER, AND C. MOSQUERA
Since E Λ ( a , . . . , a ℓ ) is a frame of L (Ω) , we have that { t λ ϕ j : λ ∈ Λ , j = 1 , . . . , ℓ } is a frame of the Paley-Wiener space P W Ω := { f ∈ L ( G ) : b f ∈ L (Ω) } = { f ∈ L ( G ) : b f ( ω ) = 0 , a.e. w ∈ b G \ Ω } . This follows from the definition of frame and the factthat for f ∈ P W Ω one has k f k L ( G ) = k b f k L (Ω) and h f, t λ ϕ j i L ( G ) = h b f , e − λ + a j i L (Ω) . In particular,
P W Ω = S Λ ( ϕ , . . . , ϕ ℓ ) := span { t λ ϕ j : λ ∈ Λ , j = 1 , . . . , ℓ } . That is, V := P W Ω is a finitely generated Λ-invariant space. Denote by J V themeasurable range function of V as given in (2.4) (see also [3], Section 3, for details).We now use the fiberization mapping T : L ( G ) −→ L ( Q Γ , ℓ (Γ)) defined in (2.2).By Proposition 2.3, for a.e. ω ∈ Q Γ the sequences {T ϕ ( ω ) , . . . , T ϕ ℓ ( ω ) } form aframe of J V ( ω ) ⊆ ℓ (Γ). Therefore, dim( J V ( ω )) ≤ ℓ, for a.e. ω ∈ Q Γ . In our particular situation there is another description of the range function J V ( ω )associated to V . For each ω ∈ Q Γ , define θ ω := { γ ∈ Γ : χ Ω ( ω + γ ) = 0 } , and ℓ ω := θ ω . Write ℓ ω = 0 if θ ω = ∅ . Then, there exist γ ( ω ) , . . . , γ ℓ ω ( ω ) ∈ Γ such that w + γ j ( ω ) ∈ Ω , for all j = 1 , . . . , ℓ ω , which implies that J V ( ω ) ⊆ ℓ ( { δ γ ( ω ) , . . . , δ γ ℓω ( ω ) } ) , for a.e. ω ∈ Q Γ . Moreover, as in Corollary 2.8. of [1], J V ( ω ) = ℓ ( { δ γ ( ω ) , . . . , δ γ ℓω ( ω ) } ) , for a.e. ω ∈ Q Γ . Thus, dim( J V ( ω )) = ℓ ω , which implies that ℓ ω ≤ ℓ, for a.e. ω ∈ Q Γ , andtherefore we obtain that F Ω , Γ ( ω ) = X γ ∈ Γ χ Ω ( ω + γ ) ≤ ℓ, for a.e. ω ∈ Q Γ . This shows that (Ω , Γ) is an m -subtiling pair for b G with m ≤ ℓ. (2) Since Ω is bounded, by Proposition 2.1 there exists a bounded set ∆ containingΩ which is an ℓ -tile of b G by Γ . Now using Theorem 4.1 of [1], there exist a , . . . , a ℓ ∈ G such that E Λ ( a , . . . , a ℓ ) is a Riesz basis of L (∆) . As a consequence, E Λ ( a , . . . , a ℓ )is a frame of L (Ω) . (cid:3) Remark 2.5.
Note that Ω does not need to be bounded: for example, E Z (0) = { e πikx : k ∈ Z } is an orthonormal basis for L (Ω) for Ω = S ∞ n =0 n + ( n +1 , n ] ⊂ R and Ω is not bounded. However, for the proof of part (2) of Theorem 1.1 we need Ω to bebounded since the proof uses Proposition 2.1. Remark 2.6.
Theorem 1.1 for the case ℓ = 1 can be found in [2] . In this case, theproof does not require making use of either the Paley-Wiener space of Ω or the rangefunction associated to it as in the proof given above. Remark 2.7.
In Part (1) of Theorem 1.1 the inequality m ≤ ℓ can be strict as thefollowing example shows: choose Ω ⊂ R d such that (Ω , Z d ) is an ℓ -tiling pair for R d andpick a , . . . , a ℓ such that E Z d ( a , . . . , a ℓ ) is a Riesz basis of L (Ω) . Let Ω ⊂ Ω be anysubset of Ω such that (Ω , Z d ) is an ( ℓ -1)-tiling pair of R d (for example, remove from Ω a fundamental domain of Z d in R d ) . Then E Z d ( a , . . . , a ℓ ) is a frame for L (Ω ) ,and (Ω , Z d ) is not an ℓ -subtiling pair for R d . RAMES OF EXPONENTIALS AND SUB–MULTITILES 7 Optimal frame bounds for sets of exponentials.
The purpose of this section is to develop another condition guaranteeing when a setof exponentials of the form E Λ ( a , . . . , a m ) := { e a j + λ : j = 1 , , . . . , m, λ ∈ Λ } forms a frame for L (Ω), where (Ω , Γ) is an ℓ -subtiling pair for b G , as well as to findoptimal frame bounds for this frame.For the ℓ -subtiling pair (Ω , Γ) of b G , let E be the set of measure zero in Q Γ suchthat F Ω , Γ > ℓ , and let Q := { ω ∈ Q Γ : F Ω , Γ ( ω ) = 0 } . Let f Q Γ := Q Γ \ ( Q ∪ E ) . For each ω ∈ f Q Γ there exist ℓ ω ≤ ℓ and γ ( ω ) , . . . , γ ℓ ω ( ω ) ∈ Γ such that ω + γ j ( ω ) ∈ Ωfor all j = 1 , . . . , ℓ ω (see the proof of Theorem 1.1). Recall that ℓ ω := { γ ∈ Γ : χ Ω ( ω + γ ) = 0 } . (3.1)Given ϕ , . . . , ϕ m ∈ P W Ω = { f ∈ L ( G ) : b f ∈ L (Ω) } , and ω ∈ e Q Γ , consider thematrix T ω = b ϕ ( ω + γ ( ω )) . . . b ϕ m ( ω + γ ( ω ))... ... b ϕ ( ω + γ ℓ ω ( ω )) . . . b ϕ m ( ω + γ ℓ ω ( ω )) (3.2)of size ℓ ω × m. Assume thatΦ Λ := { t λ ϕ j : λ ∈ Λ , j = 1 , . . . , m } is a frame for S Λ ( ϕ , · · · , ϕ m ). By Proposition 2.3, this is equivalent to having thatfor a.e. ω ∈ Q Γ the set Φ ω := {T ϕ j ( ω ) : j = 1 , . . . , m } ⊂ ℓ (Γ)is a frame for J ( ω ) = span {T ϕ ( ω ) , . . . , T ϕ m ( ω ) } ⊂ ℓ (Γ) . Moreover, as in the proofof Theorem 1.1, for a. e. ω ∈ Q Γ , J ( ω ) = ℓ ( { δ γ ( ω ) , . . . , δ γ ℓω ( ω ) } ) is a subspace of ℓ (Γ) of dimension ℓ ω . (Notice that this implies m ≥ ℓ .)It is well known (see, for example, Proposition 3.18 in [7]) that a frame in a finitedimensional Hilbert space is nothing but a generating set. Since the non-zero elementsof T ϕ j ( ω ) are precisely the j -th column of T ω , j = 1 , . . . , m , it follows that Φ Λ is aframe for S Λ ( ϕ , · · · , ϕ m ) if and only if rank ( T ω ) = ℓ ω for a.e. ω ∈ e Q Γ . For ω ∈ f Q Γ , let λ min ( T ω T ∗ ω ) and λ max ( T ω T ∗ ω ) respectively the minimal and maximaleigenvalues of T ω T ∗ ω . It is well known (see Proposition 3.27 in [7]) that the opti-mal lower and upper frame bounds of Φ ω are precisely λ min ( T ω T ∗ ω ) and λ max ( T ω T ∗ ω )respectively. By Proposition 2.3 the optimal frame bounds for Φ Λ are A = | Q Γ | ess inf ω ∈ e Q Γ λ min ( T ω T ∗ ω ) and B = | Q Γ | ess sup ω ∈ e Q Γ λ max ( T ω T ∗ ω ) . (3.3)We have proved the following result: Proposition 3.1.
With the notation and definitions as above, the following are equiv-alent:(i) The set Φ Λ := { t λ ϕ j : λ ∈ Λ , j = 1 , . . . , m } is a frame for S Λ ( ϕ , . . . , ϕ m ) . D. BARBIERI, C. CABRELLI, E. HERN ´ANDEZ, P. LUTHY, U. MOLTER, AND C. MOSQUERA (ii) The matrix T ω given in (3.2) has rank ℓ ω (see (3.1) ) for a.e. ω ∈ e Q Γ . Moreover, in this situation, the optimal frame bounds A and B of Φ Λ are given by (3.3) . Consider now the set of exponentials E Λ ( a , . . . , a m ) := { e λ + a j : λ ∈ Λ , j = 1 , . . . , m } with a , . . . , a m ∈ G . Let ϕ ∈ L ( G ) given by b ϕ = χ Ω . Consider ϕ j := t − a j ϕ , j = 1 , . . . , m. As in the proof of Theorem 1.1, E Λ ( a , . . . , a m ) is a frame for L (Ω) with frame bounds A and B if and only if the setΦ Λ := { t λ ϕ j : λ ∈ Λ , j = 1 , . . . , m } is a frame for P W Ω = S Λ ( ϕ , · · · , ϕ m ) with the same frame bounds.For our particular situation, if ω ∈ f Q Γ , T ω = e a ( ω + γ ( ω )) . . . e a m ( ω + γ ( ω ))... ... e a ( ω + γ ℓ ω ( ω )) . . . e a m ( ω + γ ℓ ω ( ω )) . (3.4)As in Theorem 2.9 of [1] the matrix T ω , for ω ∈ e Q Γ , can be factored as T ω = E ω U ω := e a ( γ ( ω )) . . . e a m ( γ ( ω ))... ... e a ( γ ℓ ω ( ω )) . . . e a m ( γ ℓ ω ( ω )) e a ( ω ) . . . . . . e a m ( w ) . (3.5)Since U ω is unitary and T ω T ∗ ω = E ω E ∗ ω , we have proved the following result: Proposition 3.2.
With the notation and definitions as above, the following are equiv-alent:(i) The set E Λ ( a , . . . , a m ) is a frame for L (Ω) .(ii) The matrix E ω given in (3.5) has rank ℓ ω (see (3.1) ) for a. e. ω ∈ f Q Γ . Moreover, in this situation, the optimal frame bounds A and B of E Λ ( a , . . . , a m ) are given by A = | Q Γ | ess inf ω ∈ e Q Γ λ min ( E ω E ∗ ω ) and B = | Q Γ | ess sup ω ∈ e Q Γ λ max ( E ω E ∗ ω ) . Remark 3.3.
Proposition 3.2 can be found in [1] when Ω is an ℓ -tile and “frame” isreplaced by “Riesz basis”. Example 3.4.
In this example we work with the additive group G = R d and the lattice Λ = Z d . Recall that b G = R d and Γ = Z d . Let Ω ⊂ Ω ⊂ [0 , d be two measurable setsin R d and let γ ∈ Z d ( γ = 0) . Take
Ω = Ω ∪ ( γ + Ω ) , so that (Ω , Z d ) is a 2-subtiling pair of R d . RAMES OF EXPONENTIALS AND SUB–MULTITILES 9
For a , a , . . . , a m ∈ R d consider the set of exponentials E Z d ( a , . . . , a m ) = { e πi h k + a j , • i : k ∈ Z d , j = 1 , . . . , m } . By factoring out e πi h a ,x i we can assume a = 0 . According to Proposition 3.2, to determine the values of a = 0 , a , . . . , a m for whichthe set E Z d (0 , a , . . . , a m ) is a frame for L (Ω) , we need to compute the ranks of thematrices E ω given in (3.5) .For ω ∈ Ω \ Ω , ℓ ω = 1 , E w = (1 , , . . . , , and rank ( E ω ) = 1 = ℓ ω . For ω ∈ Ω , ℓ ω = 2 , and E ω = (cid:18) . . . e πi h a ,γ i . . . e πi h a m ,γ i (cid:19) . (3.6) Let H := S k ∈ Z { x ∈ R d : h x, γ i = k } , that is a countable union of hyperplanes in R d perpendicular to the vector γ . The rank of the matrix given in (3.6) is 2 when atleast one of the a j does not belong to H . In this case, E Z d (0 , a , . . . , a m ) is a framefor L (Ω) as an application of Proposition 3.2.We now compute the optimal frame bounds. For ω ∈ Ω \ Ω , E ω E ∗ ω = ( m ) , so that λ min ( E ω E ∗ ω ) = λ max ( E ω E ∗ ω ) = m . For ω ∈ Ω ,E ω E ∗ ω = (cid:18) m P mj =2 e − πi h a j ,γ i P mj =2 e πi h a j ,γ i m (cid:19) . The eigenvalues of this matrix are λ = m ± (cid:12)(cid:12)(cid:12) m X j =2 e πi h a j ,γ i (cid:12)(cid:12)(cid:12) . Therefore, the optimal lower and upper frame bounds of E Z d (0 , a , . . . , a m ) in L (Ω) are A = m − (cid:12)(cid:12)(cid:12) m X j =2 e πi h a j ,γ i (cid:12)(cid:12)(cid:12) and B = m + (cid:12)(cid:12)(cid:12) m X j =2 e πi h a j ,γ i (cid:12)(cid:12)(cid:12) when a j / ∈ H for some j ∈ { , . . . , m } . Observe that the frame E Z d (0 , a , . . . , a m ) in L (Ω) is tight (with tight frame bound m ) if and only if m X j =2 e πi h a j ,γ i = 0 .This occurs, for example, if the complex numbers { , e πi h a ,γ i , . . . , e πi h a m ,γ i } are thevertices of a regular m -gon inscribed in the unit circle. References [1] E. Agora, J. Antezana, and C. Cabrelli,
Muti-tiling sets, Riesz bases, and sampling near thecritical density in LCA groups . Advances in Math., 285 (2015), 454–477.[2] D. Barbieri, E. Hern´andez and A. Mayeli.
Lattice sub-tilings and frames in LCA groups.
C. R.Acad. Sci. Paris, Ser. 1, 356 (2), (2017), 193-199.[3] C. Cabrelli and V. Paternostro.
Shift-invariant spaces on LCA groups.
J. Funct. Anal., 258(6),(2010), 2034–2059.[4] J. Feldman and F.P. Greenleaf.
Existence of Borel transversals in groups.
Pacific J. Math. 25(1968) 455-461.[5] B. Fuglede,
Commuting self-adjoint partial differential operators and a group theoretic problem .J. Funct. Anal. 16 (1974), 101–121. [6] S. Grepstad, N. Lev,
Multi-tiling and Riesz basis . Advances in Math., 252 (15), (2014), 1–6.[7] D. Han, K. Kornelson, D. Larson, E. Weber,
Frames for undergraduates . AMS, Student Math-ematical Library, Vol. 40, (2007).[8] E. Hewitt, K. A. Ross,
Abstract harmonic analysis. Vol. I: Structure of topological groups,integration theory, group representations.
Springer, 2nd Ed. (1979).[9] M. Kolountzakis,
Multiple lattice tiles and Riesz bases of exponentials . Proc. Amer. Math. Soc.143 (2015), 741–747.[10] S. Pedersen,
Spectral Theory of Commuting Self-Adjoint Partial Differential Operators . Journalof Functional Analysis 73 (1987), 122–134 .[11] R. M. Young,
Introduction to nonharmonic Fourier series . Academic Press, (1980).
Davide Barbieri, Departamento de Matem´aticas, Universidad Aut´onoma de Madrid,28049, Madrid, Spain
E-mail address : [email protected] Carlos Cabrelli, Departamento de Matem´atica, FCEyN, Universidad de BuenosAires and IMAS-UBA-CONICET, Argentina
E-mail address : [email protected] Eugenio Hern´andez, Departamento de Matem´aticas, Universidad Aut´onoma de Madrid,28049, Madrid, Spain
E-mail address : [email protected] Peter Luthy, College of Mount Saint Vincent, Bronx, NY, USA
E-mail address : [email protected] Ursula Molter, Departamento de Matem´atica, FCEyN, Universidad de BuenosAires and IMAS-UBA-CONICET, Argentina
E-mail address : [email protected] Carolina Mosquera, Departamento de Matem´atica, FCEyN, Universidad de BuenosAires and IMAS-UBA-CONICET, Argentina
E-mail address ::