Frames for weighted shift-invariant spaces
aa r X i v : . [ m a t h . F A ] M a y Frames for weighted shift-invariant spaces
Stevan Pilipovi´c , Suzana Simi´c August 4, 2010
Abstract.
In this paper we prove the equivalence of the frame property andthe closedness for a weighted shift-invariant space V pµ (Φ) = n r X i =1 X j ∈ Z d c i ( j ) φ i ( · − j ) (cid:12)(cid:12)(cid:12) { c i ( j ) } j ∈ Z d ∈ ℓ pµ o , p ∈ [1 , ∞ ] , which corresponds to Φ = Φ r = ( φ , φ , . . . , φ r ) T ∈ ( W ω ) r . We, also, constructa sequence Φ k +1 and the sequence of spaces V pµ (Φ k +1 ), k ∈ N , on R , with theuseful properties in sampling, approximations and stability. : 42C15, 42C40, 42C99, 46B15,46B35, 46B20 Key Words and Phrases : p -frame; Banach frame; weighted shift-invariantspace. In this paper, we investigate weighted shift-invariant spaces quoted in theabstract by following the methods from [2] and [25]. Such spaces figure in sev-eral areas of applied mathematics, notably in wavelet theory and approximationtheory ([2], [8]). In recent years, they have been extensively studied by many au-thors (see [1]-[8], [14]-[16], [19], [20], [25], [26]). Sampling with non-bandlimitedfunctions in shift-invariant spaces is a suitable and realistic model for manyapplications, such as modeling signals with the spectrum that is smoother thenin the case of bandlimited functions, or for the numerical implementation (see[6], [9], [10], [12], [13], [17]). These requirements can often be met by choosingappropriate functions in Φ. This means that the functions in Φ have a shapecorresponding to a particular impulse response of a device, or that they arecompactly supported or that they have a Fourier transform decaying smoothlyto zero as | ξ | → ∞ .Weighted shift-invariant spaces V pµ (Φ), p ∈ [1 , ∞ ], where µ is a weight, wereintroduced for the non uniform sampling as a direct generalization of the space V p (Φ) ([1], [26]). The determination of p and the signal smoothness are usedfor optimal compression and coding signals and images (see [9]).1he first aim of this paper is to show that the main result of [2] holds inthe case of weighted shift-invariant spaces which correspond to L pµ and ℓ pµ , i.e.,weighted L p and ℓ p spaces, respectively. Namely, we follow [2] and [25] andprove assertions which need additional arguments depending on the weights.We show that under the appropriate conditions on the frame vectors, there isan equivalence between the concept of p -frames, Banach frames with respectto ℓ pµ and closedness of the space which they generate. A weighted analog ofCorollary 3.2 from [25] simplifies a part of the proof of our main result. Althoughanother part of the proof follows, step by step, the proof of the correspondingtheorem in [2], we think that it is not simple at all, and that it is worth to bedone.The second aim of this paper is to construct V pµ (Φ k +1 ) spaces with speciallychosen functions, φ , φ , . . . , φ k , that generate a Banach frame for the shift-invariant space V pµ (Φ k +1 ). Actually, we take functions from a sequence { φ i } i ∈ Z so that the sequence of Fourier transforms b φ i = θ ( · + iπ ), i ∈ Z , θ ∈ C ∞ ( R ),makes a partition of unity in the frequency domain ( Z = N ∪ − N , N is the set ofnatural numbers and N = N ∪ { } ). We note that properties of the constructedframe guarantee the feasibility of a stable and continuous reconstruction algo-rithm in V pµ (Φ) ([26]). Also, we note that { φ i ( · − k ) | k ∈ Z , i = 1 , . . . , r } formsa Riesz basis for V pµ (Φ) when the spectrum of the Gram matrix [ b Φ , b Φ]( ξ ) isbounded and bounded away from zero (see [8]). The d -dimensional case, d > , is technically more complicated and because of that it is not considered in thispaper.The paper is organized as follows. In Section 2 we quote basic propertiesof subspaces of weighted L p and ℓ p spaces. The weighted shift-invariant spacesare investigated in Section 3, where we presented our first result quoted in theabstract, Theorem 3.10. In Section 4 we show relations between the dual of theFr´echet space T s ∈ N V p (1+ | x | ) s/ (Φ) and the space of periodic distributions. Thecase of periodic ultradistributions is obtain by using subexponential growthfunctions. In Section 5, we use a special sequence of functions { φ k | k ∈ N } to construct a sequence of p -frames. Our construction shows that the samplingand reconstruction problem in the shift-invariant spaces is robust. In the finalremark of Section 5, we list good properties of these frames. Denote by L loc ( R d ) the space of measurable functions integrable over com-pact subsets of R d . For a nonnegative function ω ∈ L loc ( R d ) we say that issubmultiplicative if ω ( x + y ) ≤ ω ( x ) ω ( y ), x , y ∈ R d and a function µ on R d is ω -moderate if µ ( x + y ) ≤ Cω ( x ) µ ( y ), x , y ∈ R d . We assume that ω is continuousand symmetric and both µ and ω call weights, as usual. The standard class ofweights on R d are of the polynomial type ω s ( x ) = (1 + | x | ) s , s ≥
0. To quan-tify faster decay of functions we use the subexponential weights ω ( x ) = e α | x | β ,for some α > < β <
1. Weighted L p spaces with moderate weights2re translation-invariant spaces (see [1]). We, also, consider weighted sequencespaces ℓ pµ ( Z d ) with ω -moderate weight µ . Recall, a sequence c belongs to ℓ pµ ( Z d )if cµ belongs to ℓ p ( Z d ).In the sequel ω is a submultipicative weight, continuous and symmetric and µ is ω -moderate. Let p ∈ [1 , ∞ ). Then (with obvious modification for p = ∞ ) L pω = n f (cid:12)(cid:12) k f k L pω = (cid:16) Z [0 , d (cid:16) X j ∈ Z d | f ( x + j ) | ω ( x + j ) (cid:17) p d x (cid:17) /p < + ∞ o ,W pω := n f (cid:12)(cid:12) k f k W pω = (cid:16) X j ∈ Z d sup x ∈ [0 , d | f ( x + j ) | p ω ( j ) p (cid:17) /p < + ∞ o . Obviously, we have W pω ⊂ W qω ⊂ L ∞ ω ⊂ L qω ⊂ L pω ⊂ L pω , W pω ⊂ W pµ ⊂ W qµ ⊂ L qµ and L pω ⊂ L pµ , where 1 < p < q ≤ + ∞ . For p = 1 and ω = 1 we have L = L . We also have ℓ ω ⊂ ℓ pω ⊂ ℓ qω ⊂ ℓ qµ , for 1 < p < q ≤ + ∞ . From [1] wehave the following properties.1) If f ∈ L pµ , g ∈ L ω and p ∈ [1 , ∞ ], then k f ∗ g k L pµ ≤ k f k L pµ k g k L ω .
2) If f ∈ L pµ , g ∈ W ω and p ∈ [1 , ∞ ], then k f ∗ g k W pµ ≤ k f k L pµ k g k W ω .
3) If c ∈ ℓ pµ and d ∈ ℓ ω , then holds the inequality k c ∗ d k ℓ pµ ≤ k c k ℓ pµ k d k ℓ ω . Denote by WC pµ , p ∈ [1 , ∞ ], a space of all 2 π -periodic functions with theirsequences of Fourier coefficients in ℓ pµ . Let D and D be the sequences of Fouriercoefficients of 2 π -periodic functions K and K , respectively. If D ∗ D ∈ ℓ pµ ,then D ∗ D is the sequence of Fourier coefficients of the product K K . For K = ( K , . . . , K r ) T ∈ ( W C pµ ) r , ( T means transpose) define k K k ℓ pµ, ∗ to be the ℓ pµ norm of its sequence of Fourier coefficients.In the sequel we use the notation Φ = ( φ , . . . , φ r ) T . Define k Φ k H = r P i =1 k φ i k H , where H = L pω , L pω or W pω , p ∈ [1 , ∞ ].We list several lemmas needed to prove our results. Their proofs are analo-gous to the proof of the corresponding lemmas in [2]. Lemma 2.1.
Let f ∈ L pµ and g ∈ W ω , p ∈ [1 , ∞ ] . Then the sequence n Z R d f ( x ) g ( x − j )d x o j ∈ Z d ∈ ℓ pµ and (cid:13)(cid:13)(cid:13)n R R d f ( x ) g ( x − j )d x o j ∈ Z d (cid:13)(cid:13)(cid:13) ℓ pµ ≤ k f k L pµ k g k W ω . Let c = { c i } i ∈ N ∈ ℓ pµ and f ∈ L pω , p ∈ [1 , ∞ ]. We define, as in [2], theirsemi-convolution f ∗ ′ c by ( f ∗ ′ c )( x ) = P j ∈ Z d c j f ( x − j ) , x ∈ R d . emma 2.2. a ) If f ∈ L pω and c ∈ ℓ pµ , p ∈ [1 , ∞ ] , then f ∗ ′ c ∈ L pµ and k f ∗ ′ c k L pµ ≤ k c k ℓ pµ k f k L pω .b ) If f ∈ L pω , p ∈ [1 , ∞ ] , and c ∈ ℓ µ , then k f ∗ ′ c k L pµ ≤ k c k ℓ µ k f k L pω .c ) If f ∈ W pω , p ∈ [1 , ∞ ] , and c ∈ ℓ µ , then k f ∗ ′ c k W pµ ≤ k c k ℓ µ k f k W pω ,d ) If f ∈ W ω and c ∈ ℓ pµ , p ∈ [1 , ∞ ] , then k f ∗ ′ c k W pµ ≤ k c k ℓ pµ k f k W ω . V pµ (Φ) In [11] Feichtinger and Gr¨ochening extended the notation of atomic decom-position to Banach spaces ([10], [12]), while Gr¨ochening [18] introduced a moregeneral concept of decomposition through Banach frames. We recall the defini-tion.Let X be a Banach space and Θ be an associated Banach space of scalarvalued sequences, indexed by I = N or I = Z . Let { f n } ⊂ X ∗ and S : Θ → X be given. The pair ( { f n } n ∈ I , S ) is called a Banach frame for E with respect toΘ if(1) { f n ( x ) } n ∈ I ∈ Θ for each x ∈ X ,(2) there exist positive constants A and B with 0 < A ≤ B < + ∞ such that A k x k X ≤ k{ f n ( x ) n ∈ I }k θ ≤ B k x k X , x ∈ X ,(3) S is a bounded linear operator such that S ( { f n ( x ) } n ∈ I ) = x , x ∈ X .It is said that a collection { φ i ( · − j ) | j ∈ Z d , ≤ i ≤ r } is a p -frame for V pµ (Φ)if there exists a positive constant C (depending on Φ, p and ω ) C − k f k L pµ ≤ r X i =1 (cid:13)(cid:13)(cid:13)n Z R d f ( x ) φ i ( x − j )d x o j ∈ Z d (cid:13)(cid:13)(cid:13) ℓ pµ ≤ C k f k L pµ , f ∈ V pµ (Φ) . (3.1)A typical application is the problem of finding a shift-invariant space modelthat describes a given class of signals or images (e.g. the class of chest X -rays).The observation set of r signals or images f , . . . , f r may be theoretical samples,or experimental data.Recall [1], the shift-invariant spaces are defined by V pµ (Φ) := n f ∈ L pµ | f ( · ) = r X i =1 X j ∈ Z d c ij φ i ( · − j ) , { c ij } j ∈ Z d ∈ ℓ pµ , ≤ i ≤ r o . Remark . If Φ ∈ W ω and µ is ω -moderate, then V pµ (Φ) is a subspace (notnecessarily closed) of L pµ and W pµ for any p ∈ [1 , ∞ ]. If r = 1 and { φ ( · − j ) | j ∈ Z d } is a p -frame for V pµ ( φ ), then V pµ ( φ ) is a closed subspace of L pµ and W pµ for p ∈ [1 , ∞ ] (see [23]). 4et [ b Φ , b Φ]( ξ ) = h P k ∈ Z d b φ i ( ξ + 2 kπ ) b φ j ( ξ + 2 kπ ) i ≤ i ≤ r, ≤ j ≤ r where we assumethat b φ i ( ξ ) b φ j ( ξ ) is integrable for any 1 ≤ i, j ≤ r. Let A = ( a ( j )) j ∈ Z d be an r × ∞ matrix and AA T = h P j ∈ Z d a i ( j ) a i ′ ( j ) i ≤ i,i ′ ≤ r . Then rank A = rank AA T .Also, since [ b Φ , b Φ]( ξ ) is continuous (as a function with r components) forany Φ ∈ ( L ω ) r , it follows that (cid:8) ξ ∈ R d | rank (cid:2)b Φ( ξ + 2 kπ ) k ∈ Z d (cid:3) > k (cid:9) is anopen set for any k > ∈ ( L ω ) r .Denote by Σ µα the family of all α -slant matrices A = [ a ( j, k ) j ∈ Z d ,k ∈ Z d ] with k A k Σ ωα = X k ∈ Z d sup j,k ∈ Z d | a ( k, j ) | χ k +[0 , d ( k − αj ) < ∞ , where µ is a weight on R d and α is a positive number. The slanted matricesappear in wavelet theory, signal processing and sampling theory (see [25]). NoteΣ µα ⊂ Σ µ α for any weight µ where µ ≡ φ , . . . , φ r ) T ∈ ( L pω ) r for p ∈ [1 , ∞ ).To prove Theorem 3.10, we need several lemmas. First we recall a resultfrom [2]. Lemma 3.2 ([2]) . The following statements are equivalent.
1) rank (cid:2)b Φ( ξ + 2 jπ ) j ∈ Z d (cid:3) is a constant function on R d .
2) rank[ b Φ , b Φ]( ξ ) is a constant function on R d . There exists a positive constant C independent of ξ such that C − [ b Φ , b Φ]( ξ ) ≤ [ b Φ , b Φ]( ξ ) [ b Φ , b Φ]( ξ ) T ≤ C [ b Φ , b Φ]( ξ ) , ξ ∈ [ − π, π ] d . The proofs of the following two lemmas are similar to proofs of the corre-sponding lemmas from [2]; hence we will not include them here. The secondone provides a localization technique in Fourier domain. It allows us to replacelocally the generator b Φ of size r by b Ψ ,λ of size k . Lemma 3.3.
All the entries of [ b Φ , b Φ]( ξ ) belong to WC ω and are continuous. Lemma 3.4.
Let the rank[ b Φ( ξ + 2 jπ ) j ∈ Z d ] = k ≥ for all ξ ∈ R d . Then thereexist a finite index set Λ , points η λ ∈ [ − π, π ] d , ≤ δ λ < / , a nonsingular π -periodic r × r matrix P λ ( ξ ) with all entries in the class WC ω and K λ ⊂ Z d with cardinality k for all λ ∈ Λ , such that: ( i ) S λ ∈ Λ B ( η λ , δ λ / ⊃ [ − π, π ] d , where B ( x , δ ) denotes the open ball in R d with center x and radius δ ; ( ii ) P λ ( ξ ) b Φ( ξ ) = " b Ψ ,λ ( ξ ) b Ψ ,λ ( ξ ) , ξ ∈ R d , λ ∈ Λ , where Ψ ,λ and Ψ ,λ arefunctions from R d to C k and C r − k , respectively, satisfying rank (cid:2) b Ψ ,λ ( ξ + 2 kπ ) k ∈ K λ (cid:3) = k , ξ ∈ B ( η λ , δ λ ) , Ψ ,λ ( ξ ) = 0 , ξ ∈ B ( η λ , δ λ /
5) + 2 π Z d . Furthermore, there exist π -periodic C ∞ functions h λ , λ ∈ Λ , on R d such that P λ ∈ Λ h λ ( ξ ) = 1 , ξ ∈ R d , and supp h λ ⊂ B ( η λ , δ λ /
2) + 2 π Z d . The next lemma is needed for the proof of Theorem 3.10. Although theformulation is not the same as [2, Lemma 3], the proof is based on the sameprocedure, and we omit it.
Lemma 3.5. ( a ) Let φ ∈ L pω s if p ∈ [1 , ∞ ) and φ ∈ W ω s if p = + ∞ . Assumethat P j ∈ Z d φ ( · + j ) = 0 . Then for any function h on R d which satisfies | h ( x ) | ≤ C (1 + | x | ) − s − d − , | h ( x ) − h ( y ) | ≤ C | x − y | (1 + min {| x | , | y |} ) s + d +1 , we have lim n → + ∞ − nd (cid:13)(cid:13)(cid:13) X j ∈ Z d h (2 − n j ) φ ( · − j ) (cid:13)(cid:13)(cid:13) L pωs = 0 . ( b ) Let µ ( x ) = e α | x | β . Let φ ∈ L pµ if p ∈ [1 , ∞ ) and φ ∈ W µ if p = + ∞ .Assume that P j ∈ Z d φ ( · + j ) = 0 . Then for any function h on R d which satisfies | h ( x ) | ≤ C e − ( α + d +1) | x | β , | h ( x ) − h ( y ) | ≤ C | x − y | e − ( α + d +1)(1+min {| x | β , | y | β } ) , we have lim n → + ∞ − nd (cid:13)(cid:13)(cid:13) X j ∈ Z d h (2 − n j ) φ ( · − j ) (cid:13)(cid:13)(cid:13) L pµ = 0 . Next, we give a result on the equivalence of ℓ pµ -stability of the synthesisoperator S Φ for a different p ∈ [1 , ∞ ] (see [25]; here we have Λ = { , , . . . , r } ). Proposition 3.6. [25, Corollary 3.2] Let
Φ = ( φ , . . . , φ r ) T ∈ ( W ω ) r , p ∈ [1 , ∞ ] and µ is ω -moderate. Define the synthesis operator S Φ : ( ℓ pµ ( Z d )) r V pµ (Φ) by S Φ : c = { c ij } j ∈ Z d , ≤ i ≤ r r X i =1 X j ∈ Z d c ij φ i ( · − j ) . If the synthesis operator S Φ has ℓ pµ -stability for some p ∈ [1 , ∞ ] , i.e., there existsa positive constant C such that C − k c k ( ℓ pµ ( Z d )) r ≤ k S Φ c k L pµ ≤ C k c k ( ℓ pµ ( Z d )) r , (3.2) for all c ∈ ( ℓ pµ ( Z d )) r , then the synthesis operator S Φ has ℓ qµ -stability for any q ∈ [1 , ∞ ] . As a consequence of the previous proposition, we have the next result.6 roposition 3.7. [25, Corollary 3.3] Let p ∈ [1 , ∞ ] and Φ = ( φ , . . . , φ r ) T ∈ ( W ω ) r , and µ ω -moderate. If the synthesis operator S Φ has ℓ pµ -stability, thenthere exists another family Ψ = ( ψ , . . . , ψ r ) T ∈ ( W ω ) r such that the inverse ofthe synthesis operator S Φ is given by ( S Φ ) − ( f ) = nZ R d f ( x ) ψ i ( x − j )d x o ≤ i ≤ r,j ∈ Z d , f ∈ V pµ . Proposition 3.6 and 3.7 imply:
Theorem 3.8.
Let
Φ = ( φ , . . . , φ r ) T ∈ ( W ω ) r , p ∈ [1 , ∞ ] , and µ is ω -moderate. Then the following three statements are equivalent. a ) The synthesis operator S Φ has ℓ p µ -stability. b ) V p µ (Φ) is closed in L p µ . c ) There exists
Ψ = ( ψ , . . . , ψ r ) T ∈ ( W ω ) r , such that f = r X i =1 X j ∈ Z d h f, ψ i ( · − j ) i φ i ( · − j ) , f ∈ V p µ (Φ) . Also we have the next assertion. d ) If the synthesis operator S Φ has ℓ p µ -stability, then the collection { φ i ( ·− j ) | j ∈ Z d , ≤ i ≤ r } is a p -frame for V p µ (Φ) .Proof. The implication a ) ⇒ c ) is a consequence of Proposition 3.6 (see Propo-sition 3.7). c ) ⇒ a ): Let f = r P i =1 P j ∈ Z d h f, ψ i ( · − j ) i φ i ( · − j ) and c i = {h f, ψ i ( · − j ) i} j ∈ Z d , ≤ i ≤ r. Then k c k ( ℓ pµ ) r = r X i =1 (cid:13)(cid:13)(cid:13)n Z R d f ( x ) ψ i ( x − j )d x o j ∈ Z d ≤ C k f k L p µ , where C = r P i =1 k ψ i k W ω . Using Lemma 2.1 and the inequality (2.2), we obtainthe right-hand side of (3.2).The equivalence a ) ⇔ b ) follows from standard functional analytic arguments(see [2, Theorem 2, Lemma 4]). d ) Lemma 2.1 implies that {h f, φ i ( · − j ) i} ∈ ℓ p µ , 1 ≤ i ≤ r , and r X i =1 (cid:13)(cid:13)(cid:13)n Z R d f ( x ) φ i ( x − j )d x o j ∈ Z d ≤ k f k L p µ r X i =1 k φ i k W ω . ℓ pµ -stability implies k f k L p µ ≤ C r X i =1 (cid:13)(cid:13)(cid:13)n Z R d f ( x ) φ i ( x − j )d x o j ∈ Z d (cid:13)(cid:13)(cid:13) ℓ p µ . Remark . Note that ℓ pµ -stability of the synthesis operator implies ℓ qµ -stability,for any q ∈ [1 , ∞ ] ([25]), so the statements b ), c ) and d ), do not depend on p ∈ [1 , ∞ ].Now, we give our main result. Theorem 3.10.
Let
Φ = ( φ , . . . , φ r ) T ∈ ( W ω ) r , p ∈ [1 , ∞ ] , and µ is ω -moderate. Then the following statements are equivalent. i ) V p µ (Φ) is closed in L p µ . ii ) { φ i ( · − j ) | j ∈ Z d , ≤ i ≤ r } is a p -frame for V p µ (Φ) . iii ) There exists a positive constant C such that C − [ b Φ , b Φ]( ξ ) ≤ [ b Φ , b Φ]( ξ )[ b Φ , b Φ]( ξ ) T ≤ C [ b Φ , b Φ]( ξ ) , ξ ∈ [ − π, π ] d .iv ) There exist positive constants C and C (depending on Φ and ω ) suchthat C k f k L p µ ≤ inf f = r P i =1 φ i ∗ ′ c i r X i =1 k{ c ij } j ∈ Z d k ℓ p µ ≤ C k f k L p µ , f ∈ V p µ (Φ) . (3.3) v ) There exists
Ψ = ( ψ , . . . , ψ r ) T ∈ ( W ω ) r , such that f = r X i =1 X j ∈ Z d h f, ψ i ( ·− j ) i φ i ( ·− j ) = r X i =1 X j ∈ Z d h f, φ i ( ·− j ) i ψ i ( ·− j ) , f ∈ V p µ (Φ) . Proof.
If the synthesis operator has ℓ pµ -stability, then the statement iv ) is sat-isfied. Conversely, if the statement iv ) is satisfied, then the right-hand side of(3.2) (with p = p ) immediately follows. Using c ) from Theorem 3.8, we ob-tain the left-hand side of (3.2). Hence, by Theorem 3.8, we have i ) ⇔ iv ) and iv ) ⇒ ii ). The equivalence iv ) ⇔ v ) follows from Lemma 2.1.We follow [2] to prove iii ) ⇒ v ) and ii ) ⇒ iii ), and carefully check the useof weights. iii ) ⇒ v ). Let B λ ( ξ ) = H λ ( ξ ) P λ ( ξ ) T h b Ψ ,λ , b Ψ ,λ i ( ξ ) −
00 I ! P λ ( ξ ),for h λ ( ξ ), P λ ( ξ ) and b Ψ ,λ as in Lemma 3.4. We have B λ ( ξ ) ∈ WC pω , for all8 ∈ [1 , + ∞ ]. Define b Ψ( ξ ) = P λ ∈ Λ h λ ( ξ ) B λ ( ξ ) b Φ( ξ ). One has Ψ ∈ W ω . Forany f ∈ V pµ (Φ), define g ( x ) = r P i =1 P j ∈ Z d h f, ψ i ( x − j ) i φ i ( x − j ), x ∈ R d . Since f ∈ V pµ (Φ), there exists a 2 π -periodic distribution A ( ξ ) ∈ WC pµ such that b f ( ξ ) = A ( ξ ) T b Φ( ξ ). By Lemma 3.4, we have b g ( ξ ) = b f ( ξ ).Since b Ψ( ξ ) = P λ ∈ Λ h λ ( ξ ) B λ ( ξ ) b Φ( ξ ), for f = r P i =1 P j ∈ Z d h f, φ i ( · − j ) i ψ ( · − j ) theproof is similar. ii ) ⇒ iii ). Let k = min ξ ∈ R d rank (cid:2)b Φ( ξ + 2 kπ ) k ∈ Z d (cid:3) and letΩ k = (cid:8) ξ ∈ R d | rank (cid:2)b Φ( ξ + 2 kπ ) k ∈ Z d (cid:3) > k (cid:9) . Then Ω k = R d . It is sufficient to prove that Ω k = ∅ (see Lemma 3.2). Supposethat Ω k = ∅ . Since Ω k is open set, then ∂ Ω k = ∅ and rank (cid:2)b Φ( ξ +2 kπ ) (cid:3) k ∈ Z d = k , for any ξ ∈ ∂ Ω k , and max ξ ∈ B ( ξ ,δ ) rank (cid:2)b Φ( ξ + 2 kπ ) (cid:3) k ∈ Z d > k , δ > . ByLemma 3.4, there exist a nonsingular 2 π -periodic r × r matrix P ξ ( ξ ) with allentries in the class WC ω , δ > K ⊂ Z d with cardinality k . Define Ψ ξ , b Ψ ξ ( ξ ) as in Lemma 3.4. The construction of Ψ ξ and (2.2) imply Ψ ξ ∈ W ω .Choose n such that 2 − n < δ and define α n ( ξ ), H n,ξ ( ξ ) and e H n,ξ ( ξ ) as in[2]. For any 2 π -periodic distribution F ∈ WC p µ define, g n , for n ≥ n + 1, as in[2]. Note that g n ∈ V p µ (Φ) and [ b g n , b Ψ ,ξ ]( ξ ) = 0. This leads to k [ b g n , b Φ]( ξ ) k ℓ p µ, ∗ ≤ C k g n k L p µ kF − ( H n,ξ ( ξ ) b Ψ ,ξ ( ξ )) k L ∞ ω . Using Lemma 3.5, we obtain lim n → + ∞ kF − ( H n,ξ ( ξ ) b Ψ ,ξ ( ξ )) k L ∞ ω = 0. Thereexists a sequence ρ n , n ≥ n , such that k [ b g n , b Φ]( ξ ) k ℓ p µ, ∗ ≤ ρ n k g n k L p µ andlim n → + ∞ ρ n = 0. This, together with the assumption ii ) and k [ b g n , b Φ]( ξ ) k ℓ p µ, ∗ = (cid:13)(cid:13)(cid:13)n Z R d g n ( ξ )Φ( ξ − j ) dx o j ∈ Z d (cid:13)(cid:13)(cid:13) ℓ p µ ≥ C k g n k L p µ , leads to g n = 0, n ≥ n + 1. Then e H n,ξ ( ξ )[ b Ψ ,ξ , b Ψ ,ξ ]( ξ )( α n ( ξ )) − b Ψ ,ξ ( ξ ) = e H n,ξ ( ξ ) b Ψ ,ξ ( ξ ) , (3.4)for any 2 π -periodic distribution F ∈ WC p µ and n ≥ n + 1. We, also, get e H n,ξ ( ξ )[ b Ψ ,ξ , b Ψ ,ξ ]( ξ )( α n ( ξ )) − b Ψ ,ξ ( ξ ) = 0 , ξ ∈ B ( ξ , − n − ) + 2 π Z d . So, from (3.4) and the fact that it is valid for all n ≥ n +1, we have b Ψ ,ξ ( ξ ) = 0, ξ ∈ B ( ξ , − n − ) + 2 π Z d . This contradicts the fact that b Ψ ,ξ ( ξ ) = 0, ∀ ξ ∈ B ( ξ , δ ) + 2 π Z d , 0 < δ < δ .With this we complete the proof ii ) ⇒ iii ) and the proof of the theorem.9 emark . Note that conditions in Theorem 3.8 and Theorem 3.10 do notdepend on p ∈ [1 , ∞ ], so we obtain the next corollary. Corollary 3.12.
Let Φ ∈ ( W ω ) r and p ∈ [1 , ∞ ] . i ) If { φ i ( ·− j ) | j ∈ Z d , ≤ i ≤ r } is a p -frame for V p µ (Φ) , then { φ i ( ·− j ) | j ∈ Z d , ≤ i ≤ r } is a p -frame for V pµ (Φ) , for any p ∈ [1 , ∞ ] . ii ) If V p µ (Φ) is closed in L p µ and W p µ , then V pµ (Φ) is closed in L pµ and W pµ ,for any p ∈ [1 , ∞ ] .Remark . ( v ) ⇒ ( ii ) implies that { ψ i ( · − j ) | ≤ i ≤ r, j ∈ Z d } is a dual p -frame of { φ i ( · − j ) | ≤ i ≤ r, j ∈ Z d } . So, the p -frame for V pµ (Φ) is a Banachframe (with respect to ℓ pµ ). We will use the notation V ps instead of V p (1+ | x | ) s/ (similarly for ℓ ps ). Since ℓ ps and V ps are isomorphic Banach spaces for all s ≥ p ∈ [1 , ∞ ], we have V ps (Φ) ⊂ V ps (Φ) for 0 ≤ s ≤ s , p ∈ [1 , ∞ ]. We define Fr´echet spaces X F,p , p ∈ [1 , ∞ ], as X F,p = T s ∈ N V ps (Φ). Clearly, X F,p is dense in V ps (Φ) for all s ∈ N .The corresponding sequence space is Q F,p = T s ∈ N ℓ ps , p ∈ [1 , ∞ ], which is thespace of rapidly decreasing sequences s . By Corollary 3.12 it follows that thedefinition of X F,p does not depend on p ∈ [1 , ∞ ]. So we use notation X F , Q F instead of X F,p , Q F,p . The set { Φ( · − k ) | k ∈ Z d } forms a F -frame for X F since it forms a Banach frame for every space in the intersection (see [23] forthe definition).Since the corresponding function space for s is the space of rapidly decreasingfunctions S = { f | k f k m = sup n ≤ m (1 + | x | ) m/ | f ( n ) ( x ) | < + ∞} , and its dual is S ′ - the space of tempered distributions, we obtain that the dual space X ′ F isisomorphic to (a complemented subspace of) S ′ .Denote by P ( − π, π ) the space of smooth 2 π - periodic functions on R d withthe family of norms | θ | k = sup {| θ ( k ) ( t ) | ; t ∈ ( − π, π ) } , k ∈ N . It is a Fr´echetspace and its dual is the space of 2 π -periodic tempered distributions. We saythat T is a 2 π -periodic distribution if it is a tempered distribution on R d and T = T ( · + 2 jπ ), for all j ∈ Z d . Denote by P ′ ( − π, π ) the space of periodictempered distributions (see [24]). Recall that F ( h ) = ˆ h = R R d e − π √− t · h ( t )d t for h ∈ L . Theorem 4.1.
Let
Φ = ( φ , . . . , φ r ) T ∈ T s ≥ ( W s ) r and Ψ = ( ψ , . . . , ψ r ) T beits dual frame ( according to v ) of Theorem 3.10 ) . Then X F = F − (cid:16) r X i =1 ˆ φ i · P ( − π, π ) (cid:17) , X ′ F = F − (cid:16) r X i =1 ˆ ψ i · P ′ ( − π, π ) (cid:17) n the topological sense. Let f = r X k =1 X p ∈ Z d c kp φ k ( · − p ) ∈ X F and F = r X i =1 X j ∈ Z d d ij ψ i ( · − j ) ∈ X ′ F . The dual pairing is given by h F, f i = r X i =1 r X k =1 D b ψ i ( ξ ) b φ k ( − ξ ) X j ∈ Z d d ij e πjξ √− , X p ∈ Z d c kp e − πpξ √− E , (4.1) where f = r P k =1 P p ∈ Z d c kp φ k ( · − p ) ∈ X F and F = r P i =1 P j ∈ Z d d ij ψ i ( · − j ) ∈ X ′ F .In particular, we have Z R d ϕ i ψ k d t = Z R d b ϕ i ( ξ ) c ψ k ( − ξ )d ξ = δ ik , ≤ i, k ≤ r .Proof. Since P p ∈ Z d c kp e π √− pξ ∈ P ( − π, π ), we obtain the structure of f ∈ X F asin the theorem. The same explanation works for X ′ F .By the fact that h F ( x ) , f ( x ) i = h b F ( ξ ) , b f ( − ξ ) i , we have that (4.1) follows.Let d i = δ ik , i = 1 , . . . , r , and d ij = 0, j = 0, and, also, let c k = δ ik for k = 1 , . . . , r and c kp = 0, p = 0. Using that, we obtain h F ( ξ ) , f ( ξ ) i = r X i =1 r X k =1 h b ψ i ( ξ ) , b φ k ( − ξ ) d i , c k i = Z R d b ψ k ( ξ ) b φ k ( − ξ )d ξ, ≤ k ≤ r. On the other hand f ( x ) = h f ( x ) , ψ k ( x ) i φ k ( x ) and f = φ k for some 1 ≤ k ≤ r , so we obtain h f, ψ k i = 1. Since F = ψ k , we get h F, f i = h f, ψ k i = 1.Finally, we have Z R d b ϕ i ( ξ ) c ψ k ( − ξ )d ξ = δ ik , 1 ≤ i, k ≤ r .Let β ∈ (0 , µ k = e k | x | β , k ∈ N , and the co-rresponding spaces V pµ k (Φ) and their intersection X ( β ) F,p = T k ∈ N V pµ k (Φ). It is aFr´echet space not depending on p , so we use notation X ( β ) F . The correspond-ing sequence space is s ( β ) = T k ∈ N ℓ pµ k , i.e., the space of subexponentially rapidlydecreasing sequences determining the space of periodic tempered ultradistribu-tions via the mapping s ( β ) ∋ ( a j ) j ∈ Z d ↔ P j ∈ Z d a j e jξ √− ∈ P ( − π, π ) (see [22]). p -frames Let θ be a smooth non negative function such that θ ( x ) = 1, x ∈ [ − π + ε, π − ε ],for 0 < ε < , and supp θ ⊆ [ − π, π ]. Let φ k ( x ) = F − ( θ ( · + kπ ))( x ), x ∈ R ,11 ∈ Z . We can divide every θ ( · + kπ ) with the sum P k ∈ Z θ ( · + kπ ) in orderto obtain the partition of unity. By the Paley-Wiener theorem, we have that φ k ∈ W µ ( R ), k ∈ Z . We say that set { φ i , φ i , . . . , φ i r } , i < i < · · · < i r , isa set of r successive functions if i n = i + ( n − n = 2 , . . . , r . Note that forevery ξ ∈ R there exist ξ ∈ ( − π, π ) and k ∈ Z such that ξ = ξ + kπ .Now, we consider the following three cases.1 ◦ The case of two successive functions.If Φ = ( φ i , φ i +1 ) T , i ∈ Z , then rank[ b Φ( ξ + 2 jπ ) j ∈ Z ], ξ ∈ R , is not a constantfunction on R . In this case, for the matrix [ b Φ( ξ + 2 jπ ) j ∈ Z ], we obtain the 2 × ∞ matrix A ( ξ ) = (cid:20) · · · α ξ · · ·· · · α ξ − α ξ · · · (cid:21) , which depends on ξ ∈ ( − π, π ), where α ξ − = θ ( ξ − π ), α ξ = θ ( ξ ) and α ξ = θ ( ξ + π ).For ξ = π , we have α ξ = 0, α ξ − = 0, and for ξ = − π , we have α ξ = 0, α ξ = 0. Since rank A ( ξ ) = 1 and rank A ( ξ ) = 2, we conclude that forsuccessive functions φ i , φ i +1 , i ∈ Z , the rank of the matrix [ b Φ( ξ + 2 jπ ) j ∈ Z ] isnot a constant function on R .2 ◦ The case of three successive functions.If Φ = ( φ i , φ i +1 , φ i +2 ) T , i ∈ Z , then rank[ b Φ( ξ + 2 jπ ) j ∈ Z ] is a constantfunction on R . We have that rank[ b Φ( ξ + 2 jπ ) j ∈ Z ] = 2, for all ξ ∈ R .Indeed, the matrix [ b Φ( ξ + 2 jπ ) j ∈ Z ], ξ ∈ R , is 3 × ∞ matrix B ( ξ ) = · · · α ξ · · ·· · · α ξ − α ξ · · ·· · · α ξ · · · , which depends on ξ ∈ ( − π, π ), where α ξ − = θ ( ξ − π ), α ξ = θ ( ξ ) and α ξ = θ ( ξ + π ). Since, θ ( ξ ) = 0 for all ξ ∈ ( − π, π ), the matrix B ( ξ ) has 2columns with non-zero elements for all ξ ∈ ( − π, π ). So, rank[ b Φ( ξ + 2 jπ ) j ∈ Z ] isa constant function on R and rank[ b Φ( ξ + 2 jπ ) j ∈ Z ] = 2, for all ξ ∈ R .3 ◦ The case of r > r + 1 successive functions φ i , φ i +1 , . . . , φ i + r , r >
2, we have dif-ferent situations described in the next lemma.
Lemma 5.1. a ) If Φ = ( φ i , φ i +1 , . . . , φ i + r ) T , for i ∈ Z , r ∈ N + 1 , then rank[ b Φ( ξ + 2 jπ ) j ∈ Z ] is not a constant function on R . b ) If Φ = ( φ i , φ i +1 , . . . , φ i + r ) T , i ∈ Z , r ∈ N , then rank[ b Φ( ξ + 2 jπ ) j ∈ Z ] is a constant function on R and we have, for all ξ ∈ R , and r = 2 n , n ∈ N , rank[ b Φ( ξ + 2 jπ ) j ∈ Z ] = n + 1 .Proof. Since supports of products b φ i ( ξ + 2 j π ) b φ i ( ξ + 2 j π ) are non-empty ifthe arguments are of the form ξ − π , ξ , ξ + π , modulo 2 jπ , j ∈ Z , we have that12nly blocks with elements (cid:20) θ ( ξ ) θ ( ξ + 2 π ) θ ( ξ − π ) θ ( ξ + π ) (cid:21) or (cid:20) θ ( ξ − π ) θ ( ξ + π ) θ ( ξ − π ) θ ( ξ ) (cid:21) , can determine the rank of the matrix [ b Φ( ξ + 2 jπ ) j ∈ Z ]. For any other choice of2 × a ) Let Φ = ( φ i , φ i +1 , . . . , φ i +(2 n − ) T , n ∈ N .For the matrix [ b Φ( ξ + 2 jπ ) j ∈ Z ], we obtain the r × ∞ matrix A r ( ξ ) = · · · α ξ · · · · · ·· · · α ξ − α ξ · · · · · ·· · · α ξ · · · · · ·· · · α ξ − α ξ · · · · · ·· · · α ξ · · · · · · ... ... ... ... ... · · · ... ... ... · · · · · · α ξ · · ·· · · · · · α ξ − α ξ · · · , where α ξ − = θ ( ξ − π ), α ξ = θ ( ξ ) and α ξ = θ ( ξ + π ), ξ ∈ ( − π, π ).For ξ = π , we have α ξ = 0, α ξ − = 0, and for ξ = − π , we obtain α ξ = 0, α ξ = 0. Since rank A r ( ξ ) = n and rank A r ( ξ ) = n + 1, we conclude that foreven number of successive functions φ i , φ i +1 , . . . , φ i +(2 n − , i ∈ Z , n ∈ N , therank of the matrix [ b Φ( ξ + 2 jπ ) j ∈ Z ] is not a constant function on R .( b ) Let Φ = ( φ i , φ i +1 , . . . , φ i +2 n ) T , i ∈ Z , n ∈ N . The matrix[ b Φ( ξ + 2 jπ ) j ∈ Z ] = · · · α ξ · · · · · ·· · · α ξ − α ξ · · · · · ·· · · α ξ · · · · · ·· · · α ξ − α ξ · · · · · · ... ... ... ... ... · · · ... ... ... · · · · · · α ξ − α ξ · · ·· · · · · · α ξ · · · , has the constant rank on R . Indeed, since α ξ = 0 for all ξ ∈ ( − π, π ), thematrix [ b Φ( ξ + 2 jπ ) j ∈ Z ] has n + 1 columns with non-zero elements for all ξ ∈ R and rank[ b Φ( ξ + 2 jπ ) j ∈ Z ] = n + 1, for all ξ ∈ R .As a consequence of Corollary 3.10 and Lemma 5.1, 1 ◦ we have the nextresult. Theorem 5.2.
Let
Φ = ( φ i , φ i +1 , . . . , φ i +2 n ) T , for i ∈ Z , n ∈ N . Then V pµ (Φ) is closed in L pµ , for any p ∈ [1 , ∞ ] , and { φ i + s ( · − j ) | j ∈ Z , ≤ s ≤ n } is a p -frame for V pµ (Φ) for any p ∈ [1 , ∞ ] . emark . In this way we obtain the sequence of closed spaces V pµ ( φ , φ , φ ), V pµ ( φ , φ , φ , φ , φ ), V pµ ( φ , φ , . . . , φ ), etc. We also conclude that spacesgenerated with even numbers of successive functions, for example V pµ ( φ , φ ), V pµ ( φ , φ , . . . , φ ), are not closed subspaces of L pµ . Theorem 5.4.
Let
Φ = ( φ k , φ k , . . . , φ k r ) T , k < k < · · · < k r , r ∈ N , k , k , . . . , k r ∈ Z , and V pµ,k ,k ,...,k r = V pµ (Φ) . We consider the following cases. i ) k i +1 − k i > , i = 1 , . . . , r − ; ii ) If for some i ∈ { , , . . . , r } holds k i +1 − k i = 1 , then there exists n ∈ N , ≤ n ≤ r , such that k i + 2 , k i + 3 ,. . . , k i + 2 n are elements ofthe set { k , . . . , k r } .In these cases the following statements hold. ◦ rank[ b Φ( ξ + 2 jπ ) j ∈ Z ] is a constant function for all ξ ∈ R . ◦ V pµ (Φ) is closed in L pµ for any p ∈ [1 , ∞ ] . ◦ { φ k i ( · − j ) | j ∈ Z , ≤ i ≤ r } is a p -frame for V pµ (Φ) for any p ∈ [1 , ∞ ] .Remark . (1) We refer to [4] and [26] for the γ -dense set X = { x j | j ∈ J } .Let φ k ( x ) = F − ( θ ( · − kπ ))( x ), x ∈ R . Following the notation of [26], weput ψ x j = φ x j where { x j | j ∈ J } is γ -dense set determined by f ∈ V ( φ ) = V ( F − ( θ )). Checking the proofs of Theorems 3.1, 3.2 and 4.1 in [26], we obtainthe same conclusions as in these theorems. These theorems show the conditionsand explicit C p and c p such that the inequality c p k f k L pµ ≤ (cid:16) X j ∈ J |h f, ψ x j i µ ( x j ) | p (cid:17) /p ≤ C p k f k L pµ holds. This inequality guarantee the feasibility of a stable and continuous re-construction algorithm in the signal spaces V pµ (Φ).(2) Since the spectrum of the Gram matrix [ b Φ , b Φ]( ξ ), for Φ defined in The-orem 5.4, is bounded and bounded away from zero (see [8]), then the family { Φ( · − j ) | j ∈ Z } forms a p -Riesz basis for V pµ (Φ).(3) For the appropriate choice of function Φ, for example Φ defined in The-orem 5.4, the associated Gram matrix satisfies a suitable Munckenhoupt A condition (see [21]), so the system { Φ( · − j ) | j ∈ Z } is stable in L µ ( R ).(4) Frames of the above type may be useful in applications since they satisfyassumptions of Theorem 3 . . φ k , . . . , φ k r .14 cknowledgment The authors are indebted to the referee for pointing out ℓ pµ -stability of thesynthesis operator which helped us to improve and simplify the proof of the maintheorem and include Theorem 3.8 in our manuscript. Also, we are grateful tothe referee for the additional useful literature suggested by him.The authors were supported in part by the Serbian Ministry of Science andTechnological Developments (Project 174024). References [1]
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J. Xian, S. Li , Sampling set conditions in weighted multiply generatedshift-invariant spaces and their applications . Appl. Comput. Harmon. Anal. (2) (2007), 171-180. Department of Mathematics and Informatics,Faculty of Science,University of Novi Sad,Trg Dositeja Obradovica 4,21000 Novi Sad,SerbiaE-mail: [email protected] Department of Mathematics and Informatics,Faculty of Science,University of Kragujevac,Radoja Domanovi´ca 12,34000 Kragujevac,SerbiaE-mail: [email protected]@kg.ac.rs