A characterization of nonhomogeneous wavelet dual frames in Sobolev spaces

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1 Zhang and Li Journal of Inequalitie and Application 016) 016:88 DOI / R E S E A R C H Open Acce A characterization of nonhomogeneou wavelet dual frame in Sobolev pace Jian-Ping Zhang * and Yun-Zhang Li * Correpondence: zhjp198@ .bjut.edu.cn College of Applied Science, Beijing Univerity of Technology, Beijing, 10014, China Abtract In recent year, nonhomogeneou wavelet frame have attracted ome mathematician interet. Thi paper invetigate uch problem in a Sobolev pace etting. A characterization of nonhomogeneou wavelet dual frame in Sobolev pace pair i obtained. MSC: 4C40; 4C15 Keyword: Beel equence; frame; nonhomogeneou wavelet dual frame; Sobolev pace 1 Introduction Wavelet frame in L ) have been widely invetigated by many author [1 8]. In particular, homogeneou wavelet dual frame in L ) were firt characterized by Han [9], and then tudied by Bownik [3]. For homogeneou wavelet dual frame, regularity and vanihing moment have been both required. However, for nonhomogeneou wavelet dual frame in Sobolev pace pair H ), H )),theycanbeeparated.itmakeiteay to contructdualframe ee [10 14] for detail). Thi paper i devoted to characterizing nonhomogeneou wavelet dual frame in Sobolev pace pair H ), H ))viaapair of equation. Before proceeding, we introduce ome notion and notation. We denote by Z and N the et of integer and the et of poitive integer, repectively. Let d N. Wedenoteby T d =[0,1) d the d-dimenional toru. For a Lebegue meaurable et E in,wedenote by E it Lebegue meaure and χ E the characteritic function of E, repectively. And we write δ for the Dirac equence, i.e., δ 0,0 =1andδ 0,k =0for0 k Z d.thefourier tranform of a function f L 1 ) L )idefinedby ˆf )= f x)e πi x, dx, andextendedtol )auual,where, denote the Euclidean inner product in. For R,wedefineSobolevpaceH ) a the pace of all tempered ditribution f uch that f H ) = ˆf ξ) 1+ ξ ) dξ <, Zhang and Li 016. Thi article i ditributed under the term of the Creative Common Attribution 4.0 International Licene which permit unretricted ue, ditribution, and reproduction in any medium, provided you give appropriate credit to the original author) and the ource, provide a link to the Creative Common licene, and indicate if change were made.

2 Zhangand LiJournal of Inequalitie and Application 016) 016:88 Page of 8 where denote the Euclidean norm on. The inner product in H )igivenby f, g H ) = ˆf ξ)ĝξ) 1+ ξ ) dξ, f, g H ). Moreover, for each g H ), f, g = ˆf ξ)ĝξ) dξ, f H ), i a linear continuou functional in H ). The H )andh ) form pair of dual pace. For function f, g : C,define [f, g] t )= f + k)g + k) 1+ +k ) t, t R. For convenience, we write f j,k )= jd f j k ) and f j,k )=j d ) f j k ) for a ditribution f, j Z, k Z d,and R. Let N 0 = N {0}.GivenL N and R,letφ, ψ 1, ψ,...,ψ L H )and φ, ψ 1, ψ,..., ψ L H ), we denote by X φ; ψ 1, ψ,...,ψ L )andx φ; ψ 1, ψ,..., ψ L ) the following two nonhomogeneou wavelet ytem in H )andh ), repectively: and X φ; ψ 1, ψ,...,ψ L )= { φ 0,k : k Z d} { ψ l,j,k : j N 0, k Z d, l =1,,...,L } 1.1) X φ; ψ 1, ψ,..., ψ L )= { φ 0,k : k Z d} { ψ l,j,k : j N 0, k Z d, l =1,,...,L }. 1.) We ay that X φ; ψ 1, ψ,...,ψ L )ianonhomogeneou wavelet frame in H )ifthere exit two poitive contant A, B uch that A f H ) f, φ0,k H ) + f, ψl,j,k H ) B f H ), f H ), 1.3) where A, B are called frame bound; it i called a nonhomogeneou wavelet Beel equence in H ) if the right-hand inequality in 1.3) hold,whereb i called a Beel bound. Furthermore, we ay that X φ; ψ 1, ψ,...,ψ L ), X φ; ψ 1, ψ,..., ψ L )) i a pair of nonhomogeneou wavelet dual frame in H ), H )) if X φ; ψ 1, ψ,...,ψ L )and X φ; ψ 1, ψ,..., ψ L )arebeelequenceinh )andh ), repectively, and f, g = f, φ 0,k φ 0,k, g + f, ψ l,j,k ψ l,j,k, g 1.4) hold for all f H )andg H ).

3 Zhangand LiJournal of Inequalitie and Application 016) 016:88 Page 3 of 8 If X φ; ψ 1, ψ,...,ψ L ), X φ; ψ 1, ψ,..., ψ L )) i a pair of dual frame in H ), H )), then it follow from 1.4)that f = f, φ 0,k φ 0,k + f, ψ l,j,k ψ l,j,k, f H ), and g = g, φ 0,k φ 0,k + g, ψ l,j,k ψ l,j,k, g H ), with the erie converging unconditionally in H )andh ), repectively. The paper i organized a follow. Section i devoted to ome lemma ued latter. Section 3 i devoted to characterizing nonhomogeneou wavelet dual frame in H ), H )) via a pair of equation. Some lemma In thi ection, we give ome auxiliary lemma which are neceary in proving Theorem 3.1 below. Definition.1 Define a function κ : Z d Z by κn)=up { j 0: n Z d} for 0 n Z d,andetκ0) = +. Lemma.1 Let R, j Z, and ψ H ). Then, for f H ) and k Z d, the kth Fourier coefficient of [ jd ˆf j ), ˆψ )] 0 ξ) i f, ψ j,k. In particular, [ jd ˆf j ), ˆψ ) ] 0 ξ)= f, ψ j,k e πi k,ξ.1) if {ψ j,k : k Z d } i a Beel equence in H ). Proof Since f H )andψ H ), we have ˆf j ) ˆψ ) L 1 ), and thu [ jd ˆf j ), ˆψ ) ] T d 0 ξ)e πi k,ξ dξ = jd ˆf j ξ + l) ) ˆψξ + l)e πi k,ξ dξ T d l Z d by the Plancherel theorem. So = jd ˆf j ξ ) ˆψξ)e πi k,ξ dξ R d = jd ˆf ξ) ˆψ ξ ) e πi k,ξ dξ R d = ˆf ξ) [ ψ j,k ) ] ξ) dξ, T d [ jd ˆf j ), ˆψ ) ] 0 ξ)e πi k,ξ dξ = f, ψ j,k..)

4 Zhangand LiJournal of Inequalitie and Application 016) 016:88 Page 4 of 8 If {ψ j,k : k Z d } i a Beel equence in H ), then { f, ψ j,k } l Z d ), and thu.1) follow by.). By a carefulobervation of the proofof [13], Propoition.1, we have the following. Lemma. Let R, φ, ψ 1, ψ,...,ψ L H ). Then X φ; ψ 1, ψ,...,ψ L ) i a Beel equence in H ) with Beel bound B if and only if g, φ0,k + g, ψl,j,k B g H for g H )..3) ) Lemma.3 Let R, φ, ψ 1, ψ,...,ψ L H ). Suppoe that X φ; ψ 1, ψ,...,ψ L ) i a Beel equence in H ) with Beel bound B, then ˆφ ) + ˆψ ) l B 1+ ).4) hold a.e. on. Proof Since X φ; ψ 1, ψ,...,ψ L )iabeelequenceinh )withbeelboundb, by Lemma.,wehave g, φ 0,k + g, ψl,j,k B g H for g H )..5) ) By Lemma.1 and an argument imilar to that of [6], Theorem 1, we get g, φ0,k + g, ψl,j,k = ˆφξ)ĝξ) ĝξ + k) ˆφξ + k) dξ + R ˆψ l ξ ) ĝξ) ĝ ξ + j k ) ˆψ l ξ + k ) dξ. d It can be rewritten a g, φ 0,k + g, ψl,j,k = ĝξ) ˆφξ) + ˆψ l ξ ) ) dξ + ĝξ) ĝξ + k) 0 κk) ˆφξ) ˆφξ + k)+ ˆψ l ξ ) ˆψ l ξ + k) )) dξ.6) by the definition of κ.

5 Zhangand LiJournal of Inequalitie and Application 016) 016:88 Page 5 of 8 Suppoe.4) doe not hold. Then there exit E with E >0uchthat ˆφ ) + ˆψ ) l > B 1+ ) on E, and thu ˆφ ) + ˆψ ) l > B 1+ ) on ome E = E [0, 1) d + k 0 )with E >0andk 0 Z d.takeg uch that ĝ ) =1+ ) / χ E in.6), then we obtain g, φ0,k + g, ψl,j,k > B E = B g H ), contradicting.5). 3 The characterization of nonhomogeneou wavelet dual frame in Sobolev pace Thi ection i devoted to characterizing nonhomogeneou wavelet dual frame in H ), H )). The following theorem provide u with a characterization via a pair of equation. Theorem 3.1 Let R, φ, ψ 1, ψ,...,ψ L H ) and φ, ψ 1, ψ,..., ψ L H ). Define wavelet ytem X φ; ψ 1, ψ,...,ψ L ) and X φ; ψ 1, ψ,..., ψ L ) a in 1.1) and 1.), repectively. Suppoe that X φ; ψ 1, ψ,...,ψ L ) i a Beel equence in H ), and X φ; ψ 1,..., ψ L ) i a Beel equence in H ). Then X φ; ψ 1, ψ,...,ψ L ), X φ; ψ 1, ψ,..., ψ L )) i a pair of dual frame in H ), H )) if and only if, for every k Z d, ˆφ ) ˆ φ + k)+ κk) ˆψ l ) ˆ ψ l + k) ) = δ 0,k a.e. on. 3.1) Proof By the definition, X φ; ψ 1, ψ,...,ψ L ), X φ; ψ 1, ψ,..., ψ L ))iapairofdualframe for H ), H )) if and only if f, φ 0,k φ 0,k, g + f, ψ l,j,k ψ l,j,k, g = f, g, f H ), g H ). 3.) By the Plancherel theorem and Lemma.1,wededucethat f, φ k) φ k), g + = T d f, ψ l,j,k ψ l,j,k, g ˆf ξ + k) ˆ φξ ) ) + k) ˆφξ + k)ĝξ + k) dξ

6 Zhangand LiJournal of Inequalitie and Application 016) 016:88 Page 6 of 8 + jd T d ˆf j ξ + k) ) ˆ ψ l ξ + k)) = ˆf ξ + k) ˆ φξ + k) ˆφξ)ĝξ) dξ + jd = ˆf ξ)ĝξ) ˆf j ξ + k) ) ˆ ψ l ξ + k) ˆψ l ξ)ĝ j ξ ) dξ ˆφξ) ˆ φξ)+ ˆψ l ξ ) ˆ ψ l ξ )) dξ + ĝξ) ˆf ξ + k) ˆφξ) ˆ φξ + k) 0 + ˆf ξ + j k ) ˆψ l ξ ) ˆ ψ l ξ + k )) dξ 0 = ˆf ξ)ĝξ) ˆφξ) ˆ φξ)+ ˆψ l ξ ) ˆ ψ l ξ )) dξ + ĝξ) ˆf ξ + k) ˆφξ) ˆ φξ + k)+ 0 ˆψ l ξ + k)ĝ j ξ + k) )) dξ κk) ˆψ l ξ ) ˆ ψ l ξ + k) )) dξ. And thu 3.)canberewrittena ˆf ξ)ĝξ) ˆφξ) ˆ φξ)+ ˆψ l ξ ) ˆ ψ l ξ )) dξ + ĝξ) ˆf ξ + k) ˆφξ) ˆ φξ κk) + k)+ ˆψ l ξ ) ˆ ψ l ξ + k) )) dξ 0 = ˆf ξ)ĝξ) dξ. 3.3) Obviouly, 3.1)implie3.3). To finih the proof, next we prove the convere implication. Suppoe 3.3) hold. By Lemma.3 and the Cauchy-Schwarzinequality, the erie ˆφ ) ˆ φ + k)+ κk) ˆψ l ) ˆ ψ l + k) ) with k Z d converge abolutely a.e. on and belong to L ), and almot all point in are Lebegue point. Let ξ 0 be uch a point. For 0 < ɛ < 1,takef and g uch that ˆf )= 1 + ) / χ Bξ0,ɛ) Bξ0, ɛ) and ĝ )= 1 + ) / χ Bξ0,ɛ) Bξ0, ɛ)

7 Zhangand LiJournal of Inequalitie and Application 016) 016:88 Page 7 of 8 in 3.3), where Bξ 0, ɛ)={ξ : ξ ξ 0 < ɛ}.then 1 ˆφξ) ˆ φξ)+ Bξ 0, ɛ) Bξ 0,ɛ) ˆψ l ξ ) ˆ ψ l ξ )) dξ =1, letting ɛ 0 and applying the Lebegue differentiation theorem, we obtain ˆφξ 0 ) ˆ φξ 0 )+ ˆψ l ξ 0 ) ˆ ψ l ξ 0 ) =1. For 0 k 0 Z d,takef and g uch that ˆf + k 0 )= 1 + ) / χ Bξ0,ɛ) Bξ0, ɛ) and ĝ )= 1 + ) / χ Bξ0,ɛ) Bξ0, ɛ) in 3.3), where 0 < ɛ < 1.Then 1 ˆφξ) ˆ φξ + k 0 )+ Bξ 0, ɛ) Bξ 0,ɛ) κk 0 ) ˆψ l ξ ) ˆ ψ l ξ + k 0 ) )) dξ =0, letting ɛ 0 and applying the Lebegue differentiation theorem, we obtain ˆφξ 0 ) ˆ φξ 0 + k 0 )+ κk 0 ) ˆψ l ) ξ 0 ˆ ψ l ξ 0 + k 0 ) ) =0. By the arbitrarine of ξ 0 and k 0,weobtain3.1). Competing interet The author declare that they have no competing interet. Author contribution All author contributed equally to the writing of thi paper. All author read and approved the final manucript. Acknowledgement The article i upported by the National Natural Science Foundation of China Grant No ). The author would like to thank the reviewer for their uggetion which greatly improved the quality of thi article. Received: 11 June 016 Accepted: 8 November 016 Reference 1. Atrea, N, Mela, A, Stavropoulo, T: Affine dual frame and extenion principle. Appl. Comput. Harmon. Anal. 36, ). Bownik, M: Tight frame of multidimenional wavelet. J. Fourier Anal. Appl. 3, ) 3. Bownik, M: A characterization of affine dual frame in L R n ). Appl. Comput. Harmon. Anal. 8, ) 4. Chritenen, O, Kim, HO, Kim, RY: On Pareval wavelet frame with two or three generator via the unitary extenion principle. Can. Math. Bull. 57, ) 5. Chritenen, O, Kim, HO, Kim, RY: On extenion of wavelet ytem to dual pair of frame. Adv. Comput. Math. 4, ) 6. Chui, CK, Shi, X: Inequalitie of Littlewood-Paley type for frame and wavelet. SIAM J. Math. Anal. 4, ) 7. Daubechie, I, Han, B, Ron, A, Shen, Z: Framelet, MRA-baed contruction of wavelet frame. Appl. Comput. Harmon. Anal. 14, ) 8. Daubechie, I, Han, B: Pair of dual wavelet frame from any two refinable function. Contr. Approx. 0, ) 9. Han, B: On dual wavelet tight frame. Appl. Comput. Harmon. Anal. 4, ) 10. Ehler, M: The multireolution tructure of pair of dual wavelet frame for a pair of Sobolev pace. Jaen J. Approx., )

8 Zhangand LiJournal of Inequalitie and Application 016) 016:88 Page 8 of Han, B: Pair of frequency-baed nonhomogeneou dual wavelet frame in the ditribution pace. Appl. Comput. Harmon. Anal. 9, ) 1. Han, B: Nonhomogeneou wavelet ytem in high dimenion. Appl. Comput. Harmon. Anal. 3, ) 13. Han, B, Shen, Z: Dual wavelet frame and Riez bae in Sobolev pace. Contr. Approx. 9, ) 14. Han, B, Shen, Z: Characterization of Sobolev pace of arbitrary moothne uing nontationary tight wavelet frame. Ir. J. Math. 17, )