In this paper, two new theorems on generalized Carleson Operator for a Walsh type wavelet packet system and for periodic Walsh type wavelet packet expansion of a function ๐๐๐ฟ๐ 0,1 , 1<๐><โ, have been established.
Generalized Carleson Operator and Convergence of Walsh Type Wavelet Packet Expansions
1. Shyam Lal and Susheel Kumar Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 7( Version 2), July 2014, pp.100-108
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Generalized Carleson Operator and Convergence of Walsh Type Wavelet Packet Expansions Shyam Lal and Susheel Kumar Department of Mathematics, Faculty of Science, Banaras Hindu University, Varansi-221005, India Abstract In this paper, two new theorems on generalized Carleson Operator for a Walsh type wavelet packet system and for periodic Walsh type wavelet packet expansion of a function ํํํฟํ 0,1 , 1<ํ<โ, have been established. Mathematics Subject Classification: 40A30, 42C15 Keywords and Phrases Walsh function, Walsh- type wavelet Packets, periodic Walsh- type wavelet packets, Cesํ ro's means of order 1 for single infinite series and for double infinite series, generalized Carleson operator for Walsh type and for periodic Walsh type wavelet packets.
I. Introduction
A new class of orthogonal expansion in ํฟ2(โ) with good time frequency and regularity approximation properties are obtained in wavelet analysis. These expansions are useful in Signal analysis, Quantum mechanics, Numerical analysis, Engineering and Technology. Wavelet analysis generalizes an orthogonal wavelet expansion with suitable time frequency properties. In transient as well as stationary phenomena, this approach have applications over wavelet and short- time Fourier analysis. The properties of orthonormal bases have been studied in ํฟ2(ํ ) for wavelet expansion. The basic wavelet packet expansions of ํฟํ-functions, 1<ํ<โ, defined on the real line and the unit interval have significant importance in wavelet analysis. Walsh system is an example of a system of basic stationary wavelet packets (Billard [1] and Sjolin [9]). Paley [7], Billard [1] Sjolin [9] and Nielsen [6] have investigated point wise convergent properties of Walsh expansion of given ํฟํ[0,1) functions. At first, in 1966, Lennart Carleson [3] introduced an operator which is presently known as Carleson operator. In this paper, generalized Carleson operators for Walsh type wavelet packet expansion and for periodic Walsh type wavelet packet expansion for any ํโํฟํ[0,1) are introduced. Two new theorems have been established (1) The generalized Carleson operator for any Walsh-type wavelet packet system is of strong type ํ,ํ , 1< ํ<โ. (2) The generalized Carleson operator for periodic Walsh-type wavelet packet expansion for a function ํโํฟํ โ , 1<ํ<โ, converges a.e..
II. Definitions and Preliminaries
The structure in which non-stationary wavelet packets live is that of a multiresolution analysis ํํ ํโโค for ํฟ2 โ .To every multiresolution analysis we have an associated scaling function ํ and a wavelet ํ with the properties that ํํ=ํ ํํํ 2 ํ 2ํ 2ํ.โํ ;ํํโค and ํํ,ํโก2 ํ 2ํ 2ํ.โํ ;ํ,ํํโค are an orthonormal basis for ํฟ2 โ . Write ํํ=ํ ํํํ 2 ํ 2ํ 2ํ.โํ ;ํํโค . Let โ be the set of natural number and ํน0 ํ ,ํน1 ํ ,ํ ํโ, be a family of bounded operators on ํ2 โค of the form ํนํ ํ ํ ํ = ํํ ํโโคํํ ํ ํโ2ํ , ํ=0,1 with ํ1 ํ ํ =(โ1)ํ ํ0 ํ (1โํ) a sequence in ํ1(โค) such that ํน0 ํ โํน0 ํ +ํน1 ํ โํน1 ํ =1 ํน0 ํ ํน1 ํ โ=0. We define the family of functions ํคํ 0โ recursively by letting ํค0=ํ ,ํค1=ํ and then for ํํโ ํค2ํ ํฅ =2 ํ0 ํ (ํ)ํคํ 2ํฅโํ ํโํซ (2.1) ํค2ํ+1 ํฅ =2 ํ1 ํ (ํ)ํคํ 2ํฅโํ ํโํซ (2.2)
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where 2ํโคํ<2ํ+1. The family ํคํ 0โ is our basic non โstationary wavelet packet. It is known that ํคํ .โํ ;ํโฅ0,ํโํซ is an orthonormal basis for ํฟ2 โ .Moreover, ํคํ .โํ ;2ํโคํ<2ํ+1,ํโํซ is an orthonormal basis for ํํ=ํ ํํํ 2 ํ 2ํค1 2ํ.โํ ;ํโํซ . Each pair ํน0 ํ ,ํน1 ํ can be chosen as a pair of quardrature mirror filters associated with a multiresolution analysis, but this is not necessary. The trigonometric polynomial given by ํ0 ํ ํ = 1 2 ํ0 ํ (ํ)ํโํํํ ํโํซ and ํ1 ํ ํ = 1 2 ํ1 ํ (ํ)ํโํํํ ํโํซ are called the symbols of filters. The Haar low- pass quadrature mirror filter ํ0(ํ) ํ is given by ํ0 0 =ํ0 1 = 1 2,ํ0 ํ =0 otherwise, and the associate high-pass filter ํ1(ํ) ํ is given by ํ1 ํ =(โ1)ํ ํ0 1โํ . 2.1 Walsh function and their properties The Walsh system ํํ ํ=0โ is defined recursively on 0,1 on considering ํ0 ํฅ = 1,0โคํฅ<10, ํํกํํํํคํํ ํ and ํ2ํ ํฅ =ํํ 2ํฅ +ํํ 2ํฅโ1 , ํ2ํ+1 ํฅ =ํํ 2ํฅ โํํ 2ํฅโ1 . Observe that the Walsh system is the family of wavelet packets obtained by considering ํ=ํ0 ํ ํฅ = 1, 0โคํฅ< 12; โ1, 12โคํฅ<1; 0,ํํกํํํํคํํ ํ. and using the Haar filters in the definition of the non- stationary wavelet packets. The Walsh system is closed under point wise multiplication. Define the binary operator โ:โ0รโ0โโ0 by mโํ= ํํโํํ 2ํโํ =0, where ํ= ํํ2ํโํ =0, ํ= ํํ2ํโํ =0 and โ0=โโช 0 . Then ํํ ํฅ ํํ ํฅ =ํํโํ ํฅ , (2.3) (Schipp et al.[8]). We can carry over the operator โ to the interval [0,1] by identifying those xโ[0,1] with a unique expansion x= ํฅํ2โํโ1โํ =0 (almost all xโ[0,1] has such a unique expansion) by there associated binary sequence ํฅํ . For two such point ํฅ,ํฆโ 0,1 , define xโํฆ= ํฅํโํฆํ 2โํโ1โํ =0. The operation โ is defined for almost all ํฅ,ํฆโ 0,1 .Using this definition we have ํํ ํฅโํฆ =ํํ ํฅ ํํ ํฆ ( 2.4) for every pair x,y for which ํฅโํฆ is defined (Golubov et al.[4]). 2.2 Walsh type wavelet packets Let ํคํ ํโฅ0,ํโโค be a family of non-stationary wavelet packets constructed by using of family ํ0 ํ (ํ) ํ=0โ of finite filters for which there is a constantํพโโ such that ํ0 ํ (ํ) is the Haar filter for every ํโฅํพ. If ํค1โํถ1(โ) and it has compact support then we call ํคํ ํ>0 a family of Walsh type wavelet packets. 2.3 Periodic Walsh type wavelet packet As Meyer[6-a], an orthonormal basis for ํฟ2[0,1) is obtained periodizing any orthonormal wavelet basis associated with a multiresolution analysis. The periodization works equally well with non stationary wavelet packets. Let ํคํ ํ=0โ be a family of non-stationary basic wavelet packets satisfying ํคํ(ํฅ) โคํถํ(1+ ํฅ )โ1โโํ for some โํ>0, nโโ0. For nโโ0 we define the corresponding periodic wavelet packets ํค ํ by ํค ํ ํฅ = ํคํ ํฅโํ ํโโค . It is important to note that the hypothesis about the point- wise decay of the wavelet packets ํคํ ensures that the periodic wavelet packets are well defined in ํฟํ[0,1) for 1โคํโคโ . Wickerhauser and Hess[11] have proved that, the family ํค ํ ํ=0โ is an orthonormal basis for ํฟํ 0,1 .
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2.4 (C,1) and (C,1,1) method The series ํํ=ํ0+ํ1+ํ2+โฏโํ =0, is said to be convergent to the sum S. If the partial sum ํํ=ํ0+ํ1+ํ2+โฏ+ํํ tends to finite limit S when nโโ; and a series which is not convergent is said to be divergent. If ํํ=ํ0+ํ1+ํ2+โฏ+ํํ, and lim ํโโ ํ0+ํ1+ํ2+โฏ+ํํ ํ+1=ํ, Then we call S the (C,1) sum of ํํ and the (C,1) limits of ํํ. (Hardy[5],p.7) The series1+0-1+1+0-1+1+0...,is not convergent but it is summable (C,1) to the sum23 , (Titchmarsh [10], p.4111). Let ํํ,ํ โํ =ํ โ ํ=0= ํ0,0+ํ0,1+ํ0,2+โฏ +ํ1,0+ํ1,1+ํ1,2+โฏ +ํ2,0+ํ2,1+ํ2,2+โฏ be a double infinite series (Bromwich[2],p.92). The partial sum of double infinite series denoted byํํ,ํ, and defined by ํํ,ํ= ํํ,ํ . ํ ํ=0 ํ ํ=0 Write ํํ,ํ= 1 ํ+1 (ํ+1) ํํ,ํ ํ ํ=1 ํ ํ=1 = 1โ ํ ํ+1 1โ ํ ํ+1 ํํ,ํ . (2.5)ํํ =1 ํํ =1 If ํํ,ํโํ as mโโ,ํโโ then we say that ํํ,ํ โํ =0โ ํ=0 is summable to S by (C,1,1) method. Consider the double infinite series (โ1)ํ+ํโํ =0โ ํ=0 in this case, ํํ,ํ= (โ1)ํ+ํ ํ ํ=0 ํ ํ=0 = โ1 (ํ)ํํ =0 โ1 (ํ)ํํ =0 = 1โ1+1โโฏ+(โ1)ํ 1โ1+1โโฏ+(โ1)ํ = 1โ(โ1)ํ+11+1 1โ(โ1)ํ+11+1 = 14 1+(โ1)ํ 1+(โ1)ํ = 1,ํํ,ํ=2ํ1,ํ=2ํ20,ํํ,ํ=2ํ1,ํ=2ํ2+10,ํํ,ํ=2ํ1+1,ํ=2ํ20,ํํ,ํ=2ํ1+1,ํ=2ํ2+1 ํคํํํํ ํ1,ํ2ํโ0. Then limํ,ํโโ,โํํ,ํ does not exist. The double infinite series โ1 (ํ+ํ)โํ =0โ ํ=0 is not convergent. Let us consider ํํ,ํ. ํํ,ํ.= 1โ ํ ํ+1 1โ ํ ํ+1 โ1 ํ+ํ ํ ํ=0 ํ ํ=0 = 3+(โ1)ํ+2ํ 3+(โ1)ํ+2ํ 16 ํ+1 (ํ+1) then ํํ,ํ.โ 14 as mโโ,ํโโ. Therefore the series โ1 (ํ+ํ)โํ =0โ ํ=0, is summable to 14 by(C,1,1) method. 2.5 Generalized Carleson operator Write ํํ,ํํ ํฅ = ํ,ํคํ(.โํ) ํํ =โํ ํํ =0ํคํ ํฅโํ ,ํโํฟํ(โ) , 1<ํ<โ,
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ํํ,ํํ ํฅ = 1 ํ+1 2 ํํ,ํํ ํฅ ํํ =0 ํํ =0 = 1โ ํ ํ+1 1โ ํ ํ+1 ํ,ํคํ(.โํ) ํํ =โํ ํํ =0ํคํ ํฅโํ The Carleson operator for the Walsh type wavelet packet system, denoted by L, is defined by ํฟํ ํฅ = ํ ํขํ ํโฅ1 ํ,ํคํ(.โํ) ํํ =โํ ํํ =0ํคํ ํฅโํ .ํโํฟํ(โ) , 1<ํ<โ, = ํ ํขํ ํโฅ1 ํํ,ํํ ํฅ . The generalized carleson operator for the Walsh type wavelet packet system denoted by ํฟํ is defined by (ํฟํf) ํฅ = ํ ํขํ ํโฅ1 1 ํ+1 2 ํํ,ํํ ํฅ ํํ =0 ํํ =0 = ํ ํขํ ํโฅ1 ํํ,ํํ ํฅ . = ํ ํขํ ํโฅ1 1โ ํ ํ+1 1โ ํ ํ+1 ํ,ํคํ(.โํ) ํํ =โํ ํํ =0ํคํ ํฅโํ (2.6). Let us define the generalized Carleson Operator for the periodic Walsh type wavelet packet system ํค ํ . Write ํ ํํ ํฅ = ํ,ํค ํ ํค ํ ํฅ ,ํโํํ =0ํฟํ(โ) , 1<ํ<โ, ํํํ ํฅ = 1 ํ+1 ํ ํํ ํฅ ํ ํ=0 = 1 ํ+1 ํ,ํค ํ (ํค ํ(ํฅ) ํ ํ=0 ํ ํ=0) = (1โ ํ ํ+1 ํ,ํค ํ ํค ํ ํฅ . ํ ํ=0 The Carleson operator for the periodic Walsh type wavelet packet system, denoted by ํพ, is defined by ํพํ ํฅ = ํ ํขํ ํโฅ1 ํ,ํค ํ ํค ํ ํฅ ํํ =0 ,ํโํฟํ(โ) , 1<ํ<โ, = ํ ํขํ ํโฅ1 ํ ํํ ํฅ . The generalized Carleson operator for the periodic Walsh type wavelet type packet system, denoted by , is defined by ํพํํ ํฅ = ํ ํขํ ํโฅ1 ํ 0ํ ํฅ + ํ 1ํ ํฅ + ํ 2ํ ํฅ +โฏ+ ํ ํํ ํฅ ํ+1 = ํ ํขํ ํโฅ1 1 ํ+1 ํ ํํ ํฅ ํํ =0 = ํ ํขํ ํโฅ1 ํํํ ํฅ = ํ ํขํ ํโฅ1 1โ ํ ํ+1 ํ,ํค ํ ํค ํ ํฅ ํํ =0 . (2.7) 2.6 Strong type (p,p) Operator An operator T defined on ํฟํ โ is of strong type(p,p) if it is sub-linear and there is a constant C such that ํํ ํโค ํถ ํ ํ for all ํโ ํฟํ โ .
III. Main Results
In this paper, two new Theorem have been established in the following form: Theorem 3.1 Let ํคํ be a family of Walsh-type wavelet packet system. Then the generalized Carleson operator ํฟํ defined by (2,6) for any Walsh-type wavelet packet system is of strong type(p,p),1<ํ<โ. Theorem 3.2 Let ํค ํ be a periodic Walsh-type wavelet packets. Then the generalized Carleson Operator ํพํ defined by (2.7) for periodic Walsh type wavelet packet expansion of any ํโํฟํ โ ,1<ํ<โ, converges a.e... 3.1 Lemmas For the proof of the theorems following lemmas are required: Lemma 3.1 (Zygmund[12],p.197),Ifํฃ1,ํฃ2,ํฃ3,โฆํฃํ are non negative and non increasing,then ํข1ํฃ1+ํข2ํฃ2+ํข3ํฃ3+โฏ+ํขํํฃํ โคํฃ1ํํํฅ ํํ Where ํํ=ํข1+ํข2+ํข3+โฏ+ํขํ for k=1,2,3,...,n. Lemma 3.2 Let ํ1โ ํฟ2(โ), and define ํํ ํโฅ2 recursively by
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ํํ ํฅ = ํํ 2ํฅ +ํํ 2ํฅโ1 ,ํ=2ํ ํํ 2ํฅ โํํ 2ํฅโ1 ,ํ=2ํ+1 Then ํํ ํฅ = ํ ํโ2ํํ2โํ 2ํโ1 ํ=0ํ1(2ํx-p) for m , j โโ,2ํโคํ<2ํ+1. Lemma 3.3 Let ํคํ ํโฅ0 be a family of Walsh type wavelet packets. If ํค1โํถ1(โ) then there exists an isomorphism ํโถ ํฟํ โ โํฟํ โ ,1<ํ<โ,ํ ํขํํ ํกํํํก ํํคํ .โํ = ํคํ .โํ ,ํโฅ0,ํโโค. Lemma (3.2) and (3.3) can be easily proved. Lemma 3.4 (Billard [1] and Sjํ lin [9]. Let ํโํฟ1[0,1) and define ํํ ํฅ,ํ = ํ ํก ํํ ํก ํํกํํ ํฅ .10 ํ ํ=0 Then the Carleson operator G defined by ํบํ ํฅ = ํ ํขํ ํ ํํ(ํฅ,ํ) is of strong type (p,p) for 1<ํ<โ. 3.2 Proof of Theorem 3.1 For, ํ,ํ,ํฟํ(โ) ํฟํ ํ+ํ ํฅ = ํ ํขํ ํโฅ1 ํํ,ํ ํ+ํ (ํฅ) = ํ ํขํ ํโฅ1 1โ ํ ํ+1 1โ ํ ํ+1 ํํ =โํ ํ+ํ,ํคํ(.โํ) ํคํ(ํฅโํ)ํํ =0 = ํ ํขํ ํโฅ1 1โ ํ ํ+1 1โ ํ ํ+1 ํํ =โํ ํ,ํคํ .โํ + ํ,ํคํ .โํ ํคํ ํฅโํ ํํ =0 ํ ํขํ โคํโฅ1 1โ ํ ํ+1 1โ ํ ํ+1 ํ ํ=โํ ํ,ํคํ(.โํ) ํคํ(ํฅโํ) ํ ํ=0 + ํ ํขํ ํโฅ1 1โ ํ ํ+1 1โ ํ ํ+1 ํ ํ=โํ ํ,ํคํ(.โํ) ํคํ(ํฅโํ) ํ ํ=0 = ํ ํขํ ํโฅ1 (ํํ,ํํ)(ํฅ) + ํ ํขํ ํโฅ1 (ํํ,ํํ)(ํฅ) = ํฟํํ ํฅ + ํฟํํ ํฅ . Also for ํผโโ ํฟํํผํ ํฅ = ํ ํขํ ํโฅ1 (ํํ,ํ(ํผํ))(ํฅ) = ํ ํขํ ํโฅ1 1โ ํ ํ+1 1โ ํ ํ+1 ํํ =โํ ํผํ,ํคํ(.โํ) ํคํ(ํฅโํ)ํํ =0 = ํผ ํ ํขํ ํโฅ1 1โ ํ ํ+1 1โ ํ ํ+1 ํ ํ=โํ ํ,ํคํ(.โํ) ํคํ(ํฅโํ) ํ ํ=0 = ํผ ํฟํํ ํฅ . Choose M โโ such that Supp(ํคํ)โ[โํ,ํ] for ํโฅ0. Fix ํโ(1,โ) and take any ํ ํฅ = ํ,ํคํ(.โํ) ํคํ ํฅโํ โํฟํ โ .ํโฅ0,ํโโค Define ํํ ํฅ = ํ,ํคํ(.โํ) ํคํ ํฅโํ ,ํํ ํฅ = ํ,ํคํ(.โํ) ํํ ํฅโํ . ํโฅ0,ํโโคํโฅ0,ํโโค We have ํํ ํโ ํํ ํ,ํคํํกํ bounds independent of k (Lemma 3.3) .Note that for {ํฅโ ํ,ํ+1 : ํฟํํ(ํฅ) >ํผ} โค ํถ ํผํ ํฟํํํ(ํฅ) ํ+1+ํ ํ=ํโํ ํ ํํฅ. Using the Marcinkiewiez interpolation theorem, it suffices to prove that ํฟํํํ ํโคํถ ํํ ํ, where C is a constant independent of k.
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Since ํํ ํ ํโค2(ํ+1) ํํ ํ ํโค2ํถ(ํ+1) ํํ ํ ํโคํถ1 ํ ํ ํ ํโโคํโโค ํ+1+ํ ํ=ํโํํโโค where ํถ1 is constant. Without loss of generality, we assume that k=0. Let K โ โ be the scale from which only the Haar filter is used to generate the wavelet packets ํคํ ํโฅ2ํ+1. Let N โโ and suppose 2ํโคํโค2ํฝ+1 for some J>ํพ+1.ํถํํํํํํฆ,ํํํ each ํฅโโ. ํฟํํ ํฅ = ํ ํขํ ํโฅ1 1โ ํ ํ+1 1โ ํ ํ+1 ํํ =โํ ํ,ํคํ(.โํ) ํคํ(ํฅโํ)ํํ =0 =ํ ํขํํโฅ1 1โ ํ ํ+1 ํ,ํคํ(.) ํคํ(ํฅ)ํํ =0 , ํ=0, โคํ ํขํ1โคํ<2ํ+1 1โ ํ ํ+1 ํ,ํคํ (.) ํคํ(ํฅ)ํํ =0 +ํ ํขํํฝ>ํพ+1 1โ ํ ํ+1 ํ,ํคํ (.) 2ํฝโ1 ํ=2ํพ+1 ํคํ (ํฅ) +ํ ํขํํฝ>ํพ+1 ํ ํขํ2ํฝโคํ<2ํฝ+1 1โ ํ ํโ2ํฝ+1 ํ,ํคํ(.) ํคํ (ํฅ)ํ ํ=2ํฝ =ํฝ1+ํฝ2+ํฝ3 say. (3.2) Using Lemma 3.1, we have ํฝ1=ํ ํขํ1โคํ<2ํ+1 1โ ํ ํ+1 ํ,ํคํ (.) ํคํ(ํฅ) 2ํ+1โ1 ํ=0 โคํ ํขํ1โคํ<2ํ+1 ํํํฅ ํ,ํคํ (.) ํคํ(ํฅ)0=ํโค2ํพ+1โ1 โค ํ,ํคํ(.) 2ํพ+1โ1 ํ=0 ํคํ (ํฅ) โค ํ,ํคํ(.) ํคํ(ํฅ) โ ํ[โํ,ํ] (ํฅ) โค ํ0 ํ ํคํ ํ 2ํพ+1โ1 ํ=0 ํคํ (ํฅ) โ ํ[โํ,ํ] ํฅ . (3.3) Next, ํฝ2= ํ ํขํํฝ>ํพ+1 1โ ํ ํ+1 2ํฝโ1 ํ=2ํพ+1 ํ,ํคํ . ํคํ(ํฅ) โค ํ ํขํ ํฝ>ํ+1max ํ,ํคํ . ํคํ(ํฅ)2ํ+1=ํโค2ํฝโ1 . Also ํ ํขํ ํฝ>ํ+1 (1โ ํ ํ+1) ํ,ํคํ . ํคํ(ํฅ)2ํฝโ1 ํ=2ํ+1 ํ โค supํํํฅ ํฝ>ํ+1 ํ,ํคํ . ํคํ(ํฅ) ํ2ํ+1=ํโค2ํฝโ1 โคํถ ํ,ํคํ . ํคํ(ํฅ) ํ โ ํ=0 =ํถ ํ0 ํ. (3.4) Consider ํฝ3, ํฝ3= ํ ํขํ 2ํฝโคํ<2ํฝ+1 (1โ ํ ํ+1) ํ,ํคํ . ํคํ(ํฅ) ํ ํ=2ํฝ โค ํ ํขํ 2ํฝ+ํ2ํฝโํพโคํ<2ํฝ+(ํ+1)2ํฝโํพ (1โ ํ ํ+1) ํ,ํคํ . ํคํ(ํฅ) ํ ํ=2ํฝ+ํ2ํฝโํพ 2ํพโ1 ํ=0, so it suffices to prove that ํ ํขํ ํ>ํ+1 ํ ํขํ 2ํฝ+ํ2ํฝโํพโคํ<2ํฝ+(ํ+1)2ํฝโํพ (1โ ํ ํ+1) ํ,ํคํ . ํคํ(ํฅ) ํ ํ=2ํฝ+ํ2ํฝโํพ ํ โคํถ ํ0 ํ for ํ=0,1,2,โฆ,2ํโ1. Fix ํฝ>ํพ+1 ,0โคํโค22ํโ1 and 2ํฝ+ํ2ํฝํพโคํ<2ํฝ+ ํ+1 2ํฝโํพ. Write,
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ํ,3= (1โ ํ ํ+1) ํ,ํคํ . ํคํ(ํฅ) ํ ํ=2ํฝ+ํ2ํฝโํพ . Using lemma 3.2, we have ํ,3 = (1โ ํ ํ+1) ํ,ํคํ . ํ ํโ2ํฝโํ2ํฝโํพ(ํ 2โ ํฝโํพ ) ํ ํ=2ํฝ+ํ2ํฝโํพ ํค2ํ+ํ(2ํฝโํพํฅโํ ) 2ํฝโํพโ1 ํ =0 . Define ํนํ ํก = 1โ ํ ํ+1 ํ,ํคํ . ํํโ2ํฝโํ2ํฝโํพ, ํก , ํ ํ=2ํฝ+ํ2ํฝโํพ and ํน ํก = ํ ํขํ ํ<2ํฝ+(ํ+1)2ํฝโํพ ํนํ ํก . We have ํโฒ 3 = 1โ ํ ํ+1 ํ,ํคํ . ํคํ ํฅ ํ ํ=2ํฝ+ํ2ํฝโํพ , โคํํํฅ ํ,ํคํ . ํคํ ํฅ 2ํฝ+ํ2ํฝโํพ=ํ<ํ , โค ํน(ํ 2โ(ํฝโํพ) ํค2ํ+ํ((2ํฝโํพ)ํฅโํ ) 2ํฝโํพโ1 ํ=0, and using the compact support of the wavelet packets, ํโฒ 3 = 1โ ํ ํ+1 ํ,ํคํ . ํคํ ํฅ ํ ํ=2ํฝ+ํ2ํฝโํพ , โคํํํฅ ํ,ํคํ . ํคํ ํฅ 2ํฝ+ํ2ํฝโํพ=ํ<ํ โค ํค2ํ+ํ โ ํน(([2ํฝโํพํ+1 ํ=โํํฅ]+ํ)2โ(ํฝโํพ)). Note that F is constant on dyadic interval of type [ํ2โ(ํโํ),(ํ+1)2โ(ํโํ)) and taking ฮํ=(( 2ํฝโํพํฅ+ํ 2โ ํฝโํพ , 2ํฝโํพํฅ+(ํ+1) 2โ ํฝโํพ ), we have ํโฒ 3 = 1โ ํ ํ+1 ํ,ํคํ . ํคํ ํฅ ํ ํ=2ํฝ+ํ2ํฝโํพ โคํํํฅ ํ,ํคํ . ํคํ ํฅ ํ 2ํฝ+ํ2ํฝโํพ=ํ<ํ โค ํค2ํ+ํ โ ฮํ โ1 ํน ํก ํํก ฮํ ํ+1 ํ=โํ . We need an estimate of F that does not depend on J . Note that for k, 0โคํ<2ํฝโํพ , using (2.3) ํ2ํฝ+ํ2ํฝโํพ ํก ํํ ํก =ํ2ํฝ+ํ2ํฝโํพ+ํ ํก , since the binary expansions of 2ํฝ+ํ2ํฝโํพ and of k have no 1โs in common. Hence, ํนํ ํก = ํ2ํฝ+ํ2ํฝโํพ ํก ํนํ ํก = 1โ ํ ํ+1 ํ,ํคํ . ํํ ํก ํ ํ=2ํฝ+ํ2ํฝโํพ
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โคํํํฅ ํ,ํคํ . ํํ ํก ํ 2ํฝ+ํ2ํฝโํพ=ํ<ํ . Using lemma (3.4) , we have F(t)โค2 ํบํํ ํก . Thus, 1โ ํ ํ+1 ํ,ํคํ . ํคํ ํฅ ํ ํ=2ํฝ+ํ2ํฝโํพ โคํํํฅ ํ,ํคํ . ํคํ ํฅ ํ 2ํฝ+ํ2ํฝโํพ=ํ<ํ โค2 ํค2ํ+ํ โ ฮํ โ1 ํบํ0 ํก ํํก ฮํ ํ+1 ํ=โํ . Let ฮํ โ be the smallest dyadic interval containing ฮํ and ํฅ , and note that ฮํ โ โค(ํ+1) ฮํ since ํฅโ ฮ0โํท . We have 1โ ํ ํ+1 ํ,ํคํ . ํคํ ํฅ ํ ํ=2ํฝ+ํ2ํฝโํพ โคํํํฅ ํ,ํคํ . ํคํ ํฅ 2ํฝ+ํ2ํฝโํพ=ํ<ํ โค2 ํค2ํ+ํ โ ฮํ โ1 ํบํ0 ํก ํํก ฮํ ํ+1 ํ=โํ โค4 ํค2ํ+ํ โ(ํ+1)2 ํโ ํบํ0 ํฅ (3.5) Where ํโ is the maximal operator of Hardy and Little wood . The right hand side of (3.5) neither depend on N nor J so we may conclude that ํฝ3โค ํ ํขํ ํฝ>ํพ+1 ํ ํขํ 2ํฝ+ํ2ํฝโํพโคํ<2ํฝ+(ํ+1)2ํฝโํพ (1โ ํ ํ+1) ํ,ํคํ . ํคํ(ํฅ) ํ ํ=2ํฝ+ํ2ํฝโํพ 2ํพโ1 ํ=0 โค ํ ํขํ 2ํฝ+ํ2ํฝโํพโคํ<2ํฝ+(ํ+1)2ํฝโํพ ํํํฅ ํ,ํคํ . ํคํ ํฅ 2ํฝ+ํ2ํฝโํพ=ํ<ํ 2ํโ1 ํ=0 โค4 ํค2ํ+ํ โ(ํ+1)2 ํโ ํบํ0 ํฅ ,ํ.ํ.. (3.6) Using (Sjํ lin [9]), ํโ and G both of strong type (p,p), hence both are bounded. ํ ํขํ ํฝ>ํพ+1ํฝ3 ํ โค ํ ํขํ ํฝ>ํพ+1 ํ ํขํ 2ํฝ+ํ2ํฝโํพโคํ<2ํฝ+(ํ+1)2ํฝโํพ (1 ํ ํ=2ํฝ+ํ2ํฝโํ 2ํพโ1 ํ=0โ ํ ํ+1) ํ,ํคํ . ํคํ(ํฅ) ํ โคํถ ํ0 ํ โคํถ1 ํ0 ํ, ํ=0,1,2,3,โฆ,2ํพ-1. Thus the Theorem 3.1 is completely established. 3.3 proof of the Theorem 3.2 Let ํโํฟํ 0,1 , and choose Mโโ such that supp(ํคํ)โ[โํ,ํ] for nโฅ0. Then ํพํํ ํฅ = ํ ํขํ ํโฅ1 1โ ํ ํ+1 ํ,ํค ํ ํค ํ(ํฅ) ํ ํ=0 = ํ ํขํ ํโฅ1 1โ ํ ํ+1 ํ, ํคํ(.โํ1)ํ+1 ํ1=โํ ํคํ(ํฅโํ2)ํ+1 ํ2=โํ ํํ =0
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= ํ ํขํ ํโฅ1 1โ ํ ํ+1 ํ,ํคํ(.โํ1) ํ+1 ํ1=โํ ํคํ(ํฅโํ2)ํ+1 ํ2=โํ ํํ =0 = ํ ํขํ ํโฅ1 1โ ํ ํ+1 ํ,ํคํ(.โํ1) ํํ =0ํคํ(ํฅโํ2) ํ+1 ํ2=โํ ํ+1 ํ1=โํ = ํ ํขํ ํโฅ1 1โ ํ ํ+1 ํ ํฆ ํคํ ํฆโํ1 ํํฆํคํ(ํฅโํ2) 10 ํํ =0 ํ+1 ํ2=โํ ํ+1 ํ1=โํ Following the proof of theorem 3.1, it can be proved that the generalized Carleson operator ํพํ for the periodic Walsh type wavelet packet expansions converges a.e..
IV. Acknowledgments
Shyam Lal, one of the author, is thankful to DST - CIMS for encouragement to this work. Susheel Kumar, one of the authors, is grateful to C.S.I.R. (India) in the form of Junior Research Fellowship vide Ref. No. 19-06/2011 (i)EU-IV Dated:03-10-2011 for this research work. Authors are grateful to the referee for his valuable suggestions and comments for the improvement of this paper. References [1] Billard, P., Sur la convergence Presque partout des series de Fourier-Walshdes fontions de lโespace ํฟ2(0,1), Studia Math, 28:363-388, 1967. [2] Bromwich, T. J .I โA., An introduction to the theory of infinite Series, The University press, Macmillan and Co; Limited, London, 1908. [3] Carleson, L. , โOn convergence and growth of partial sums of Fourier seriesโ , Acta Mathematica 116(1): 135157, (1966). [4] Golubov , B., Efimov, A. And Skvortson, V., Walsh Series and Transforms, Dordrcct: Kluwer Academic Publishers Group ,Theory and application, Translated Form the 1987 Russian Original by W.R.. Wade, 1991. [5] Hardy, G. H., Divergent Series, Oxford at the Clarendon Press, 1949. [6-a] Meyer .Y. Wavelets and Operators. Cambridge University Press, 1992. [6] Morten. N. Walsh Type Wavelet Packet Expansions. [7] Paley R. E. A. C., A remarkable system of orthogonal functions. Proc. Lond. Math. Soc., 34:241279, 1932. [8] Schipp, F., Wade, W. R. and Simon, P., Walsh Series, Bristol: AdamHilger Ltd. An Introduction to Dyadic Harmonic Analysis, with the Collaboration of J. P_al, 1990. [9] Sjํ lin P., An inequality of paley and convergence a.e. of Walsh-FourierSeries, Arkiv fํ r Matematik, 7:551-570, 1969. [10] Titchmarsh, E.C., The Theory of functions, Second Edition, OxfordUniversity Press, (1939). [11] Wickerhauser, M. V. and Hess,N. Wavelets and Time-Frequency Analysis, Proceedings of the IEEE, 84:4(1996), 523-540. [12] Zygmund A., Trigonometric Series Volume I, Cambridge UniversityPress, 1959.