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 𝑓𝜖𝐿𝑝 0,1 , 1<𝑝><∞, have been established.

1. Shyam Lal and Susheel Kumar Int. Journal of Engineering Research and Applications www.ijera.com
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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 푓휖퐿푝 0,1 , 1<푝<∞, have been established. Mathematics Subject Classification: 40A30, 42C15 Keywords and Phrases Walsh function, Walsh- type wavelet Packets, periodic Walsh- type wavelet packets, Ces푎 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 퐿2(ℝ) 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 퐿2(푅) for wavelet expansion. The basic wavelet packet expansions of 퐿푝-functions, 1<푝<∞, 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 퐿푝[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 푓∈퐿푝[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 푝,푝 , 1< 푝<∞. (2) The generalized Carleson operator for periodic Walsh-type wavelet packet expansion for a function 푓∈퐿푝 ℝ , 1<푝<∞, converges a.e..
II. Definitions and Preliminaries
The structure in which non-stationary wavelet packets live is that of a multiresolution analysis 푉푗 푗∈ℤ for 퐿2 ℝ .To every multiresolution analysis we have an associated scaling function 휙 and a wavelet 휓 with the properties that 푉푗=푠푝푎푛 2 푗 2휙 2푗.−푘 ;푘휖ℤ and 휓푗,푘≡2 푗 2휓 2푗.−푘 ;푗,푘휖ℤ are an orthonormal basis for 퐿2 ℝ . Write 푊푗=푠푝푎푛 2 푗 2휓 2푗.−푘 ;푘휖ℤ . Let ℕ be the set of natural number and 퐹0 푝 ,퐹1 푝 ,푝 휖ℕ, be a family of bounded operators on 푙2 ℤ of the form 퐹휖 푝 푎 푘 = 푎푛 푛∈ℤ푕휖 푝 푛−2푘 , 휖=0,1 with 푕1 푝 푛 =(−1)푛 푕0 푝 (1−푛) a sequence in 푙1(ℤ) such that 퐹0 푝 ∗퐹0 푝 +퐹1 푝 ∗퐹1 푝 =1 퐹0 푝 퐹1 푝 ∗=0. We define the family of functions 푤푛 0∞ recursively by letting 푤0=휙 ,푤1=휓 and then for 푛휖ℕ 푤2푛 푥 =2 푕0 푝 (푞)푤푛 2푥−푞 푞∈핫 (2.1) 푤2푛+1 푥 =2 푕1 푝 (푞)푤푛 2푥−푞 푞∈핫 (2.2)
RESEARCH ARTICLE OPEN ACCESS

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where 2푝≤푛<2푝+1. The family 푤푛 0∞ is our basic non –stationary wavelet packet. It is known that 푤푛 .−푘 ;푛≥0,푘∈핫 is an orthonormal basis for 퐿2 ℝ .Moreover, 푤푛 .−푘 ;2푗≤푛<2푗+1,푘∈핫 is an orthonormal basis for 푊푗=푠푝푎푛 2 푗 2푤1 2푗.−푘 ;푘∈핫 . Each pair 퐹0 푝 ,퐹1 푝 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 푚0 푝 휉 = 1 2 푕0 푝 (푘)푒−푖푘휉 푘∈핫 and 푚1 푝 휉 = 1 2 푕1 푝 (푘)푒−푖푘휉 푘∈핫 are called the symbols of filters. The Haar low- pass quadrature mirror filter 푕0(푘) 푘 is given by 푕0 0 =푕0 1 = 1 2,푕0 푘 =0 otherwise, and the associate high-pass filter 푕1(푘) 푘 is given by 푕1 푘 =(−1)푘 푕0 1−푘 . 2.1 Walsh function and their properties The Walsh system 푊푛 푛=0∞ is defined recursively on 0,1 on considering 푊0 푥 = 1,0≤푥<10, 표푡푕푒푟푤푖푠푒 and 푊2푛 푥 =푊푛 2푥 +푊푛 2푥−1 , 푊2푛+1 푥 =푊푛 2푥 −푊푛 2푥−1 . Observe that the Walsh system is the family of wavelet packets obtained by considering 휑=푊0 휓 푥 = 1, 0≤푥< 12; −1, 12≤푥<1; 0,표푡푕푒푟푤푖푠푒. 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 ⊕:ℕ0×ℕ0→ℕ0 by m⊕푛= 푚푖−푛푖 2푖∞푖 =0, where 푚= 푚푖2푖∞푖 =0, 푛= 푛푖2푖∞푖 =0 and ℕ0=ℕ∪ 0 . Then 푊푚 푥 푊푛 푥 =푊푚⊕푛 푥 , (2.3) (Schipp et al.[8]). We can carry over the operator ⊕ to the interval [0,1] by identifying those x∈[0,1] with a unique expansion x= 푥푗2−푗−1∞푗 =0 (almost all x∈[0,1] has such a unique expansion) by there associated binary sequence 푥푖 . For two such point 푥,푦∈ 0,1 , define x⊕푦= 푥푖−푦푖 2−푗−1∞푗 =0. The operation ⊕ is defined for almost all 푥,푦∈ 0,1 .Using this definition we have 푊푛 푥⊕푦 =푊푛 푥 푊푛 푦 ( 2.4) for every pair x,y for which 푥⊕푦 is defined (Golubov et al.[4]). 2.2 Walsh type wavelet packets Let 푤푛 푛≥0,푘∈ℤ be a family of non-stationary wavelet packets constructed by using of family 푕0 푝 (푛) 푝=0∞ of finite filters for which there is a constant퐾∈ℕ such that 푕0 푝 (푛) is the Haar filter for every 푝≥퐾. If 푤1∈퐶1(ℝ) and it has compact support then we call 푤푛 푛>0 a family of Walsh type wavelet packets. 2.3 Periodic Walsh type wavelet packet As Meyer[6-a], an orthonormal basis for 퐿2[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 푤푛 푛=0∞ be a family of non-stationary basic wavelet packets satisfying 푤푛(푥) ≤퐶푛(1+ 푥 )−1−∈푛 for some ∈푛>0, n∈ℕ0. For n∈ℕ0 we define the corresponding periodic wavelet packets 푤 푛 by 푤 푛 푥 = 푤푛 푥−푘 푘∈ℤ . It is important to note that the hypothesis about the point- wise decay of the wavelet packets 푤푛 ensures that the periodic wavelet packets are well defined in 퐿푃[0,1) for 1≤푝≤∞ . Wickerhauser and Hess[11] have proved that, the family 푤 푛 푛=0∞ is an orthonormal basis for 퐿푃 0,1 .

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2.4 (C,1) and (C,1,1) method The series 푎푛=푎0+푎1+푎2+⋯∞푛 =0, is said to be convergent to the sum S. If the partial sum 푆푛=푎0+푎1+푎2+⋯+푎푛 tends to finite limit S when n→∞; and a series which is not convergent is said to be divergent. If 푆푛=푎0+푎1+푎2+⋯+푎푛, and lim 푛→∞ 푆0+푆1+푆2+⋯+푆푛 푛+1=푆, Then we call S the (C,1) sum of 푎푛 and the (C,1) limits of 푆푛. (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 푎푚,푛 ∞푛 =표 ∞ 푚=0= 푎0,0+푎0,1+푎0,2+⋯ +푎1,0+푎1,1+푎1,2+⋯ +푎2,0+푎2,1+푎2,2+⋯ be a double infinite series (Bromwich[2],p.92). The partial sum of double infinite series denoted by푆푚,푛, and defined by 푆푚,푛= 푎푖,푗 . 푛 푗=0 푚 푖=0 Write 휎푚,푛= 1 푚+1 (푛+1) 푆푖,푗 푛 푗=1 푚 푖=1 = 1− 푖 푚+1 1− 푗 푛+1 푎푖,푗 . (2.5)푛푗 =1 푚푖 =1 If 휎푚,푛→푆 as m→∞,푛→∞ then we say that 푎푚,푛 ∞푛 =0∞ 푚=0 is summable to S by (C,1,1) method. Consider the double infinite series (−1)푚+푛∞푛 =0∞ 푚=0 in this case, 푆푚,푛= (−1)푖+푗 푛 푗=0 푚 푖=0 = −1 (푖)푚푖 =0 −1 (푗)푛푗 =0 = 1−1+1−⋯+(−1)푚 1−1+1−⋯+(−1)푛 = 1−(−1)푚+11+1 1−(−1)푛+11+1 = 14 1+(−1)푚 1+(−1)푛 = 1,푖푓,푚=2푛1,푛=2푛20,푖푓,푚=2푛1,푛=2푛2+10,푖푓,푚=2푛1+1,푛=2푛20,푖푓,푚=2푛1+1,푛=2푛2+1 푤푕푒푟푒 푛1,푛2휖ℕ0. Then lim푚,푛→∞,∞푆푚,푛 does not exist. The double infinite series −1 (푚+푛)∞푛 =0∞ 푚=0 is not convergent. Let us consider 휎푚,푛. 휎푚,푛.= 1− 푖 푚+1 1− 푗 푛+1 −1 푖+푗 푛 푗=0 푚 푖=0 = 3+(−1)푚+2푚 3+(−1)푛+2푛 16 푚+1 (푛+1) then 휎푚,푛.→ 14 as m→∞,푛→∞. Therefore the series −1 (푚+푛)∞푛 =0∞ 푚=0, is summable to 14 by(C,1,1) method. 2.5 Generalized Carleson operator Write 푆푁,푁푓 푥 = 푓,푤푛(.−푘) 푁푘 =−푁 푁푛 =0푤푛 푥−푘 ,푓∈퐿푝(ℝ) , 1<푝<∞,

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휎푁,푁푓 푥 = 1 푁+1 2 푆푖,푗푓 푥 푁푗 =0 푁푖 =0 = 1− 푛 푁+1 1− 푘 푁+1 푓,푤푛(.−푘) 푁푘 =−푁 푁푛 =0푤푛 푥−푘 The Carleson operator for the Walsh type wavelet packet system, denoted by L, is defined by 퐿푓 푥 = 푠푢푝 푁≥1 푓,푤푛(.−푘) 푁푘 =−푁 푁푛 =0푤푛 푥−푘 .푓∈퐿푝(ℝ) , 1<푝<∞, = 푠푢푝 푁≥1 푆푁,푁푓 푥 . The generalized carleson operator for the Walsh type wavelet packet system denoted by 퐿푐 is defined by (퐿푐f) 푥 = 푠푢푝 푁≥1 1 푛+1 2 푆푖,푗푓 푥 푁푗 =0 푁푖 =0 = 푠푢푝 푁≥1 푆푁,푁푓 푥 . = 푠푢푝 푁≥1 1− 푛 푁+1 1− 푘 푁+1 푓,푤푛(.−푘) 푁푘 =−푁 푁푛 =0푤푛 푥−푘 (2.6). Let us define the generalized Carleson Operator for the periodic Walsh type wavelet packet system 푤 푛 . Write 푠푁푓 푥 = 푓,푤 푛 푤 푛 푥 ,푓∈푁푛 =0퐿푝(ℝ) , 1<푝<∞, 휎푁푓 푥 = 1 푁+1 푠푁푓 푥 푁 푛=0 = 1 푁+1 푓,푤 푛 (푤 푖(푥) 푛 푖=0 푁 푛=0) = (1− 푛 푁+1 푓,푤 푛 푤 푛 푥 . 푁 푛=0 The Carleson operator for the periodic Walsh type wavelet packet system, denoted by 픾, is defined by 픾푓 푥 = 푠푢푝 푁≥1 푓,푤 푛 푤 푛 푥 푁푛 =0 ,푓∈퐿푝(ℝ) , 1<푝<∞, = 푠푢푝 푁≥1 푠푁푓 푥 . The generalized Carleson operator for the periodic Walsh type wavelet type packet system, denoted by , is defined by 픾푐푓 푥 = 푠푢푝 푁≥1 푠0푓 푥 + 푠1푓 푥 + 푠2푓 푥 +⋯+ 푠푁푓 푥 푁+1 = 푠푢푝 푁≥1 1 푁+1 푠푛푓 푥 푁푛 =0 = 푠푢푝 푁≥1 휎푁푓 푥 = 푠푢푝 푁≥1 1− 푛 푁+1 푓,푤 푛 푤 푛 푥 푁푛 =0 . (2.7) 2.6 Strong type (p,p) Operator An operator T defined on 퐿푝 ℝ is of strong type(p,p) if it is sub-linear and there is a constant C such that 푇푓 푝≤ 퐶 푓 푝 for all 푓∈ 퐿푝 ℝ .
III. Main Results
In this paper, two new Theorem have been established in the following form: Theorem 3.1 Let 푤푛 be a family of Walsh-type wavelet packet system. Then the generalized Carleson operator 퐿푐 defined by (2,6) for any Walsh-type wavelet packet system is of strong type(p,p),1<푝<∞. Theorem 3.2 Let 푤 푛 be a periodic Walsh-type wavelet packets. Then the generalized Carleson Operator 픾푐 defined by (2.7) for periodic Walsh type wavelet packet expansion of any 푓∈퐿푝 ℝ ,1<푝<∞, converges a.e... 3.1 Lemmas For the proof of the theorems following lemmas are required: Lemma 3.1 (Zygmund[12],p.197),If푣1,푣2,푣3,…푣푛 are non negative and non increasing,then 푢1푣1+푢2푣2+푢3푣3+⋯+푢푛푣푛 ≤푣1푚푎푥 푈푘 Where 푈푘=푢1+푢2+푢3+⋯+푢푘 for k=1,2,3,...,n. Lemma 3.2 Let 푓1∈ 퐿2(ℝ), and define 푓푛 푛≥2 recursively by

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푓푛 푥 = 푓푚 2푥 +푓푚 2푥−1 ,푛=2푚 푓푚 2푥 −푓푚 2푥−1 ,푛=2푚+1 Then 푓푚 푥 = 푊 푚−2푗푝2−푗 2푗−1 푝=0푓1(2푗x-p) for m , j ∈ℕ,2푗≤푚<2푗+1. Lemma 3.3 Let 푤푛 푛≥0 be a family of Walsh type wavelet packets. If 푤1∈퐶1(ℝ) then there exists an isomorphism 휓∶ 퐿푝 ℝ →퐿푝 ℝ ,1<푝<∞,푠푢푐푕 푡푕푎푡 휓푤푛 .−푘 = 푤푛 .−푘 ,푛≥0,푘∈ℤ. Lemma (3.2) and (3.3) can be easily proved. Lemma 3.4 (Billard [1] and Sj표 lin [9]. Let 푓∈퐿1[0,1) and define 푆푛 푥,푓 = 푓 푡 푊푘 푡 푑푡푊푘 푥 .10 푛 푘=0 Then the Carleson operator G defined by 퐺푓 푥 = 푠푢푝 푛 푆푛(푥,푓) is of strong type (p,p) for 1<푝<∞. 3.2 Proof of Theorem 3.1 For, 푓,푔,퐿푝(ℝ) 퐿푐 푓+푔 푥 = 푠푢푝 푁≥1 휎푁,푁 푓+푔 (푥) = 푠푢푝 푁≥1 1− 푛 푁+1 1− 푘 푁+1 푁푘 =−푁 푓+푔,푤푛(.−푘) 푤푛(푥−푘)푁푛 =0 = 푠푢푝 푁≥1 1− 푛 푁+1 1− 푘 푁+1 푁푘 =−푁 푓,푤푛 .−푘 + 푔,푤푛 .−푘 푤푛 푥−푘 푁푛 =0 푠푢푝 ≤푁≥1 1− 푛 푁+1 1− 푘 푁+1 푁 푘=−푁 푓,푤푛(.−푘) 푤푛(푥−푘) 푁 푛=0 + 푠푢푝 푁≥1 1− 푛 푁+1 1− 푘 푁+1 푁 푘=−푁 푔,푤푛(.−푘) 푤푛(푥−푘) 푁 푛=0 = 푠푢푝 푁≥1 (휎푁,푁푓)(푥) + 푠푢푝 푁≥1 (휎푁,푁푔)(푥) = 퐿푐푓 푥 + 퐿푐푔 푥 . Also for 훼∈ℝ 퐿푐훼푓 푥 = 푠푢푝 푁≥1 (휎푁,푁(훼푓))(푥) = 푠푢푝 푁≥1 1− 푛 푁+1 1− 푘 푁+1 푁푘 =−푁 훼푓,푤푛(.−푘) 푤푛(푥−푘)푁푛 =0 = 훼 푠푢푝 푁≥1 1− 푛 푁+1 1− 푘 푁+1 푁 푘=−푁 푓,푤푛(.−푘) 푤푛(푥−푘) 푁 푛=0 = 훼 퐿푐푓 푥 . Choose M ∈ℕ such that Supp(푤푛)⊂[−푀,푀] for 푛≥0. Fix 푝∈(1,∞) and take any 푓 푥 = 푓,푤푛(.−푘) 푤푛 푥−푘 ∈퐿푝 ℝ .푛≥0,푘∈ℤ Define 푓푘 푥 = 푓,푤푛(.−푘) 푤푛 푥−푘 ,푔푘 푥 = 푓,푤푛(.−푘) 푊푛 푥−푘 . 푛≥0,푘∈ℤ푛≥0,푘∈ℤ We have 푓푘 푝≈ 푔푘 푝,푤푖푡푕 bounds independent of k (Lemma 3.3) .Note that for {푥∈ 푙,푙+1 : 퐿푐푓(푥) >훼} ≤ 퐶 훼푝 퐿푐푓푘(푥) 푙+1+푛 푘=푙−푛 푝 푑푥. Using the Marcinkiewiez interpolation theorem, it suffices to prove that 퐿푐푓푘 푝≤퐶 푓푘 푝, where C is a constant independent of k.

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Since 푓푘 푝 푝≤2(푛+1) 푓푘 푝 푝≤2퐶(푛+1) 푔푘 푝 푝≤퐶1 푓 푝 푝 푘∈ℤ푘∈ℤ 푙+1+푛 푘=푙−푛푙∈ℤ where 퐶1 is constant. Without loss of generality, we assume that k=0. Let K ∈ ℕ be the scale from which only the Haar filter is used to generate the wavelet packets 푤푛 푛≥2푘+1. Let N ∈ℕ and suppose 2푗≤푁≤2퐽+1 for some J>퐾+1.퐶푙푒푎푟푙푦,푓표푟 each 푥∈ℝ. 퐿푐푓 푥 = 푠푢푝 푁≥1 1− 푛 푁+1 1− 푘 푁+1 푁푘 =−푁 푓,푤푛(.−푘) 푤푛(푥−푘)푁푛 =0 =푠푢푝푁≥1 1− 푛 푁+1 푓,푤푛(.) 푤푛(푥)푁푛 =0 , 푘=0, ≤푠푢푝1≤푁<2푘+1 1− 푛 푁+1 푓,푤푛 (.) 푤푛(푥)푁푛 =0 +푠푢푝퐽>퐾+1 1− 푛 푁+1 푓,푤푛 (.) 2퐽−1 푛=2퐾+1 푤푛 (푥) +푠푢푝퐽>퐾+1 푠푢푝2퐽≤푁<2퐽+1 1− 푛 푁−2퐽+1 푓,푤푛(.) 푤푛 (푥)푁 푛=2퐽 =퐽1+퐽2+퐽3 say. (3.2) Using Lemma 3.1, we have 퐽1=푠푢푝1≤푁<2푘+1 1− 푛 푁+1 푓,푤푛 (.) 푤푛(푥) 2푘+1−1 푛=0 ≤푠푢푝1≤푁<2푘+1 푚푎푥 푓,푤푛 (.) 푤푛(푥)0=푛≤2퐾+1−1 ≤ 푓,푤푛(.) 2퐾+1−1 푛=0 푤푛 (푥) ≤ 푓,푤푛(.) 푤푛(푥) ∞ 휒[−푁,푁] (푥) ≤ 푓0 푝 푤푛 푞 2퐾+1−1 푛=0 푤푛 (푥) ∞ 휒[−푁,푁] 푥 . (3.3) Next, 퐽2= 푠푢푝퐽>퐾+1 1− 푛 푁+1 2퐽−1 푛=2퐾+1 푓,푤푛 . 푤푛(푥) ≤ 푠푢푝 퐽>푘+1max 푓,푤푛 . 푤푛(푥)2푘+1=푛≤2퐽−1 . Also 푠푢푝 퐽>푘+1 (1− 푛 푁+1) 푓,푤푛 . 푤푛(푥)2퐽−1 푛=2푘+1 푃 ≤ sup푚푎푥 퐽>푘+1 푓,푤푛 . 푤푛(푥) 푝2푘+1=푛≤2퐽−1 ≤퐶 푓,푤푛 . 푤푛(푥) 푝 ∞ 푛=0 =퐶 푓0 푝. (3.4) Consider 퐽3, 퐽3= 푠푢푝 2퐽≤푁<2퐽+1 (1− 푛 푁+1) 푓,푤푛 . 푤푛(푥) 푁 푛=2퐽 ≤ 푠푢푝 2퐽+푗2퐽−퐾≤푁<2퐽+(푗+1)2퐽−퐾 (1− 푛 푁+1) 푓,푤푛 . 푤푛(푥) 푁 푛=2퐽+푗2퐽−퐾 2퐾−1 푗=0, so it suffices to prove that 푠푢푝 푗>푘+1 푠푢푝 2퐽+푗2퐽−퐾≤푁<2퐽+(푗+1)2퐽−퐾 (1− 푛 푁+1) 푓,푤푛 . 푤푛(푥) 푁 푛=2퐽+푗2퐽−퐾 푝 ≤퐶 푓0 푝 for 푗=0,1,2,…,2푘−1. Fix 퐽>퐾+1 ,0≤푗≤22푘−1 and 2퐽+푗2퐽퐾≤푁<2퐽+ 푗+1 2퐽−퐾. Write,

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푗,3= (1− 푛 푁+1) 푓,푤푛 . 푤푛(푥) 푁 푛=2퐽+푗2퐽−퐾 . Using lemma 3.2, we have 푗,3 = (1− 푛 푁+1) 푓,푤푛 . 푊 푛−2퐽−푗2퐽−퐾(푠2− 퐽−퐾 ) 푁 푛=2퐽+푗2퐽−퐾 푤2푘+푗(2퐽−퐾푥−푠) 2퐽−퐾−1 푠=0 . Define 퐹푁 푡 = 1− 푛 푁+1 푓,푤푛 . 푊푛−2퐽−푗2퐽−퐾, 푡 , 푁 푛=2퐽+푗2퐽−퐾 and 퐹 푡 = 푠푢푝 푁<2퐽+(푗+1)2퐽−퐾 퐹푁 푡 . We have 푗′ 3 = 1− 푛 푁+1 푓,푤푛 . 푤푛 푥 푁 푛=2퐽+푗2퐽−퐾 , ≤푚푎푥 푓,푤푛 . 푤푛 푥 2퐽+푗2퐽−퐾=푛<푁 , ≤ 퐹(푠2−(퐽−퐾) 푤2푘+푗((2퐽−퐾)푥−푠) 2퐽−퐾−1 푆=0, and using the compact support of the wavelet packets, 푗′ 3 = 1− 푛 푁+1 푓,푤푛 . 푤푛 푥 푁 푛=2퐽+푗2퐽−퐾 , ≤푚푎푥 푓,푤푛 . 푤푛 푥 2퐽+푗2퐽−퐾=푛<푁 ≤ 푤2푘+푗 ∞ 퐹(([2퐽−퐾푀+1 푙=−푀푥]+푙)2−(퐽−퐾)). Note that F is constant on dyadic interval of type [푙2−(푗−푘),(푙+1)2−(푗−푘)) and taking Δ푙=(( 2퐽−퐾푥+푙 2− 퐽−퐾 , 2퐽−퐾푥+(푙+1) 2− 퐽−퐾 ), we have 푗′ 3 = 1− 푛 푁+1 푓,푤푛 . 푤푛 푥 푁 푛=2퐽+푗2퐽−퐾 ≤푚푎푥 푓,푤푛 . 푤푛 푥 푁 2퐽+푗2퐽−퐾=푛<푁 ≤ 푤2푘+푗 ∞ Δ푙 −1 퐹 푡 푑푡 Δ푙 푀+1 푙=−푀 . We need an estimate of F that does not depend on J . Note that for k, 0≤푘<2퐽−퐾 , using (2.3) 푊2퐽+푗2퐽−퐾 푡 푊푘 푡 =푊2퐽+푗2퐽−퐾+푘 푡 , since the binary expansions of 2퐽+푗2퐽−퐾 and of k have no 1’s in common. Hence, 퐹푁 푡 = 푊2퐽+푗2퐽−퐾 푡 퐹푁 푡 = 1− 푛 푁+1 푓,푤푛 . 푊푛 푡 푁 푛=2퐽+푗2퐽−퐾

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≤푚푎푥 푓,푤푛 . 푊푛 푡 푁 2퐽+푗2퐽−퐾=푛<푁 . Using lemma (3.4) , we have F(t)≤2 퐺푔표 푡 . Thus, 1− 푛 푁+1 푓,푤푛 . 푤푛 푥 푁 푛=2퐽+푗2퐽−퐾 ≤푚푎푥 푓,푤푛 . 푤푛 푥 푁 2퐽+푗2퐽−퐾=푛<푁 ≤2 푤2푘+푗 ∞ Δ푙 −1 퐺푔0 푡 푑푡 Δ푙 푀+1 푙=−푀 . Let Δ푙 ∗ be the smallest dyadic interval containing Δ푙 and 푥 , and note that Δ푙 ∗ ≤(푀+1) Δ푙 since 푥∈ Δ0−퐷 . We have 1− 푛 푁+1 푓,푤푛 . 푤푛 푥 푁 푛=2퐽+푗2퐽−퐾 ≤푚푎푥 푓,푤푛 . 푤푛 푥 2퐽+푗2퐽−퐾=푛<푁 ≤2 푤2푘+푗 ∞ Δ푙 −1 퐺푔0 푡 푑푡 Δ푙 푀+1 푙=−푀 ≤4 푤2푘+푗 ∞(푀+1)2 푀∗ 퐺푔0 푥 (3.5) Where 푀∗ 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 퐽3≤ 푠푢푝 퐽>퐾+1 푠푢푝 2퐽+푗2퐽−퐾≤푁<2퐽+(푗+1)2퐽−퐾 (1− 푛 푁+1) 푓,푤푛 . 푤푛(푥) 푁 푛=2퐽+푗2퐽−퐾 2퐾−1 푗=0 ≤ 푠푢푝 2퐽+푗2퐽−퐾≤푁<2퐽+(푗+1)2퐽−퐾 푚푎푥 푓,푤푛 . 푤푛 푥 2퐽+푗2퐽−퐾=푛<푁 2푘−1 푗=0 ≤4 푤2푘+푗 ∞(푀+1)2 푀∗ 퐺푔0 푥 ,푎.푒.. (3.6) Using (Sj표 lin [9]), 푀∗ and G both of strong type (p,p), hence both are bounded. 푠푢푝 퐽>퐾+1퐽3 푃 ≤ 푠푢푝 퐽>퐾+1 푠푢푝 2퐽+푗2퐽−퐾≤푁<2퐽+(푗+1)2퐽−퐾 (1 푁 푛=2퐽+푗2퐽−푘 2퐾−1 푗=0− 푛 푁+1) 푓,푤푛 . 푤푛(푥) 푝 ≤퐶 푔0 푝 ≤퐶1 푓0 푝, 푗=0,1,2,3,…,2퐾-1. Thus the Theorem 3.1 is completely established. 3.3 proof of the Theorem 3.2 Let 푓∈퐿푝 0,1 , and choose M∈ℕ such that supp(푤푛)⊂[−푀,푀] for n≥0. Then 픾푐푓 푥 = 푠푢푝 푁≥1 1− 푛 푁+1 푓,푤 푛 푤 푛(푥) 푁 푛=0 = 푠푢푝 푁≥1 1− 푛 푁+1 푓, 푤푛(.−푘1)푀+1 푘1=−푀 푤푛(푥−푘2)푀+1 푘2=−푀 푁푛 =0

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= 푠푢푝 푁≥1 1− 푛 푁+1 푓,푤푛(.−푘1) 푀+1 푘1=−푀 푤푛(푥−푘2)푀+1 푘2=−푀 푁푛 =0 = 푠푢푝 푁≥1 1− 푛 푁+1 푓,푤푛(.−푘1) 푁푛 =0푤푛(푥−푘2) 푀+1 푘2=−푀 푀+1 푘1=−푀 = 푠푢푝 푁≥1 1− 푛 푁+1 푓 푦 푤푛 푦−푘1 푑푦푤푛(푥−푘2) 10 푁푛 =0 푀+1 푘2=−푀 푀+1 푘1=−푀 Following the proof of theorem 3.1, it can be proved that the generalized Carleson operator 픾푐 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 l’espace 퐿2(0,1), Studia Math, 28:363-388, 1967. [2] Bromwich, T. J .I ’A., An introduction to the theory of infinite Series, The University press, Macmillan and Co; Limited, London, 1908. [3] Carleson, L. , “On convergence and growth of partial sums of Fourier series” , 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] Sj표 lin P., An inequality of paley and convergence a.e. of Walsh-FourierSeries, Arkiv f표 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.