# On Generalized Carleson Operators of Periodic  Wavelet Packet Expansions

Authors: Shyam Lal; Manoj Kumar
Journal: The Scientific World Journal
 (2013)

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## Abstract

Three new theorems based on the generalized Carleson operators for the periodic Walsh-type wavelet packets have been established. An application of these theorems as convergence a.e. for the periodic Walsh-type wavelet packet expansion of block function with the help of summation by arithmetic means has been studied.


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## Body

## 1. Introduction

Wavelet packet expansions have wide applications in engineering and technology. The Walsh-type wavelet packet expansions play an important role in signal processing, numerical analysis, and quantum mechanics. A family of nonstationary wavelet packets considered the smooth generalization of the Walsh functions having some of the same nice convergence properties for expansion ofLp-function, 1<p<∞, as the Walsh-Fourier series. Walsh-type wavelet packet expansion has been studied by the researchers Billard [1], Nielsen [2], Sjölin [3] and others. In 1966, at first, Carleson operator has been introduced by Lennart Carleson (Carleson [4]). Several important properties of this operator has been studied by researcher Nielsen [2]. In this paper, the pointwise convergence almost everywhere by arithmetic means or (C,1) summability method of the partial sum operator for Walsh-type wavelet packet expansion of functions from the block space, 𝔹q,1<q≤∞,p-1+q-1=1 has been studied. Generalized Carleson operators are introduced and some new properties of generalized Carleson operators are investigated. Specific convergence properties of Walsh-type wavelet packet expansions of block functions using (C,1) method and generalized Carleson operator have been obtained.


## 2. Definitions and Preliminaries

Walsh-Type Wavelet Packets.To every multiresolution analysis {Vj}j∈ℤ for L2(ℝ), an associated scaling function φ and a wavelet ψ are given with the properties that
(1)Vj=span¯{2j/2φ(2j·-k):k∈ℤ},j∈ℤ,{ψj,k≡2j/2ψ(2j·-k):j,k∈ℤ}
is an orthonormal basis for L2(ℝ).We write(2)Wj=span¯{2j/2ψ(2j·-k):k∈ℤ},j∈ℤ.Letℕ be the set of natural numbers. Let (F0(p),F1(p)),p∈ℕ, be a family of bounded operators on l2(ℤ) of the form
(3)(Fϵ(p)a)k=∑n∈ℤ‍anhϵ(p)(n-2k),ϵ=0,1
with h1(p)(n)=(-1)nh0(p)(1-n) a real-valued sequence in l1(ℤ) such that
(4)F0(p)*F0(p)+F1(p)*F1(p)=1,F0(p)F1(p)*=0.Define the family of functions{wn}n=0∞ recursively by letting w0=φ,   w1=ψ and then for n∈ℕ,
(5)w2n(x)=2∑l∈ℤ‍h0(p)(l)wn(2x-l),w2n+1(x)=2∑l∈ℤ‍h1(p)(l)wn(2x-l),
where 2p≤n<2p+1.The family{wn}n=0∞ is basic non stationary wavelet packets. {wn(·-k):n≥0,k∈ℤ} is an orthonormal basis for L2(ℝ).Moreover,(6){wn(·-k):2j≤n<2j+1,k∈ℤ}
is an orthonormal basis for Wj=span¯{2j/2ψ(2j·-k):k∈ℤ}.Each pair(F0(p),F1(p)) can be chosen as a pair of quadrature mirror filters associated with a multiresolution analysis, but this is not necessary.The trigonometric polynomials given by(7)m0(p)(ξ)=12∑k∈ℤ‍h0(p)(k)e-ikξ,m1(p)(ξ)=12∑k∈ℤ‍h1(p)(k)e-ikξ
are called the symbols of the filters.The Fourier transforms of (5) are given by
(8)w^2n(ξ)=m0(p)(ξ2)w^n(ξ2),w^2n+1(ξ)=m1(p)(ξ2)w^n(ξ2).The Haar low-pass quadrature mirror filter{h0(k)}k is given by h0(0)=h0(1)=1/2,h0(k)=0  otherwise, and the associated high-pass filter {h1(k)}k is given by
(9)h1(k)=(-1)kh0(1-k).Definition 1.
Let{wn}n≥0,k∈ℤ be a family of non-stationary wavelet packets constructed by using a family {h0(p)(n)}p=1∞ of finite filters for which there is a constant, K∈ℤ such that h0(p)(n) is the Haar filter for every p≥K. If w1∈C1(ℝ) is compactly supported then {wn}n≥0 is called a family of Walsh-type wavelet packets.Definition 2.
Let{wn}n=0∞ be a family of Walsh-type basic wavelet packets. For n∈ℕ0, define the corresponding periodic Walsh-type wavelet packets w~n by
(10)w~n(x)=∑k∈ℤ‍wn(x-k).From Fubini’s theorem, it follows that{w~n}n=0∞ is an orthonormal basis for L2[0,1).Block Spaces. A dyadic q-block is a function β∈Lq[0,1) which is supported on some dyadic interval I such that ∥β∥q≤|I|1/q-1, where ∥β∥q=[∫01‍|β(t)|qdt]1/q,1<q<∞. Let 𝔹q denote the space of measurable functions f on [0,1) which has an expansion
(11)f=∑k=1∞‍ckβk,
where each βk is a q-block and the coefficients ck,k∈ℤ satisfy
(12)∥|{ck}|∥=∑k:ck≠0‍|ck|[1+log∑j=1∞|cj||ck|]<∞.The quasi norm off∈𝔹q is given as the infimum of ∥|·|∥ over all possible decompositions of f into blocks
(13)∥f∥𝔹q=inff=∑‍ckβk∥|{ck}|∥.Letf∈𝔹q; then
(14)∥f∥1≤∑k=1∞‍|ck|∥βk∥1≤∑k=1∞‍|ck|<∞,
using (12) and the fact that for each k,∥βk∥q≤|I|1/q-1 which implies that ∥βk∥1≤1; that is, 𝔹q⊂L1[0,1). Moreover, for
(15)f∈Lq[0,1),1<q<∞,β=∥f∥q-1f
is a q-block supported on I=[0,1) so Lq[0,1)⊂𝔹q.The classical example to show that for eachq>1 there exists f∈𝔹q which belongs to none of the Lp[0,1)-space is the following.Let(16)βk(x)={2k,12k<x≤32(k+1),0,otherwise.Thenf=∑k=1∞‍k-2βk∈𝔹q, but ∥f∥pp=∑k=1∞‍(1/2)k-2p2k(p-1)=∞ for every p>1.Summation of Series by Arithmetic Means. If a series u0+u1+u2+⋯ is not convergent, that is, if sn=u0+u1+u2+⋯+un does not tend to a limit, it is some time possible to associate with the series a “sum” in a less direct way. The simplest such method is “summation by arithmetic means”. Let
(17)σn=s0+s1+s2+⋯+snn+1
be the arithmetic mean of the partial sums of the given series.Ifsn→s, then also σn→s; for if sn=s+δn, then
(18)σn=s+δ0+δ1+δ2+δ2+⋯+δnn+1,
and the last term tends to zero if δn→0. Consider
(19)σn=s0+s1+s2+⋯+snn+1=(u0+(u0+u1)+⋯+(u0+u1+⋯+uk)+⋯+(u0+u1+⋯+un))×(n+1)-1=∑k=0n‍(1-kn+1)uk.Ifσn→s as n→∞,∑n=0∞‍un is said to be summable to s by Cesàro’s means of order 1. We write
(20)∑n=0∞‍un=s(C,1).Butσn may tend to a limit even though sn does not, for example, the series
(21)1-1+1-1+⋯.Here the partial sumssn are alternately 1 and 0, and it is easily seen that σn→1/2.


### 2.1. Generalized Carleson Operators

Let{w~n} be a periodic Walsh-type wavelet packet basis. For any function f∈L1[0,1), define
(22)(SNf)(x)=∑n=0N‍〈f,w~n〉w~n(x).The Carleson operator𝔾 is defined by
(23)𝔾f(x)=supN≥0|∑n=0N‍〈f,w~n〉w~n(x)|=supN≥0|(SNf)(x)|.The generalized Carleson operator𝔾c is defined by
(24)𝔾cf(x)=supN≥0|(S0f)(x)+(S1f)(x)+⋯+(SNf)(x)N+1|=supN≥0|1N+1∑ν=0N‍∑n=0ν‍〈f,w~n〉w~n(x)|=supN≥0|∑n=0N‍(1-nN+1)〈f,w~n〉w~n(x)|.The weak Carleson operatorG is defined by
(25)Gf(x)=limsupN≥0|∑n=0N‍〈f,w~n〉w~n(x)|.=limsupN≥0|(SNf)(x)|.The generalized weak Carleson operatorGc is define by
(26)Gcf(x)=limsupN≥0|(S0f)(x)+(S1f)(x)+⋯+(SNf)(x)N+1|=limsupN≥0|1N+1∑ν=0N‍∑n=0ν‍〈f,w~n〉w~n(x)|=limsupN≥0|∑n=0N‍(1-nN+1)〈f,w~n〉w~n(x)|.The dyadic Carleson operator𝔾d is defined by
(27)𝔾df(x)=supN≥0|∑n=02N-1‍〈f,w~n〉w~n(x)|=supN≥0|(S2Nf)(x)|.The generalized dyadic Carleson operator𝔾cd is define by
(28)𝔾cdf(x)=supN≥0|(S0f)(x)+(S1f)(x)+⋯+(S2N-1f)(x)2N|=supN≥0|12N∑ν=02N-1‍∑n=0ν‍〈f,w~n〉w~n(x)|=supN≥0|∑n=02N-1‍(1-n2N)〈f,w~n〉w~n(x)|.It is easy to prove that𝔾c,Gc and 𝔾cd are sublinear operators.Walsh Functions and Their Properties. The Walsh system {Wn}n=0∞ is defined recursively on [0,1) by letting
(29)W0(x)={1,0≤x<1;0,otherwise,W2n(x)=Wn(2x)+Wn(2x-1),W2n+1(x)=Wn(2x)-Wn(2x-1).Observe that the Walsh system is the family of wavelet packets obtained by consideringφ=W0,
(30)ψ(x)={1,0≤x<12;-1,12≤x<1;0,otherwise
and using the Haar filters in the definition of the nonstationary wavelet packets.The Walsh system is closed under pointwise multiplication. Define the binary operator⊕:ℕ0×ℕ0→ℕ0 by
(31)m⊕n=∑i=0∞‍|mi-ni|2i,
where m=∑i=0∞‍mi2i and n=∑i=0∞‍ni2i. Then(32)Wm(x)Wn(x)=Wm⊕n(x),(see Schipp et al. [5]).We can carry over the operator⊕ to the interval [0,1] by identifying those x∈[0,1] with a unique expansion x=∑j=0∞‍xj2-j-1 (almost all x∈[0,1] has such a unique expansion) by their associated binary sequence {xi}. For two such points x,y∈[0,1], define(33)x⊕y=∑j=0∞‍|xj-yj|2-j-1.The operation⊕ is defined for almost all x,y∈[0,1]. With this definition, we have
(34)Wn(x⊕y)=Wn(x)Wn(y)
for every pair x,y for which x⊕y is defined, (Golubov et al. [6], page 11).


## 2.1. Generalized Carleson Operators

Let{w~n} be a periodic Walsh-type wavelet packet basis. For any function f∈L1[0,1), define
(22)(SNf)(x)=∑n=0N‍〈f,w~n〉w~n(x).The Carleson operator𝔾 is defined by
(23)𝔾f(x)=supN≥0|∑n=0N‍〈f,w~n〉w~n(x)|=supN≥0|(SNf)(x)|.The generalized Carleson operator𝔾c is defined by
(24)𝔾cf(x)=supN≥0|(S0f)(x)+(S1f)(x)+⋯+(SNf)(x)N+1|=supN≥0|1N+1∑ν=0N‍∑n=0ν‍〈f,w~n〉w~n(x)|=supN≥0|∑n=0N‍(1-nN+1)〈f,w~n〉w~n(x)|.The weak Carleson operatorG is defined by
(25)Gf(x)=limsupN≥0|∑n=0N‍〈f,w~n〉w~n(x)|.=limsupN≥0|(SNf)(x)|.The generalized weak Carleson operatorGc is define by
(26)Gcf(x)=limsupN≥0|(S0f)(x)+(S1f)(x)+⋯+(SNf)(x)N+1|=limsupN≥0|1N+1∑ν=0N‍∑n=0ν‍〈f,w~n〉w~n(x)|=limsupN≥0|∑n=0N‍(1-nN+1)〈f,w~n〉w~n(x)|.The dyadic Carleson operator𝔾d is defined by
(27)𝔾df(x)=supN≥0|∑n=02N-1‍〈f,w~n〉w~n(x)|=supN≥0|(S2Nf)(x)|.The generalized dyadic Carleson operator𝔾cd is define by
(28)𝔾cdf(x)=supN≥0|(S0f)(x)+(S1f)(x)+⋯+(S2N-1f)(x)2N|=supN≥0|12N∑ν=02N-1‍∑n=0ν‍〈f,w~n〉w~n(x)|=supN≥0|∑n=02N-1‍(1-n2N)〈f,w~n〉w~n(x)|.It is easy to prove that𝔾c,Gc and 𝔾cd are sublinear operators.Walsh Functions and Their Properties. The Walsh system {Wn}n=0∞ is defined recursively on [0,1) by letting
(29)W0(x)={1,0≤x<1;0,otherwise,W2n(x)=Wn(2x)+Wn(2x-1),W2n+1(x)=Wn(2x)-Wn(2x-1).Observe that the Walsh system is the family of wavelet packets obtained by consideringφ=W0,
(30)ψ(x)={1,0≤x<12;-1,12≤x<1;0,otherwise
and using the Haar filters in the definition of the nonstationary wavelet packets.The Walsh system is closed under pointwise multiplication. Define the binary operator⊕:ℕ0×ℕ0→ℕ0 by
(31)m⊕n=∑i=0∞‍|mi-ni|2i,
where m=∑i=0∞‍mi2i and n=∑i=0∞‍ni2i. Then(32)Wm(x)Wn(x)=Wm⊕n(x),(see Schipp et al. [5]).We can carry over the operator⊕ to the interval [0,1] by identifying those x∈[0,1] with a unique expansion x=∑j=0∞‍xj2-j-1 (almost all x∈[0,1] has such a unique expansion) by their associated binary sequence {xi}. For two such points x,y∈[0,1], define(33)x⊕y=∑j=0∞‍|xj-yj|2-j-1.The operation⊕ is defined for almost all x,y∈[0,1]. With this definition, we have
(34)Wn(x⊕y)=Wn(x)Wn(y)
for every pair x,y for which x⊕y is defined, (Golubov et al. [6], page 11).


## 3. Main Results

In this paper, three new theorems for the generalized Carleson operators on the periodic Walsh-type wavelet packets have been determined in the following form.Theorem 3.
Let{w~n} be a periodic Walsh-type wavelet packet basis. Then for every q-block β,1<q≤∞,
(35)|{𝔾cdβ>α}|≤Cqα,α>0,
where 𝔾cd is the generalized dyadic Carleson operator defined by (28) and Cq is a positive finite constant.Theorem 4.
Let{w~n} be a periodic Walsh-type wavelet packet basis. Then for every q-block β,1<q≤∞,
(36)|{Gcβ>α}|≤Cqα,α>0,
where Gc is the generalized weak Carleson operator defined by (26) and Cq is a positive finite constant.Theorem 5.
If a functionf belongs to 𝔹q-class, 1<q≤∞, then
(37)|{Gcf>α}|=O(∥f∥𝔹q),forα>0,
where Gc is the generalized weak Carleson operator.


## 4. Lemmas

For the proof of our theorems, the following lemmas are required.Lemma 6 (Nielsen [7]).
Letf1∈L2(ℝ), and define {fn}n≥2 recursively by
(38)f2n(x)=fn(2x)+fn(2x-1),f2n+1(x)=fn(2x)-fn(2x-1).
Then(39)fn(x)=∑s=02J-1‍Wn-2J(s2-J)f1(2Jx-s),
where n,J∈ℕ,2J≤n<2J+1.Lemma 7 (Zygmund [8], page 3).
Consider(40)∑ν=1n‍uνvν=∑ν=1n-1‍(vν-vν+1)Uν+Unvn,
where Uk=u1+u2+⋯+uk for k=1,2,…,n; it is also called Abel’s transformation.Lemma 8.
Let{Wn}n=0∞ be the Walsh system. Then
(41)|∑n=2Km‍(1-nm-2k+1)Wn-2K([2Kx]2-K)∑n=2Km‍×Wn-2K([2Ky]2-K)|≤Cx⊕y,
where C is a finite positive constant, K≥1,2K≤n<2K+1, and for all pairs x,y∈[0,1) for which x⊕y is defined.Proof.
The Dirichlet kernel,Dn(x)=∑k=0n-1‍Wk(x), for the Walsh system satisfies
(42)|Dn(x⊕y)|≤1x⊕y (see Golubov et al.  [6], page 21).
Hence,(43)|∑n=2Km‍(1-nm-2k+1)Wn-2K([2Kx]2-K)∑n=2Km‍×Wn-2K([2Ky]2-K)|=|∑n=2Km‍(1-nm-2k+1)Wn-2K∑n=2Km‍×([2Kx]2-K⊕[2Ky]2-K)|=|∑n=0m-2K‍(1-nm-2k+1)Wn∑n=0m-2K‍×([2Kx]2-K⊕[2Ky]2-K)|=|∑n=0m-2K-1‍{(1-nm-2K+1)-(1-n+1m-2K+1)}×∑r=0n‍Wr([2Kx]2-K⊕[2Ky]2-K)+(1-m-2Km-2K+1)×∑n=0m-2K‍Wn([2Kx]2-K⊕[2Ky]2-K)|,byLemma7,=|∑n=0m-2K-1‍1m-2K+1×∑r=0n‍Wr([2Kx]2-K⊕[2Ky]2-K)+1m-2K+1×∑n=0m-2K‍Wn([2Kx]2-K⊕[2Ky]2-K)|≤|∑n=0m-2K-1‍Wn([2Kx]2-K⊕[2Ky]2-K)+1m-2K+1×∑n=0m-2K‍Wn([2Kx]2-K⊕[2Ky]2-K)|=|∑n=2Km-1‍Wn-2K([2Kx]2-K⊕[2Ky]2-K)+1m-2K+1×∑n=2Km‍Wn-2K([2Kx]2-K⊕[2Ky]2-K)|≤|∑n=2Km-1‍Wn-2K([2Kx]2-K⊕[2Ky]2-K)|+1m-2K+1×|∑n=2Km‍Wn-2K([2Kx]2-K⊕[2Ky]2-K)|=|W2K([2Kx]2-K⊕[2Ky]2-K)Dm-2K×([2Kx]2-K⊕[2Ky]2-K)|+1m-2K+1×|W2K([2Kx]2-K⊕[2Ky]2-K)Dm-2K+1×([2Kx]2-K⊕[2Ky]2-K)|=|Dm-2K(x⊕y)|+1m-2K+1|Dm-2K+1(x⊕y)|≤1(x⊕y)+1m-2k+11(x⊕y),x⊕y≠0=(1+1m-2k+1)1(x⊕y)≤C(x⊕y),
where (32), (34), and the fact that Dν+1-2K is a constant on dyadic intervals of the form [l2-K,(l+1)2-K) are used. This completes the proof of Lemma 8.Lemma 9.
If(44)KJ,m(σ)(x,y)=∑n=2Jm‍(1-nm-2J+1)wn(x)wn(y),for2J≤m<2J+1,
then
(45)|KJ,m(σ)(x,y)|≤∑l=-2N2N‍C|x-y+2K-Jl|,
where C is an arbitrary constant.Proof.
The kernel can be expanded as(46)KJ,m(σ)(x,y)=∑n=2Jm‍(1-nm-2J+1)wn(x)wn(y)=∑n=2Jm‍(1-nm-2J+1)×(∑l=02J-K-1‍Wn-2J-K(l2-(J-K))w2K(2J-Kx-l)×∑k=02J-K-1‍Wn-2J-K(k2-(J-K))∑l=02J-K-1‍×w2K(2J-Ky-k)),byLemma6,=∑l=02J-K-1‍∑k=02J-K-1‍{∑n=2Jm‍(1-nm-2J+1)×(Wn-2J-K(l2-(J-K))×Wn-2J-K(k2-(J-K)))×w2K(2J-Kx-l)∑n=2Jm‍×w2K(2J-Ky-k)}.
Therefore, using Lemma8,(47)|KJ,m(σ)(x,y)|≤∑l=-NN‍′∑k=-NN‍′|∑n=2Jm‍(1-nm-2J+1)×Wn-2J-K([2J-K(x+2K-Jl)]2-(J-K))∑n=2Jm‍×Wn-2J-K([2J-K(y+2K-Jk)]2-(J-K))|×∥w2K∥∞2≤∑l=-NN‍′∑k=-NN‍′C(x+2K-Jl)⊕(y+2K-Jk),
where ∑‍′ indicates that only the terms for which x+2K-Jl∈[0,1) and y+2K-Jk∈[0,1), respectively, should be included in the sum. This implies the estimate
(48)|KJ,m(σ)(x,y)|≤∑l=-NN‍∑k=-NN‍C~|x-y+2K-J(l-k)|,
since a⊕b≥2-log2[|a-b|]≥|a-b|/2. This completes the proof of Lemma 9.


## 5. Proof of Theorem3

The dyadic arithmetic mean of partial sums for the expansion of a measurable (integrable) functionf in the periodic Walsh-type wavelet packets,
(49)(σ2Nf)(x)=12N∑n=02N-1‍(Snf)(x)=12N∑n=02N-1‍(∑k=0n‍〈f,w~k〉w~k(x)),by(22),=∑n=02N-1‍(1-n2N)〈f,w~n〉w~n(x),
holds everywhere with the arithmetic mean of the projection onto the (periodized) scaling space V~N associated with the underlying multiresolution analysis (Hess-Nielsen and Wickerhauser [9]). Therefore, it suffices to consider the arithmetic mean of the projection operators PV~N on to the space V~N.Suppose that theq-block β is associated with the dyadic interval I⊂[0,1). If 1<α|I|, then |I|1-q/αq≤1/α, and using the fact that the operator f→supN∑n=0N(1-n/(N+1))PV~nf(x) (and thus f→𝔾cdf(x)) is of strong type (q,q). We have
(50)|{𝔾cdf(x)>α}|≤Cq(∥β∥qα)q≤Cq|I|1-qαq≤Cqα.Now suppose that1≥α|I| with I=[a,b). Put I~=[(3a-b)/2,(3b-a)/2]∩[0,1), and define I-=[0,1)-I~. We have
(51)|{𝔾cdf(x)>α}|≤2|I|+|I~∩{𝔾cdf(x)>α}|≤2α+|I~∩{𝔾cdf(x)>α}|.Fixx∈I-, and let KN(x,y) denote the operator kernel associated with the projection operators PV~N. Then there exists a finite constant C (independent of N) such that
(52)|KN(x,y)|≤C|x-y| (see Terence [10]).Using the estimate (52) on the kernel KN, we obtain
(53)|(σ2Nβ)(x)|=|∑n=02N-1‍(1-n2N)〈β,w~n〉w~n(x)|=|∑n=02N-2‍{(1-n2N)-(1-n+12N)}×∑r=0n‍〈β,w~r〉w~r(x)+(1-2N-12N)∑n=02N-1‍〈β,w~n〉w~n(x)|,byLemma7,=|∑n=02N-2‍12N∑r=0n‍〈β,w~r〉w~r(x)+12N∑r=02N-1‍〈β,w~n〉w~n(x)|=|∑n=02N-2‍12N(Snβ)(x)+12N(S2Nβ)(x)|≤∑n=02N-2‍12N|(Snβ)(x)|+12N|(S2Nβ)(x)|≤∑n=02N-2‍12N|∫I‍Kn(x,y)β(y)dy|+12N|∫I‍K2N(x,y)β(y)dy|≤∑n=02N-2‍12N(C|x-a|+C|x-b|)∥β∥1+12N(C|x-a|+C|x-b|)∥β∥1=(2N-12N+12N)(C|x-a|+C|x-b|)∥β∥1=(C|x-a|+C|x-b|)∥β∥1.Since∥β∥1≤1 and x∈I- implies that |x-a|,|x-b|≥|I|/2, therefore,
(54)|(σ2Nβ)(x)|≤{2C|I|+2C|I|}=4C|I|≤C~α.
Finally we obtain
(55)|{x∈I-:supN|(σ2Nβ)(x)|>α}|≤C~α,
where C~ is independent of I and β and hence Theorem 3 follows.


## 6. Proof of Theorem4

Fixα>0 and a q-block β supported on the dyadic interval I⊂[0,1); two cases are considered.Case I.
If1<α|I|, then |I|1-q/αq≤1/α. Therefore, using Theorem 5.1. [7], page 275, we have
(56)|{Gcβ>α}|≤Cq(∥β∥qα)q≤Cq(|I|1/q-1)qαq=Cq|I|1-qαq≤Cqα.Case II.
Let1≥α|I| with I=[a,b). Let
(57)I~=(∪j=-11(j+[3a-b2,3b-a2)))∩[0,1),
and define I-=[0,1)∖I~. Then
(58)|{Gcβ>α}|≤|I~|+|I-∩{Gcβ>α}|≤3|I|+|I-∩{Gcβ>α}|≤6α+|I-∩{Gcβ>α}|.
Notice that(59)|I-∩{Gcβ>α}|≤|I~∩{Gcdβ>α2}|+|I-∩{limsupJMJβ>α2}|,
with
(60)MJβ(x)=max2J≤m<2J+1-1MJmβ(x),MJmβ(x)=|∑n=2Jm‍(1-nm-2J+1)〈β,w~n〉w~n(x)|.
Forx∈[0,1), we have
(61)limsupJ,mMJmβ(x)=limsupJ,m|∑n=2Jm‍(1-nm-2J+1)〈β,w~n〉w~n(x)|=limsupJ,m|∑l1=-NN‍∑l2=-NN‍∑n=2Jm‍(1-nm-2J+1)∑l1=-NN‍∑l2=-NN‍∑n=2Jm‍×〈β,wn(·-l1)〉w(x-l2)|≤∑l1=-NN‍∑l2=-NN‍limsupJ,m|∑n=2Jm‍(1-nm-2J+1)∑n=2Jm‍×〈β,wn(·-l1)〉w(x-l2)|.
Hence, it suffices to estimate|Eαl1,l2| with
(62)Eαl1,l2={x∈I-:limsupJ,m|∑n=2Jm‍(1-nm-2J+1)∑n=2Jm‍×〈β,wn(·-l1)〉w(x-l2)|>α}.
Fixx∈ℝ∖I; then
(63)|∫-∞∞‍KJ,mσ(x-l1,y-l2)β(y)dy|≤C~∑l=-2N2N‍∫-∞∞‍β(y)dy|x-y+l2-l1+2K-Jl|,
which implies that whenever x∈Eαl1,l2, there is an increasing sequence Jk→∞ for which
(64)(1|x-a+l2-l1+2K-Jkl|+1|x-b+l2-l1+2K-Jkl|)>Cα,
for some fixed C>0 and for k=1,2,…. Since Jk→∞, therefore
(65)(1|x-a+l2-l1|+1|x-b+l2-l1|)>Cα.
Using thatI-=[0,1)∖I~ and the same technique as in the proof of Lemma 9, we complete the proof to conclude that |Eαl1,l2|≤1/α and consequently
(66)|I-∩{limsupJMJβ>α2}|≤C~α,
which completes the proof of Theorem 4.


## 7. Proof of Theorem5

Letf=∑k=1∞‍ckβk be a function of 𝔹q. Then
(67)σNf=∑n=0N‍(1-nN+1)〈f,w~n〉w~n=∑n=0N‍(1-nN+1)〈∑k=1∞‍ckβk,w~n〉w~n=∑n=0N‍(1-nN+1)∑k=1∞‍ck〈βk,w~n〉w~n=∑k=1∞‍ck(∑n=0N‍(1-nN+1)〈βk,w~n〉w~n)=∑k=1∞‍ck(σNβk),
due to the L1 convergence of the average sum defining f. Since
(68)∑n=0N‍(1-nN+1)〈f,w~n〉w~n=∑k=1∞‍ck∑n=0N‍(1-nN+1)〈βk,w~n〉w~n,limsupN|∑n=0N‍(1-nN+1)〈f,w~n〉w~n|≤∑k=1∞‍|ck|limsupN|∑n=0N‍(1-nN+1)〈βk,w~n〉w~n|Gcf≤∑k=1∞|ck|‍Gcβktherefore(69)|{Gcf>α}|≤|{∑k=1∞‍|ck|Gcβk>α}|≤Cqα∑k=1∞‍|ck|,byTheorem4,≤Cqα∥f∥𝔹q,by(12),=O(∥f∥𝔹q).This completes the proof of Theorem5.


## 8. Applications

Following corollary can be deduced from our theorems.Corollary 10.
Let{w~n} be a periodic Walsh-type wavelet packet basis. Then the Fourier expansion of any function f∈𝔹q,1<q<∞, in {w~n} is summable by arithmetic means pointwise a.e.Proof.
Let us write(SNf)(x)=∑n=0N‍〈f,w~n〉w~n(x) and
(70)(σNf)(x)=1N+1∑n=0N‍(Snf)(x)=∑n=0N‍(1-nN+1)〈f,w~n〉w~n(x).
Withf=∑k=1∞‍ckβk∈𝔹q, let gK=∑k=1K‍ckβk, and observe that ∥f-gK∥𝔹q→0. For each x∈[0,1), write
(71)f-σNf=(f-gK)+(gK-σNgK)+(σNgK-σNf).
Thus(72)|{x:limsupN→∞|f(x)-(σNf)(x)|>α}|≤|{x:limsupN→∞|f(x)-gK(x)|>α3}|+|{x:limsupN→∞|gK(x)-(σNgK)(x)|>α3}|+|{x:limsupn→∞|(σNgK)(x)-(σNf)(x)|>α3}|≤|{x:limsupN→∞|f(x)-gK(x)|>α3}|+|{x:limsupN→∞|gK(x)∑n=0N‍-∑n=0N‍(1-nN+1)∑n=0N‍×〈gK,w~n〉w~n(x)|>α3}|+|{x:limsupN→∞|∑n=0N‍(1-nN+1)∑n=0N‍×〈gK-f,w~n〉w~n(x)|>α3}|≤|{x:limsupN→∞|f(x)-gK(x)|>α3}|+|{x:limsupN→∞|gK(x)-∑n=0N‍〈gK,w~n〉w~n(x)|>α3}|+|{x:limsupN→∞|∑n=0N‍〈gK-f,w~n〉w~n(x)|>α3}|≤|{x:limsupN→∞|f(x)-gK(x)|>α3}|+|{x:limsupN→∞|gK(x)-(SNgK)(x)|>α3}|+|{x:limsupN→∞|(SNgK)(x)-(SNf)(x)|>α3}|≤3α∥f-gK∥𝔹q+0+3αCq∥f-gK∥𝔹q,byTheorem5.
From this it follows that(73)|{x:limsupn→∞|f(x)-(σNf)(x)|>α}|=0.
This completes the proof of the corollary.



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*Source: 379861-2013-08-20.xml*