# Convergence and Characterization of Wavelets Associated with Dilation Matrix in VariableLp Spaces

Authors: Devendra Kumar
Journal: Journal of Mathematics
 (2013)

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## Abstract

We study the convergence of wavelet expansions associated with dilation matrix in the variableLp(·)(ℝd) spaces using the approximate identity. Also, we obtain conditions for the wavelet characterizations of Lp(·)(ℝd) with respect to norm estimates. Moreover, the results of Izuki (2008) and Kumar (2009) have been extended.


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## Body

## 1. Introduction

Various authors such as Cruz-Uribe et al. [1, 2], Diening [3, 4], Nekvinda [5], Izuki [6], and Kopaliani [7] studied the conditions for the boundedness of the Hardy-Littlewood maximal function M on variable Lp spaces Lp(·)(Ω). In the case Ω=ℝd, Diening [4] has proved that the boundedness of M on Lp(·)(ℝd) is equivalent to that on Lp′(·)(ℝd), where p′(·) is the conjugate exponent function. Cruz-Uribe et al. [8] used the extrapolation technique to give a different proof of the result obtained by Izuki [6].The convergence of the wavelet series has been studied by Meyer [9], Walter [10], and Kelly et al. [11, 12]. Meyer was amongst the first to study convergence results for wavelet expansions. Wavelet theory is based on analyzing signals to their components by using a set of bases functions. An important characteristic of wavelet bases functions is that they relate to each other by simple scaling and translation. Meyer [9] showed that the regular wavelet expansions converge in Lp, 1≤p≤∞, for expansions of uniformly continuous functions; the expansions of continuous functions converge everywhere. His results are based on the assumption of the regularity for basic wavelets and their derivatives. Walter [10] studied the results for regular wavelet expansions of continuous functions. Kelly et al. [11, 12] extended and obtained results analogous to those obtained by Carleson [13] and Hunt [14] for the Fourier series. The results of Kelly et al. [11, 12] only assumed that the wavelets are bounded by radial decreasing L1-functions. The behavior of wavelet expansions outside the Lebesgue set is discussed by Reyes [15], for 1≤p<∞, whereas Kelly et al. [11] proved that the wavelet expansion of a function in Lp spaces converges pointwise everywhere on the Lebesgue set of a given function. Tao [16] has extended the results of Meyer [9] and Kelly et al. [11, 12] and showed that the wavelet expansion of any Lp-function converges pointwise almost everywhere under the wavelet projections, hard sampling, and soft sampling summation methods, for 1<p<∞.The pointwise convergence of wavelet expansions associated with a dilation matrix has been studied by Manchanda et al. [17]. For the applications point of view, sometimes it is reasonable to study the characterization and convergence of wavelets associated with a dilation matrix in variable Lp(·)(ℝd) spaces. The construction of wavelet bases in variable exponent spaces was first studied by Sharapudinov [18, 19] who gave a basis of Lp(·)([0,1]) using the Haar system. In this paper, we study the convergence and characterization of wavelet expansions associated with dilation matrix in variable Lp(·)(Ω) spaces where Ω⊆ℝd.


## 2. Definitions and Auxiliary Results

LetA denote a dilation matrix in ℤd; that is, A is a d×d matrix such that(i)
A(ℤd)⊂ℤd;(ii)
all eigen valuesλ of A satisfy |λ|>1.Property (i) implies thatA has integer entries, and hence, |detA| is an integer, and from property (ii) we know that |detA|>1. Considering ℤd as an additive group, we find that Aℤd is a normal subgroup of ℤd; so, we can form the cosets of Aℤd in ℤd. It is a well-known fact that the number of distinct cosets of Aℤd in ℤd is equal to |detA| [20].


### 2.1. VariableLp Spaces and Hardy-Littlewood Maximal Function

First, we define variableLp spaces on Ω, Ω⊆ℝd, and introduce some known results on the boundedness of the Hardy-Littlewood maximal operator. Throughout the paper, we consider a measurable function p:ℝd→[1,∞), and p′(·) means the conjugate exponent function of p(·); that is, p′(·) satisfies 1/p(x)+1/p′(x)=1(x∈Ω). χE denotes the characteristic function of a measurable set E⊂ℝd.Definition 1.
The variableLp space Lp(·)(Ω) consists of all measurable functions f defined on Ω such that ρp(f/λ)<∞ for some constant λ>0, where ρp(f)=∫Ω‍|f(x)|p(x)dx.L
p
(
·
)
(
Ω
) is a Banach space with the norm defined by
(1)∥f∥Lp(·)(Ω)=inf{λ>0:ρp(fλ)≤1}.
Now, we define two classes of exponent functions. We use the following notations:
(2)p-=essinf{p(x):x∈Ω},p+=esssup{p(x):x∈Ω}.Definition 2.
The set℘(Ω) consists of all p(·) satisfying p->1 and p+<∞.Given a functionf∈Lloc1(Ω), the Hardy-Littlewood maximal operator M is defined by
(3)Mf(x)=supx∈Q1|Q|∫Q‍|f(y)|dy(x∈Ω), where the supremum is taken over all cubes Q containing x.Definition 3.
B
(
Ω
) is the set of p(·)∈℘(Ω) satisfying the condition that M is bounded on Lp(·)(Ω).Cruz-Uribe et al. [1] and Nekvinda [5] proved the following sufficient conditions independently. Note that Nekvinda [5] gave a more general condition in place of (5).Proposition 4 (see [1, 5]).
Letp(·)∈℘(Ω). Suppose that
(4)|p(x)-p(y)|≤C-log(|x-y|)if|x-y|≤12,(5)|p(x)-p(y)|≤Clog(e+|x|)if|y|≥|x|,
where C>0 is a constant independent of x and y. Then, one has p(·)∈B(Ω). If p(·) satisfies (4) and (5), then so does p′(·). Diening [4] proved the following equivalence.Proposition 5 (see [4, Theorem 8.1]).
Letp(·)∈℘(ℝd). Then, the following four conditions are equivalent:(1)
p(·)∈B(ℝd);(2)
p′(·)∈B(ℝd);(3)
there exists a constantq∈(1,p-) such that p(·)/q∈B(ℝd);(4)
there exists a constantq∈(1,p-) such that (p(·)/q)′∈B(ℝd).Definition 6 (wavelets).
Let{φl:l=1,2,…,|detA|-1} be a set of functions belonging to L2(ℝd). Define
(6)φj,kl(x)=|detA|j/2φl(Ajx-k)=|detA|j/2φl(Ajx1-k1,…,Ajxd-kd),(x=(x1,…,xd)∈ℝd)
for each l=1,2,…,|detA|-1, j∈ℤ, and k=(k1,…,kd)∈ℤd. The sequence {φl:l=1,2,…,|detA|-1} is a wavelet set if {φj,kl:l=1,2,…,|detA|-1}, j∈ℤ, k∈ℤd, forms an orthonormal basis in L2(ℝd), and each φl is a wavelet.We generally need suitable smoothness or decay on wavelets in order to obtain wavelet characterizations of function spaces. Now, we define the classS(ℝd) of rapidly decreasing C∞-functions on ℝd as
(7)S(ℝd)={f:ℝd→ℝd:supx∈ℝd(xndmdxmf)(x)<∞,n∈N0,m∈N0d,N0=N∪{0}supx∈ℝd(xndmdxmf)}.In this paper, we use a wavelet set which consists of wavelets inS(ℝd). We can construct it by means of a multiresolution analysis (MRA) [9, 20, 21].Definition 7.
An MRA is a sequence{Vj}j∈ℤ of closed subspaces of L2(ℝd) associated with dilation matrix A such that the following conditions hold:(1)
Vj⊂Vj+1 for all j∈ℤ;(2)
∪j∈ℤVj is dense in L2(ℝd);(3)
∩j∈ℤVj={0};(4)
f∈Vj holds if and only if f(Ax)∈Vj+1 for all j∈ℤ;(5)
f∈V0 holds if and only if f(x-k)∈V0 for every k∈ℤd;(6)
there exists a functionϕ∈V0 such that the system {ϕ(x-k)}k∈ℤd is an orthonormal basis in V0. ϕ is called a scaling function of {Vj}j∈ℤ.Denoting the orthogonal complement ofVj in Vj+1 by Wj, it is easy to show that the integer translations {ϕ(x-k):k∈ℤd} of the mother wavelet ϕ form an orthonormal basis for W0. Then, we can construct a wavelet set {φl:l=1,2,…,|detA|-1} associated with {Vj}j∈ℤ and satisfying the assumption that each wavelet φl is in S(ℝd), and {φj,kl(x)=|detA|j/2φl(Ajx-k):k∈ℤd} forms an orthonormal basis in Wj for every j∈ℤ. Since ∪j∈ℤVj is dense in L2(ℝd), the collection {φj,kl} is an orthonormal basis of L2(ℝd).Given an MRA{Vj}j∈ℤ with scaling function ϕ, let Pj and Qj, respectively, be the orthogonal projections onto the spaces Vj and Wj with the kernels Pj(x,y) and Qj(x,y), defined as
(8)Pj(x,y)=∑l=1|detA|-1‍∑k∈ℤd‍ϕj,kl(x)ϕj,kl(y)¯,
where ϕj,kl(x)=|detA|j/2ϕl(Ajx-k) and
(9)Qj(x,y)=∑l=1|detA|-1‍∑k∈ℤd‍φj,kl(x)φj,kl(y)¯.
In view of Vj+1=Vj⊕Wj, Pj(x,y) can be written as
(10)Pj(x,y)=∑m<j‍Qm(x,y)=∑l=1|detA|-1‍∑m<j‍∑k∈ℤd‍φm,kl(x)φm,kl(y)¯.
The sequence of projections {Pjf(x)} is called the multiresolution expansion of f. The scaling expansion of any function f∈L2(ℝd) is defined as
(11)f(x)=∑l=1|detA|-1‍∑k∈ℤd‍al,j,k|detA|jϕl(Ajx-k)+∑l=1|detA|-1‍∑j≤m‍∑k∈ℤd‍bl,j,kφj,kl(x),
where
(12)al,j,k=∫ℝd‍f(x)|detA|jϕl(Ajx-k)dx,bl,j,k=∫ℝd‍f(x)φj,kl(x)dx.
Therefore the wavelet expansion associated with dilation matrix A of f is given by
(13)f(x)~∑l=1|detA|-1‍∑j∈ℤ‍∑k∈ℤd‍bl,j,kφj,kl(x),
where the wavelet coefficients bl,j,k=〈f,φj,kl〉 are defined as earlier.By considering convergence in the sense ofL2(ℝd), we may write
(14)f(x)=∑l=1|detA|-1‍∑j∈ℤ‍∑k∈ℤd‍bl,j,kφj,kl(x),f(x)=∑l=1|detA|-1‍∑k∈ℤd‍al,j,k|detA|jϕl(Ajx-k)+∑l=1|detA|-1‍∑j≤m‍∑k∈ℤd‍bl,m,kφj,kl(x)=fjl(x)+rml(x).The functionfml(x) is in the space Vm. It is the projection of f onto Vm and can be written as
(15)fml(x)=∫ℝd‍qml(x,t)f(t)dt,
where
(16)qml(x,t)=|detA|mq0l(Amx,Amt)=λdq0(λx,λt),(λ=Am)=∑l=1|detA|-1‍∑k∈ℤd‍ϕm,kl(x)ϕm,kl(t),q0l(x,t)=∑l=1|detA|-1‍∑k∈ℤd‍ϕl(x-k)ϕl(t-k).qml(x,t) will be called the reproducing kernel of Vm. Now, (15) takes the form
(17)fml(x)=∫ℝd‍λdq0l(λx,λt)f(t)dt.
We will assume that the scaling function ϕ does not generate an orthonormal basis of V0 or of W0. This leads to the fact that condition (6) of MRA can be replaced by the weaker condition that {ϕ(x-k):k∈ℤd} is a Riesz basis of V0. We also assume that Fourier transform of ϕ has a compact support. Since {ϕ(x-k):k∈ℤ} is a Riesz basis of V0, it has a biorthonormal basis {ϕk*(x)} such that for any function f∈V0, we have
(18)f(x)=∑k∈ℤd‍〈f,ϕk*〉ϕ0,k(x).
Similarly, for f∈Vm, we have
(19)fml(x)=∑l=1|detA|-1‍∑k∈ℤd‍〈f,ϕm,k*l〉ϕm,kl(x),
where {ϕm,k*l(x)}k∈ℤd is the biorthonormal basis of {ϕm,kl(x)}k∈ℤd and {l=1,2,…,|detA|-1}.Definition 8.
Ifϕ∈L1(ℝd) with ϕ^(0)=1 and one defines ϕn(x)=ndϕ(nx), where n→∞, and ϕ^ denotes the Fourier transform of ϕ, then the sequence of functions {ϕn}n=1∞ is an approximate identity if (1)
∫ℝd‍ϕn(x)dx=1 for all n,(2)
supn∫ℝd‍|ϕn(x)|dx<∞,(3)
limn→∞∫|x|>δ‍|ϕn(x)|dx=0 for every δ>0.Remark 9.
If0≤ϕ(x)∈S(ℝd), then ϕn(x)=ndϕ(nx) is an approximate identity.Lemma 10.
Ifq0l(x,t)∈L1(ℝd) with ∫ℝd‍q0l(x,t)dt=1, then qml(x,t) is an approximate identity if (1)
∫ℝd‍qml(x,t)dt=1 for all m,(2)
supm∫ℝd‍|qml(x,t)|dt<∞,(3)
limm→∞∫|x-t|>δ‍|qml(x,t)|dt=0 for every δ>0.Proof.
From [22, pages 64-65], we can easily obtain that q0l(x,t)=q0l(|x-t|) almost everywhere for {l=1,2,…,|detA|-1}. Now, following the lines of proof of [22, Lemma 2.2], the proof of Lemma 10 is immediate.Remark 11.
Let0≤q0l(x,t)∈S(ℝd). Then, qml(x,t)=λdq0(λ|x-t|) is an approximate identity.Lemma 12.
Iff∈L1(ℝd) and qml(x,t)∈S(ℝd), then qml(x,t)*f∈S(ℝd), where * denotes the convolution operator.Proof.
The proof follows on proceeding on the lines of proof of [22, Lemma 1.3].Lemma 13.
Suppose thatp+<∞. Then, S(ℝd) is dense in Lp(·)(ℝd).Proof.
SetCc∞(ℝd)={ϕ∈C∞(ℝd)} such that supp(ϕ) is compact.Obviously,S(ℝd) is a vector space and Cc∞(ℝd)⊆S(ℝd) and S(ℝd) is dense in Lp(ℝd). So, we have
(20)Cc∞(ℝd)⊆S(ℝd)⊆Lp(ℝd),1≤p<∞.
It has been proved by Kováčik and Rákosník [23, Theorem 2.11] that if p+<∞, then Cc∞(ℝd) is dense in Lp(·)(ℝd). By virtue of (20), it follows that S(ℝd) is dense in Lp(·)(ℝd).Definition 14.
Let1<p0<∞ be a constant, and let w be a weight on ℝd such that w-1/(p0-1) is locally integrable; w is said to be an Ap0 weight if w satisfies
(21)Ap0,w=supQ:cube1|Q|∫Q‍w(x)dx×(1|Q|∫Q‍w(x)-1/(p0-1)dx)p0-1<∞,
where |Q| means the Lebesgue measure of Q. Ap0 denotes the set of all Ap0 weights.Definition 15.
(1) A weightw is said to be an A∞ weight if for all 0<α<1, there exists a constant 0<β<1 such that for every cube Q and measurable set E⊂Q satisfying |E|≥α|Q|,
(22)β∫Q‍w(x)dx≤∫E‍w(x)dx.A∞ denotes the set of all A∞ weights.
(2) A family of weights{wt}t>0 is said to be a family of A∞ weights uniformly in t if each wt is an A∞ weight and the constant β is independent of t. It is well known that A∞=∪1<p0<∞Ap0 [24]. Lerner [24] gave the following characterization of families of A∞ weights.Lemma 16 (see [24, Lemma 2.1]).
Letp(·) be a nonnegative and measurable function on ℝd. The family of weights {tp(·)}t>0 is a family of A∞ weights uniformly in t if and only if p(·)≡p0 for some constant p0.Definition 17.
Let1≤p0<∞ be a constant, and let w be a weight on ℝd. The weighted Lp0 space Lwp0(ℝd) is the space of all measurable functions f with
(23)∥f∥Lwp0(ℝd)=(∫ℝd‍|f(x)|pw(x)dx)1/p0<∞.Lwp0(ℝd) is a Banach space with the norm ∥.∥Lwp0(ℝd).The following lemma also holds for variableLp spaces on an open set of ℝd.Lemma 18 (see [25, Corollary 1.11]).
Letp(·)∈℘(ℝd), and let F be a family of ordered pairs of non-negative and measurable functions (f,g). Suppose that
(24)(p(·)p1)′∈B(ℝd)forsomeconstantp1∈(1,p-),
and that there exists a constant p0∈(1,∞) such that for all w∈Ap0,w and all (f,g)∈F with f∈Lwpo(R),
(25)∥f∥Lwpo(Rn)≤Co∥g∥Lwpo(Rn),
where Co is a positive constant depending only on n, po, and Apo,w. Then, it follows that for all (f,g)∈F such that f∈Lp(·)(ℝd),
(26)∥f∥Lp(·)(ℝd)≤C1∥g∥Lp(·)(ℝd),
where C1 is a positive constant independent of (f,g).


### 2.2. Banach Function Spaces

Definition 19.
LetM(ℝd) be the set of all measurable functions on ℝd. (1)
A linear spaceX⊂M(ℝd) is said to be a Banach function space if there exists a functional ∥·∥:M(ℝd)→ℝ which has the norm property and satisfies the following conditions:
(a)
f∈X if and only if ∥f∥<∞;(b)
∥f∥=∥|f|∥ for all f∈M(ℝd);(c)
if0≤f1≤f2≤⋯ and limn→∞fn=f, then ∥f1∥≤∥f2∥≤⋯ and limn→∞∥fn∥=∥f∥;(d)
for allE⊂ℝd with |E|<∞, it follows that ∥χE∥<∞;(e)
for allE⊂ℝd with |E|<∞, there exists a constant CE>0 such that ∫E‍|f(x)|dx≤CE∥f∥ for all f∈X.(2)
LetX be a Banach function space with the norm ∥·∥. The norm ∥·∥ is said to be an absolutely continuous norm if limj→∞∥fχEj∥=0 for all f∈X and all sequences of measurable sets {Ej}j=1 such that limj→∞χEj=0 almost everywhere.The following two lemmas imply thatLp(·)(ℝd) is a Banach function space with the absolutely continuous norm ∥·∥Lp(·)(ℝd) (cf. [26]).Lemma 20 (see [27]).
Let(X,∥·∥) be a Banach function space. Then, X is separable if and only if the norm ∥·∥ is absolutely continuous on X.Lemma 21 (see [23]).
Suppose thatp+<∞. Then, Lp(·)(ℝd) is separable.


## 2.1. VariableLp Spaces and Hardy-Littlewood Maximal Function

First, we define variableLp spaces on Ω, Ω⊆ℝd, and introduce some known results on the boundedness of the Hardy-Littlewood maximal operator. Throughout the paper, we consider a measurable function p:ℝd→[1,∞), and p′(·) means the conjugate exponent function of p(·); that is, p′(·) satisfies 1/p(x)+1/p′(x)=1(x∈Ω). χE denotes the characteristic function of a measurable set E⊂ℝd.Definition 1.
The variableLp space Lp(·)(Ω) consists of all measurable functions f defined on Ω such that ρp(f/λ)<∞ for some constant λ>0, where ρp(f)=∫Ω‍|f(x)|p(x)dx.L
p
(
·
)
(
Ω
) is a Banach space with the norm defined by
(1)∥f∥Lp(·)(Ω)=inf{λ>0:ρp(fλ)≤1}.
Now, we define two classes of exponent functions. We use the following notations:
(2)p-=essinf{p(x):x∈Ω},p+=esssup{p(x):x∈Ω}.Definition 2.
The set℘(Ω) consists of all p(·) satisfying p->1 and p+<∞.Given a functionf∈Lloc1(Ω), the Hardy-Littlewood maximal operator M is defined by
(3)Mf(x)=supx∈Q1|Q|∫Q‍|f(y)|dy(x∈Ω), where the supremum is taken over all cubes Q containing x.Definition 3.
B
(
Ω
) is the set of p(·)∈℘(Ω) satisfying the condition that M is bounded on Lp(·)(Ω).Cruz-Uribe et al. [1] and Nekvinda [5] proved the following sufficient conditions independently. Note that Nekvinda [5] gave a more general condition in place of (5).Proposition 4 (see [1, 5]).
Letp(·)∈℘(Ω). Suppose that
(4)|p(x)-p(y)|≤C-log(|x-y|)if|x-y|≤12,(5)|p(x)-p(y)|≤Clog(e+|x|)if|y|≥|x|,
where C>0 is a constant independent of x and y. Then, one has p(·)∈B(Ω). If p(·) satisfies (4) and (5), then so does p′(·). Diening [4] proved the following equivalence.Proposition 5 (see [4, Theorem 8.1]).
Letp(·)∈℘(ℝd). Then, the following four conditions are equivalent:(1)
p(·)∈B(ℝd);(2)
p′(·)∈B(ℝd);(3)
there exists a constantq∈(1,p-) such that p(·)/q∈B(ℝd);(4)
there exists a constantq∈(1,p-) such that (p(·)/q)′∈B(ℝd).Definition 6 (wavelets).
Let{φl:l=1,2,…,|detA|-1} be a set of functions belonging to L2(ℝd). Define
(6)φj,kl(x)=|detA|j/2φl(Ajx-k)=|detA|j/2φl(Ajx1-k1,…,Ajxd-kd),(x=(x1,…,xd)∈ℝd)
for each l=1,2,…,|detA|-1, j∈ℤ, and k=(k1,…,kd)∈ℤd. The sequence {φl:l=1,2,…,|detA|-1} is a wavelet set if {φj,kl:l=1,2,…,|detA|-1}, j∈ℤ, k∈ℤd, forms an orthonormal basis in L2(ℝd), and each φl is a wavelet.We generally need suitable smoothness or decay on wavelets in order to obtain wavelet characterizations of function spaces. Now, we define the classS(ℝd) of rapidly decreasing C∞-functions on ℝd as
(7)S(ℝd)={f:ℝd→ℝd:supx∈ℝd(xndmdxmf)(x)<∞,n∈N0,m∈N0d,N0=N∪{0}supx∈ℝd(xndmdxmf)}.In this paper, we use a wavelet set which consists of wavelets inS(ℝd). We can construct it by means of a multiresolution analysis (MRA) [9, 20, 21].Definition 7.
An MRA is a sequence{Vj}j∈ℤ of closed subspaces of L2(ℝd) associated with dilation matrix A such that the following conditions hold:(1)
Vj⊂Vj+1 for all j∈ℤ;(2)
∪j∈ℤVj is dense in L2(ℝd);(3)
∩j∈ℤVj={0};(4)
f∈Vj holds if and only if f(Ax)∈Vj+1 for all j∈ℤ;(5)
f∈V0 holds if and only if f(x-k)∈V0 for every k∈ℤd;(6)
there exists a functionϕ∈V0 such that the system {ϕ(x-k)}k∈ℤd is an orthonormal basis in V0. ϕ is called a scaling function of {Vj}j∈ℤ.Denoting the orthogonal complement ofVj in Vj+1 by Wj, it is easy to show that the integer translations {ϕ(x-k):k∈ℤd} of the mother wavelet ϕ form an orthonormal basis for W0. Then, we can construct a wavelet set {φl:l=1,2,…,|detA|-1} associated with {Vj}j∈ℤ and satisfying the assumption that each wavelet φl is in S(ℝd), and {φj,kl(x)=|detA|j/2φl(Ajx-k):k∈ℤd} forms an orthonormal basis in Wj for every j∈ℤ. Since ∪j∈ℤVj is dense in L2(ℝd), the collection {φj,kl} is an orthonormal basis of L2(ℝd).Given an MRA{Vj}j∈ℤ with scaling function ϕ, let Pj and Qj, respectively, be the orthogonal projections onto the spaces Vj and Wj with the kernels Pj(x,y) and Qj(x,y), defined as
(8)Pj(x,y)=∑l=1|detA|-1‍∑k∈ℤd‍ϕj,kl(x)ϕj,kl(y)¯,
where ϕj,kl(x)=|detA|j/2ϕl(Ajx-k) and
(9)Qj(x,y)=∑l=1|detA|-1‍∑k∈ℤd‍φj,kl(x)φj,kl(y)¯.
In view of Vj+1=Vj⊕Wj, Pj(x,y) can be written as
(10)Pj(x,y)=∑m<j‍Qm(x,y)=∑l=1|detA|-1‍∑m<j‍∑k∈ℤd‍φm,kl(x)φm,kl(y)¯.
The sequence of projections {Pjf(x)} is called the multiresolution expansion of f. The scaling expansion of any function f∈L2(ℝd) is defined as
(11)f(x)=∑l=1|detA|-1‍∑k∈ℤd‍al,j,k|detA|jϕl(Ajx-k)+∑l=1|detA|-1‍∑j≤m‍∑k∈ℤd‍bl,j,kφj,kl(x),
where
(12)al,j,k=∫ℝd‍f(x)|detA|jϕl(Ajx-k)dx,bl,j,k=∫ℝd‍f(x)φj,kl(x)dx.
Therefore the wavelet expansion associated with dilation matrix A of f is given by
(13)f(x)~∑l=1|detA|-1‍∑j∈ℤ‍∑k∈ℤd‍bl,j,kφj,kl(x),
where the wavelet coefficients bl,j,k=〈f,φj,kl〉 are defined as earlier.By considering convergence in the sense ofL2(ℝd), we may write
(14)f(x)=∑l=1|detA|-1‍∑j∈ℤ‍∑k∈ℤd‍bl,j,kφj,kl(x),f(x)=∑l=1|detA|-1‍∑k∈ℤd‍al,j,k|detA|jϕl(Ajx-k)+∑l=1|detA|-1‍∑j≤m‍∑k∈ℤd‍bl,m,kφj,kl(x)=fjl(x)+rml(x).The functionfml(x) is in the space Vm. It is the projection of f onto Vm and can be written as
(15)fml(x)=∫ℝd‍qml(x,t)f(t)dt,
where
(16)qml(x,t)=|detA|mq0l(Amx,Amt)=λdq0(λx,λt),(λ=Am)=∑l=1|detA|-1‍∑k∈ℤd‍ϕm,kl(x)ϕm,kl(t),q0l(x,t)=∑l=1|detA|-1‍∑k∈ℤd‍ϕl(x-k)ϕl(t-k).qml(x,t) will be called the reproducing kernel of Vm. Now, (15) takes the form
(17)fml(x)=∫ℝd‍λdq0l(λx,λt)f(t)dt.
We will assume that the scaling function ϕ does not generate an orthonormal basis of V0 or of W0. This leads to the fact that condition (6) of MRA can be replaced by the weaker condition that {ϕ(x-k):k∈ℤd} is a Riesz basis of V0. We also assume that Fourier transform of ϕ has a compact support. Since {ϕ(x-k):k∈ℤ} is a Riesz basis of V0, it has a biorthonormal basis {ϕk*(x)} such that for any function f∈V0, we have
(18)f(x)=∑k∈ℤd‍〈f,ϕk*〉ϕ0,k(x).
Similarly, for f∈Vm, we have
(19)fml(x)=∑l=1|detA|-1‍∑k∈ℤd‍〈f,ϕm,k*l〉ϕm,kl(x),
where {ϕm,k*l(x)}k∈ℤd is the biorthonormal basis of {ϕm,kl(x)}k∈ℤd and {l=1,2,…,|detA|-1}.Definition 8.
Ifϕ∈L1(ℝd) with ϕ^(0)=1 and one defines ϕn(x)=ndϕ(nx), where n→∞, and ϕ^ denotes the Fourier transform of ϕ, then the sequence of functions {ϕn}n=1∞ is an approximate identity if (1)
∫ℝd‍ϕn(x)dx=1 for all n,(2)
supn∫ℝd‍|ϕn(x)|dx<∞,(3)
limn→∞∫|x|>δ‍|ϕn(x)|dx=0 for every δ>0.Remark 9.
If0≤ϕ(x)∈S(ℝd), then ϕn(x)=ndϕ(nx) is an approximate identity.Lemma 10.
Ifq0l(x,t)∈L1(ℝd) with ∫ℝd‍q0l(x,t)dt=1, then qml(x,t) is an approximate identity if (1)
∫ℝd‍qml(x,t)dt=1 for all m,(2)
supm∫ℝd‍|qml(x,t)|dt<∞,(3)
limm→∞∫|x-t|>δ‍|qml(x,t)|dt=0 for every δ>0.Proof.
From [22, pages 64-65], we can easily obtain that q0l(x,t)=q0l(|x-t|) almost everywhere for {l=1,2,…,|detA|-1}. Now, following the lines of proof of [22, Lemma 2.2], the proof of Lemma 10 is immediate.Remark 11.
Let0≤q0l(x,t)∈S(ℝd). Then, qml(x,t)=λdq0(λ|x-t|) is an approximate identity.Lemma 12.
Iff∈L1(ℝd) and qml(x,t)∈S(ℝd), then qml(x,t)*f∈S(ℝd), where * denotes the convolution operator.Proof.
The proof follows on proceeding on the lines of proof of [22, Lemma 1.3].Lemma 13.
Suppose thatp+<∞. Then, S(ℝd) is dense in Lp(·)(ℝd).Proof.
SetCc∞(ℝd)={ϕ∈C∞(ℝd)} such that supp(ϕ) is compact.Obviously,S(ℝd) is a vector space and Cc∞(ℝd)⊆S(ℝd) and S(ℝd) is dense in Lp(ℝd). So, we have
(20)Cc∞(ℝd)⊆S(ℝd)⊆Lp(ℝd),1≤p<∞.
It has been proved by Kováčik and Rákosník [23, Theorem 2.11] that if p+<∞, then Cc∞(ℝd) is dense in Lp(·)(ℝd). By virtue of (20), it follows that S(ℝd) is dense in Lp(·)(ℝd).Definition 14.
Let1<p0<∞ be a constant, and let w be a weight on ℝd such that w-1/(p0-1) is locally integrable; w is said to be an Ap0 weight if w satisfies
(21)Ap0,w=supQ:cube1|Q|∫Q‍w(x)dx×(1|Q|∫Q‍w(x)-1/(p0-1)dx)p0-1<∞,
where |Q| means the Lebesgue measure of Q. Ap0 denotes the set of all Ap0 weights.Definition 15.
(1) A weightw is said to be an A∞ weight if for all 0<α<1, there exists a constant 0<β<1 such that for every cube Q and measurable set E⊂Q satisfying |E|≥α|Q|,
(22)β∫Q‍w(x)dx≤∫E‍w(x)dx.A∞ denotes the set of all A∞ weights.
(2) A family of weights{wt}t>0 is said to be a family of A∞ weights uniformly in t if each wt is an A∞ weight and the constant β is independent of t. It is well known that A∞=∪1<p0<∞Ap0 [24]. Lerner [24] gave the following characterization of families of A∞ weights.Lemma 16 (see [24, Lemma 2.1]).
Letp(·) be a nonnegative and measurable function on ℝd. The family of weights {tp(·)}t>0 is a family of A∞ weights uniformly in t if and only if p(·)≡p0 for some constant p0.Definition 17.
Let1≤p0<∞ be a constant, and let w be a weight on ℝd. The weighted Lp0 space Lwp0(ℝd) is the space of all measurable functions f with
(23)∥f∥Lwp0(ℝd)=(∫ℝd‍|f(x)|pw(x)dx)1/p0<∞.Lwp0(ℝd) is a Banach space with the norm ∥.∥Lwp0(ℝd).The following lemma also holds for variableLp spaces on an open set of ℝd.Lemma 18 (see [25, Corollary 1.11]).
Letp(·)∈℘(ℝd), and let F be a family of ordered pairs of non-negative and measurable functions (f,g). Suppose that
(24)(p(·)p1)′∈B(ℝd)forsomeconstantp1∈(1,p-),
and that there exists a constant p0∈(1,∞) such that for all w∈Ap0,w and all (f,g)∈F with f∈Lwpo(R),
(25)∥f∥Lwpo(Rn)≤Co∥g∥Lwpo(Rn),
where Co is a positive constant depending only on n, po, and Apo,w. Then, it follows that for all (f,g)∈F such that f∈Lp(·)(ℝd),
(26)∥f∥Lp(·)(ℝd)≤C1∥g∥Lp(·)(ℝd),
where C1 is a positive constant independent of (f,g).


## 2.2. Banach Function Spaces

Definition 19.
LetM(ℝd) be the set of all measurable functions on ℝd. (1)
A linear spaceX⊂M(ℝd) is said to be a Banach function space if there exists a functional ∥·∥:M(ℝd)→ℝ which has the norm property and satisfies the following conditions:
(a)
f∈X if and only if ∥f∥<∞;(b)
∥f∥=∥|f|∥ for all f∈M(ℝd);(c)
if0≤f1≤f2≤⋯ and limn→∞fn=f, then ∥f1∥≤∥f2∥≤⋯ and limn→∞∥fn∥=∥f∥;(d)
for allE⊂ℝd with |E|<∞, it follows that ∥χE∥<∞;(e)
for allE⊂ℝd with |E|<∞, there exists a constant CE>0 such that ∫E‍|f(x)|dx≤CE∥f∥ for all f∈X.(2)
LetX be a Banach function space with the norm ∥·∥. The norm ∥·∥ is said to be an absolutely continuous norm if limj→∞∥fχEj∥=0 for all f∈X and all sequences of measurable sets {Ej}j=1 such that limj→∞χEj=0 almost everywhere.The following two lemmas imply thatLp(·)(ℝd) is a Banach function space with the absolutely continuous norm ∥·∥Lp(·)(ℝd) (cf. [26]).Lemma 20 (see [27]).
Let(X,∥·∥) be a Banach function space. Then, X is separable if and only if the norm ∥·∥ is absolutely continuous on X.Lemma 21 (see [23]).
Suppose thatp+<∞. Then, Lp(·)(ℝd) is separable.


## 3. Main Results

First, we prove the convergence of wavelet associated with dilation matrix inLp(·)(ℝd) in reference of MRA.Theorem 22.
Letp(·)∈B(ℝd) and ϕ, ϕ*∈S(ℝd). Define
(27)Pj*f=∑k∈ℤd‍〈f,ϕj,k〉ϕj,k*.
If {qj(x,t)}j=1∞ is an approximate identity, then for every f∈Lp(·)(ℝd) and j∈ℤ,
(28)limj→∞∥Pj*(f)-f∥Lp(·)(ℝd)=0.Remark 23.
(1) Ifϕ is a scaling function of an MRA and ϕ=ϕ*, then we see that Pj*(f)=Pj(f), where Pj is the orthogonal projection for each j∈ℤ.
(2) In view of generalized Holder inequality [11, Theorem 2.1], we have
(29)∫ℝd‍|f(x)g(x)|dx≤rp∥f∥Lp(·)(ℝd)∥g∥Lp′(·)(ℝd)
for all f∈Lp(·)(ℝd) and g∈Lp′(·)(ℝd), where rp=1+1/p--1/p+. Hence, the previous L2-inner product 〈f,ϕj,k〉 makes sense.Proof of Theorem22.
The nonorthogonal projection off∈L2(ℝd) on the space Vj is given by
(30)Pj*(f)=∑k∈ℤd‍〈f,ϕj,k〉ϕj,k*=∑k∈ℤd‍∫ℝd‍f(t)ϕj,k(x)ϕj,k*(t)dt=∫ℝd‍f(t)∑k∈ℤd‍ϕj,k(x)ϕj,k*(t)dt=∫ℝd‍f(t)qj(x,t)dt=∫ℝd‍f(t)qj(x-t)dt=(f*qj)(x)→f(x)(byLemma12).
Thus, if f∈Lp(ℝd) and qj(x,t)∈S(ℝd), then qj*f∈S(ℝd). Now, applying Lemma 18, we obtain that if f∈Lp(·)(ℝd) and qj(x,t)∈S(ℝd), then qj*f∈S(ℝd). By virtue of Lemma 13 we get qj*f∈Lp(·)(ℝd). Hence, for f∈Lp(·)(ℝd), qj*f→f∈Lp(·)(ℝd) for j→∞. Hence, the proof is completed.Corollary 24.
Letp(·)∈B(ℝd) and ϕ, ϕ*∈S(ℝd). If {qj(x,t)}j=1∞ is an approximate identity, then for every f∈Lp(·)(ℝd), the wavelet series
(31)fj(x)=∑k∈ℤd‍〈f,ϕj,k〉ϕj,k*(x)
converges to f(x) as j→∞ at every point of continuity of f(x).To characterize the wavelet associated with dilation matrixA in Lp(·)(ℝd) with respect to the norm ∥·∥Lℝdp(·), we define a cube
(32)Ij,k=Πi=1d(|detA|-jki,|detA-j|(ki+1))
and denote χj,k=|detA|j/2χIj,k for j∈ℤ and k∈ℤd. Given a wavelet set {φl:l=1,2,…,|detA|-1}, we consider the following two square functions in order to obtain wavelet characterizations:
(33)V*f=(∑l=1|detA|-1‍∑j=-∞∞‍∑k∈ℤd‍|〈f,φj,kl〉φj,kl|2)1/2,W*f=(∑l=1|detA|-1‍∑j=-∞∞‍∑k∈ℤd‍|〈f,φj,kl〉χj,kl|2)1/2.Lemma 25.
Letp0∈(1,∞) be a constant, w∈Ap0, and {φl} a wavelet set such that each wavelet φl∈S(ℝd). Then, there exist constants c,C,c′,C′>0 depending only on d, p0, {φl}, and Ap0,w such that for all f∈Lwp0(ℝd),
(34)c∥f∥Lwp0(ℝd)≤∥V*f∥Lwp0(ℝd)≤C∥f∥Lwp0(ℝd),c′∥f∥Lwp0(ℝd)≤∥W*f∥Lwp0(ℝd)≤C′∥f∥Lwp0(ℝd).Proof.
In view of [28, 29], the estimates (34) hold for φl∈ℝ0(ℝd) [3]. The class ℝ0(ℝd) is contained in S(ℝd); therefore, estimates (34) also hold for φl∈S(ℝd).Now, we prove the following.Theorem 26.
Letp(·)∈B(ℝd), and let {φl:l=1,2,…,|detA|-1} be a wavelet set such that wavelet φl∈S(ℝd). Then, there exist constants c,C,c′,C′>0 such that for all f∈Lp(·)(ℝd),
(35)c∥f∥Lp(·)(ℝd)≤∥V*f∥Lp(·)(ℝd)≤C∥f∥Lp(·)(ℝd),c′∥f∥Lp(·)(ℝd)≤∥W*f∥Lp(·)(ℝd)≤C′∥f∥Lp(·)(ℝd).Proof.
We see thatp(·)∈B(ℝd) is equivalent to condition (24) in view of Proposition 5. Thus, Lemma 25 together with Lemma 18 implies that the estimates (35) hold for all f∈S(ℝd) and φl∈S(ℝd). By using Lemma 13, these inequalities also hold for f∈Lp(·)(ℝd).Applying the wavelet characterization (Theorem26) and [27, Proposition 3.6, Ch. 1], we can construct unconditional basis of Lp(·)(ℝd) in terms of wavelets.Theorem 27.
Letp(·)∈B(ℝd), and let {φl:l=1,2,…,|detA|-1} be a wavelet set such that each wavelet φl is in S(ℝd). Then, the wavelet basis {φj,kl:l=1,2,…,|detA|-1,j∈ℤ,k∈ℤd} forms an unconditional basis on Lp(·)(ℝd).Proof.
Let us denoteΛ={1,2,…,|detA|-1}×ℤ×ℤd and
(36)SDf=∑(l,j,k)∈D‍〈f,φj,kl〉φj,klforD⊂Λ.
In order to complete the proof, we have to check the following two conditions:(1)
there exists a constantC>0 such that ∥SDf∥Lp(·)(ℝd)≤C∥f∥Lp(·)(ℝd) for all D⊂Λ and all f∈Lp(·)(ℝd);(2)
span{ϕj,kl:(l,j,k)∈Λ} is dense in Lp(·)(ℝd).
To check condition (1), using Theorem26 and the orthonormality, we obtain for all f∈Lp(·)(ℝd),
(37)∥SDf∥Lp(·)(ℝd)≤C0∥W*(SDf)∥Lp(·)(ℝd)≤C0∥W*f∥Lp(·)(ℝd)≤C0C1∥f∥Lp(·)(ℝd),
where C0,C1>0 are constants independent of f. Using monotone convergence theorem as finite sets D exhaust Λ, we get that (37) holds for (l,j,k)∈Λ.
To check condition (2), it suffices to show thatlimD→Λ∥f-SDf∥Lp(·)(ℝd)=0. By Lemmas 20 and 21, Lp(·)(ℝd) is a Banach function space with absolutely continuous norm ∥·∥Lp(·)(ℝd). For any f∈Lp(·)(ℝd)∩L2(ℝd), we have f=∑(l,j,k)∈D‍〈f,φj,kl〉φj,kl, with convergence in L2(ℝd), and there exists a subsequence of partial sums ∑(l,j,ki)∈D‍〈f,φj,kil〉φj,kil for i∈N that converge almost everywhere on ℝd. Using Fatou’s lemma, we get
(38)∥f-∑i=1,(l,j,ki)∈DN〈f,φj,kil〉φj,kil∥Lp(·)(ℝd)≤limm→∞inf∥∑i=N+1,(l,j,ki)∈DNm‍〈f,φj,kil〉φj,kil∥Lp(·)(ℝd),m∈N,≤Cp∥(∑i=N+1,(l,j,ki)∈D∞‍|〈f,φj,kil〉φj,kil|2)1/2∥Lp(·)(ℝd).
The result [27, Proposition 3.6, Ch. 1] can be applied to Lp(·)(ℝd) since Lp(·)(ℝd) is a Banach function space with the absolutely continuous norm (cf. [26]). Using this result, the last term is zero, and by virtue of (37), ∥(∑(l,j,k)∈Λ‍|〈f,φj,kl〉φj,kl|2)1/2∥Lp(·)(ℝd)<∞; we can use the dominated convergence theorem to obtain
(39)f=∑i=1,(l,j,ki)∈D∞‍〈f,φj,kil〉φj,kil
with convergence in Lp(·)(ℝd). Therefore, limD→Λ∥f-SDf∥Lp(·)(ℝd)=0.
Hence, the proof is completed.



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*Source: 254694-2013-03-07.xml*