#### Abstract

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

#### 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 on variable spaces . In the case , Diening [4] has proved that the boundedness of on is equivalent to that on , where 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 , , 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 -functions. The behavior of wavelet expansions outside the Lebesgue set is discussed by Reyes [15], for , whereas Kelly et al. [11] proved that the wavelet expansion of a function in 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 -function converges pointwise almost everywhere under the wavelet projections, hard sampling, and soft sampling summation methods, for .

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 spaces. The construction of wavelet bases in variable exponent spaces was first studied by Sharapudinov [18, 19] who gave a basis of using the Haar system. In this paper, we study the convergence and characterization of wavelet expansions associated with dilation matrix in variable spaces where .

#### 2. Definitions and Auxiliary Results

Let denote a dilation matrix in ; that is, is a matrix such that(i);(ii)all eigen values of satisfy .

Property (i) implies that has integer entries, and hence, is an integer, and from property (ii) we know that . Considering as an additive group, we find that is a normal subgroup of ; so, we can form the cosets of in . It is a well-known fact that the number of distinct cosets of in is equal to [20].

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

First, we define variable spaces on , , and introduce some known results on the boundedness of the Hardy-Littlewood maximal operator. Throughout the paper, we consider a measurable function , and means the conjugate exponent function of ; that is, satisfies . denotes the characteristic function of a measurable set .

*Definition 1. *The variable space consists of all measurable functions defined on such that for some constant , where .

is a Banach space with the norm defined by Now, we define two classes of exponent functions. We use the following notations:

*Definition 2. *The set consists of all satisfying and .

Given a function , the Hardy-Littlewood maximal operator is defined by , where the supremum is taken over all cubes containing .

*Definition 3. * is the set of satisfying the condition that is bounded on .

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]). *Let . Suppose that
**
where is a constant independent of and . Then, one has . If satisfies (4) and (5), then so does . Diening [4] proved the following equivalence.*

Proposition 5 (see [4, Theorem 8.1]). *Let . Then, the following four conditions are equivalent:*(1);(2);(3)there exists a constant such that ;(4)there exists a constant such that .

*Definition 6 (wavelets). *Let be a set of functions belonging to . Define
for each , , and . The sequence is a wavelet set if , , , forms an orthonormal basis in , and each 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 class of rapidly decreasing -functions on as

In this paper, we use a wavelet set which consists of wavelets in . We can construct it by means of a multiresolution analysis (MRA) [9, 20, 21].

*Definition 7. *An MRA is a sequence of closed subspaces of associated with dilation matrix such that the following conditions hold:(1) for all ;(2) is dense in ;(3);(4) holds if and only if for all ;(5) holds if and only if for every ;(6)there exists a function such that the system is an orthonormal basis in . is called a scaling function of .

Denoting the orthogonal complement of in by , it is easy to show that the integer translations of the mother wavelet form an orthonormal basis for . Then, we can construct a wavelet set associated with and satisfying the assumption that each wavelet is in , and forms an orthonormal basis in for every . Since is dense in , the collection is an orthonormal basis of .

Given an MRA with scaling function , let and , respectively, be the orthogonal projections onto the spaces and with the kernels and , defined as where and In view of , can be written as The sequence of projections is called the multiresolution expansion of . The scaling expansion of any function is defined as where Therefore the wavelet expansion associated with dilation matrix of is given by where the wavelet coefficients are defined as earlier.

By considering convergence in the sense of , we may write

The function is in the space . It is the projection of onto and can be written as where will be called the reproducing kernel of . Now, (15) takes the form We will assume that the scaling function does not generate an orthonormal basis of or of . This leads to the fact that condition of MRA can be replaced by the weaker condition that is a Riesz basis of . We also assume that Fourier transform of has a compact support. Since is a Riesz basis of , it has a biorthonormal basis such that for any function , we have Similarly, for , we have where is the biorthonormal basis of and .

*Definition 8. *If with and one defines , where , and denotes the Fourier transform of , then the sequence of functions is an approximate identity if (1) for all , (2), (3) for every .

*Remark 9. *If , then is an approximate identity.

Lemma 10. *If with , then is an approximate identity if *(1)* for all , *(2)*, *(3)* for every . *

*Proof. *From [22, pages 64-65], we can easily obtain that almost everywhere for . Now, following the lines of proof of [22, Lemma 2.2], the proof of Lemma 10 is immediate.

*Remark 11. *Let . Then, is an approximate identity.

Lemma 12. *If and , then , where denotes the convolution operator.*

*Proof. *The proof follows on proceeding on the lines of proof of [22, Lemma 1.3].

Lemma 13. *Suppose that . Then, is dense in .*

*Proof. *Set such that is compact.

Obviously, is a vector space and and is dense in . So, we have It has been proved by Kováčik and Rákosník [23, Theorem 2.11] that if , then is dense in . By virtue of (20), it follows that is dense in .

*Definition 14. *Let be a constant, and let be a weight on such that is locally integrable; is said to be an weight if satisfies
where means the Lebesgue measure of . denotes the set of all weights.

*Definition 15. *(1) A weight is said to be an weight if for all , there exists a constant such that for every cube and measurable set satisfying ,
denotes the set of all weights.

(2) A family of weights is said to be a family of weights uniformly in if each is an weight and the constant is independent of . It is well known that [24]. Lerner [24] gave the following characterization of families of weights.

Lemma 16 (see [24, Lemma 2.1]). *Let be a nonnegative and measurable function on . The family of weights is a family of weights uniformly in if and only if for some constant .*

*Definition 17. *Let be a constant, and let be a weight on . The weighted space is the space of all measurable functions with
is a Banach space with the norm .

The following lemma also holds for variable spaces on an open set of .

Lemma 18 (see [25, Corollary 1.11]). *Let , and let be a family of ordered pairs of non-negative and measurable functions . Suppose that
**
and that there exists a constant such that for all and all with ,
**
where is a positive constant depending only on , , and . Then, it follows that for all such that ,
**
where is a positive constant independent of . *

##### 2.2. Banach Function Spaces

*Definition 19. *Let be the set of all measurable functions on . (1)A linear space is said to be a Banach function space if there exists a functional which has the norm property and satisfies the following conditions:(a) if and only if ;(b) for all ;(c)if and , then and ;(d)for all with , it follows that ;(e)for all with , there exists a constant such that for all .(2)Let be a Banach function space with the norm . The norm is said to be an absolutely continuous norm if for all and all sequences of measurable sets such that almost everywhere.

The following two lemmas imply that is a Banach function space with the absolutely continuous norm (cf. [26]).

Lemma 20 (see [27]). *Let be a Banach function space. Then, is separable if and only if the norm is absolutely continuous on . *

Lemma 21 (see [23]). *Suppose that . Then, is separable.*

#### 3. Main Results

First, we prove the convergence of wavelet associated with dilation matrix in in reference of MRA.

Theorem 22. *Let and , . Define
**
If is an approximate identity, then for every and ,
*

*Remark 23. *(1) If is a scaling function of an MRA and , then we see that , where is the orthogonal projection for each .

(2) In view of generalized Holder inequality [11, Theorem 2.1], we have
for all and , where . Hence, the previous -inner product makes sense.

*Proof of Theorem 22. *The nonorthogonal projection of on the space is given by
Thus, if and , then . Now, applying Lemma 18, we obtain that if and , then . By virtue of Lemma 13 we get . Hence, for , for . Hence, the proof is completed.

Corollary 24. *Let and , . If is an approximate identity, then for every , the wavelet series
**
converges to as at every point of continuity of .*

To characterize the wavelet associated with dilation matrix in with respect to the norm , we define a cube and denote for and . Given a wavelet set , we consider the following two square functions in order to obtain wavelet characterizations:

Lemma 25. *Let be a constant, , and a wavelet set such that each wavelet . Then, there exist constants depending only on , , , and such that for all ,
*

*Proof. *In view of [28, 29], the estimates (34) hold for [3]. The class is contained in ; therefore, estimates (34) also hold for .

Now, we prove the following.

Theorem 26. *Let , and let be a wavelet set such that wavelet . Then, there exist constants such that for all ,
*

*Proof. *We see that 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 and . By using Lemma 13, these inequalities also hold for .

Applying the wavelet characterization (Theorem 26) and [27, Proposition 3.6, Ch. 1], we can construct unconditional basis of in terms of wavelets.

Theorem 27. *Let , and let be a wavelet set such that each wavelet is in . Then, the wavelet basis forms an unconditional basis on .*

*Proof. *Let us denote and
In order to complete the proof, we have to check the following two conditions:(1)there exists a constant such that for all and all ;(2)span is dense in .

To check condition (1), using Theorem 26 and the orthonormality, we obtain for all ,
where are constants independent of . Using monotone convergence theorem as finite sets exhaust , we get that (37) holds for .

To check condition (2), it suffices to show that . By Lemmas 20 and 21, is a Banach function space with absolutely continuous norm . For any , we have , with convergence in , and there exists a subsequence of partial sums for that converge almost everywhere on . Using Fatou’s lemma, we get
The result [27, Proposition 3.6, Ch. 1] can be applied to since 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), ; we can use the dominated convergence theorem to obtain
with convergence in . Therefore, .

Hence, the proof is completed.

#### Acknowledgments

The author is extremely thankful to the learned reviewers for taking pains and giving fruitful comments to improve the paper. This work is supported by University Grants Commission, New Delhi, India, under the Grant of Major Research Project no. 41-792/2012(SR).