An application of nonstationary wavelets
Journal of Inequalities and Applicationsvolume 2014, Article number: 153 (2014)
Using nonstationary wavelets, we investigate the wavelet expansion in the standard Besov spaces. Especially, the nonstationary wavelets’ characterization for Besov spaces is given.
MSC:42C40, 35Q30, 41A15.
In this paper, we shall study the nonstationary wavelet  expansion in the standard Besov spaces. Nonstationary wavelet systems are generally obtained from a sequence of nonstationary refinable functions.
Definition 1.1 A sequence of functions in is said to consist of nonstationary refinable functions if, for all ,
where are 2π-periodic measurable functions, called refinement masks, or simply masks.
The classical Fourier transform is defined by for . The standard extension can be made to functions. Wavelet functions , and (quite often ), are generally obtained from nonstationary refinable functions by
where are 2π-periodic measurable functions called wavelet masks. The masks and satisfy . When , let , and , where . Define . Then the following theorem holds.
Theorem 1.1 (i) (Theorem 1.3, ). Let (mask for the pseudo-spline of type I with order ). Then is a compactly supported and wavelet frame in for arbitrary ;
(ii) (Theorem 1.4, ). When , then () is a compactly supported orthonormal basis in , and () has vanishing moments. More precisely, and with uniform constant .
In this paper, we use to denote the classical -Sobolev spaces with the smoothness parameter s. It is well known that Besov spaces contain a large number of fundamental spaces, such as Sobolev spaces, Hölder spaces, Lipschitz spaces etc. [3, 4]. They are frequently used in certain PDEs as the solution spaces. To extend the result of (i) in Theorem 1.1, we shall characterize Besov spaces by using nonstationary wavelets in this paper. It should be pointed out that Bittner and Urban  study the following standard Besov spaces: Let , , and let stand for the largest integer less than or equal to s,
Here, with and denotes the M th order smooth modulus of a function f, defined by as usual. The classical difference operator is defined by , as well as for a positive integer . The Besov (quasi-)norm is given by and the two integers yield equivalent norms (, Remark 3.2.2).
Based on Hermite multiwavelets, Bittner and Urban characterize by using sequence norms,
for , . However, due to the regularity restrictions of the Hermite splines, their characterization requires in the quadratic case and in the cubic one (e.g. [5, 7, 8]). In , we remove that restriction of s by using the B-spline wavelets with weak duals as introduced in , but the supports of the wavelets become larger as s increases. So, the main result of this paper is to characterize Besov spaces via nonstationary wavelets because of their arbitrary smoothness and uniform support.
Let N, Z, and R be the set of positive integers, the set of integers, and the set of real numbers, respectively, as well as . Throughout this paper, we use to abbreviate that A is bounded by a constant multiple of B, is defined as and means and . Write
for , with Lebesgue measurable set , , and . For a Lebesgue measurable function f, the support of f means the set , which is well defined up to a set of measure 0. Define throughout this paper.
Now, we state the Main Theorem of this paper.
Main Theorem Let , ; and are from (ii) of Theorem 1.1. When , then
for . Moreover, when , , with and , then
for , .
2 Proof of Main Theorem
This section is devoted to proving the Main Theorem. We begin with three lemmas for proving upper and lower bounds of the characterization.
Lemma 2.1 Let , , be arbitrary, then and with a uniform constant for .
Proof First, we will show with a uniform constant C for all j. By Lemma 2.1, Theorem 2.8 in , and Theorem 2.1 in , are all compactly supported and for a uniform constant . Therefore, for all j because of . Note that (Lemma 2.2 in ). This with
for leads to for all . holds similarly.
Second, let , by (2.1),
because of Corollary 3.3 in  and the orthonormality property of , . This with (2.1) leads to
as well as for with a uniform constant . Thus, the result holds. □
Lemma 2.2 Suppose is compactly supported with and . Then
Note that the constants are uniform because of .
Lemma 2.3 If , , define σ as a closed interval of R and let be the set of m-order polynomials, then
where and with .
Now, we are in the position to show the Main Theorem.
By Lemma 2.1, and . This, with Lemma 2.2, shows that
To prove the lower bound, one finds that ,
where . Let . This with shows that
where the equality comes from the vanishing moments of by Theorem 1.1 and the second inequality holds due to Lemma 2.3. Then by the same proof as of (3.3) in . Therefore,
Remark 2.1 In conclusion, we have a characterization of Besov spaces by
with , , , , , and .
Remark 2.2 Some questions are left to be considered. Note that we assume in our Main Theorem. Then a natural question is to study the case for . Another one is to discuss whether or not the wavelet frames of (i) in Theorem 1.1 can characterize Besov spaces. The last question is to relax the restriction .
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The author would like to thank the referees for their helpful comments. This work is supported by the National Natural Science Foundation of China, Tian Yuan Foundation (Grant No. 11226106) and the National Natural Science Foundation of China (Grant No. 11201094).
The author declares that he has no competing interests.