Abstract
For the past few years, wavelet and multi-wavelet frames have attracted interest from researchers. In this paper, we address some of these problems in the setting of the Sobolev space, and characterize of multi-wavelet dual frames in these spaces by using a pair of equations.
1. Introduction
In view of the great design freedom and the efficient application in practice, such as image restoration, signal de-noising, the numerical solution of operator equations, wavelet, and multi-wavelet frames have been extensively investigated by many researchers (see [1–8] for details). In particular, many of these applications show that it is highly desirable to have a wavelet frame for Sobolev spaces [9–15]. Recently, Han and Shen in [10, 11] generalized the MEP of Ron and Shen [16] from to Sobolev spaces pairs for the construction of wavelet dual frames; the vanishing moment and smoothness requirements of two systems were separated completely from each other. Consequently, the construction of wavelet dual frames in Sobolev spaces is much easier. Li et al. in [13] further generalized the conclusions of Han and Shen; the matrix associated with the wavelet frame was extended to an isotropic expansive matrix, the Bessel properties of multi-wavelet sequences, and multi-wavelet dual frames were studied in Sobolev spaces. In the present paper, we will use a pair of equations to study the multi-wavelet dual frames which are not derived from refinable function vectors as in [13] in Sobolev spaces.
We first give some necessary notations and notions. We use , , and to denote the set of integers, the set of positive integers, and the set of nonegative integers, respectively. Let , we use to denote a -dimensional torus, and, given a set on , to denote the Lebesgue measure, and to denote the characteristic function on . Throughout this paper, we write as the Dirac sequences so that , for . For a function in , it’s Fourier transform is defined by , and is naturally extended to tempered distributions, where denotes the Euclidean inner product on .
Given , the Sobolev space consists of all distributions such that
where denotes the Euclidean norm on . Note that is a separable Hilbert space under the following definition of inner product:
Furthermore, for every ,
gives a linear continuous functional in . The Sobolev space is the dual space of , and vice versa.
For , we define
We use to denote the conjugate transpose of a order matrix . It is called an expansive matrix if the modulus of all its eigenvalues is greater than 1. In a vast body of the literature, is required to be an expansive integer matrix so that ( is the identity matrix). Moreover, for the sake of convenience, we always write
and write
for a tempered distribution .
Given , let and , and, we denote by and the following two multi-wavelet systems in and :
and
If there exist such that
then we say is a multi-wavelet frame ( MWF) in , where and are called upper and lower frame bounds, respectively; If the inequality on the right side of (9) holds, then we say is a multi-wavelet Bessel sequence (MWBS) in , where is called a Bessel bound. In addition, if the multi-wavelet systems and are two Bessel sequences in and , respectively, and the identity
holds, then we say
is a pair of multi-wavelet dual frames (MWDF) in .
If is a pair of MWDF in , then it follows from (10) that
and
where the series in (12) and (13) converge absolutely in and , respectively.
In the following section, we provide some necessary lemmas used latter. In Section 3, we provide a characterization of MWDF of the form (11) in by using a pair of equations.
2. Some Necessary Lemmas
In this section, we provide some necessary lemmas which are used for proving the main theorem below.
Definition 1. Let be a expansive matrix. Define a function byand set .
Lemma 1. Let , and . Then, for , the th Fourier coefficient of is . In particular,if is a Bessel sequence in .
Proof. Since , , we have , and thus
By the Plancherel theorem, we getIf is a Bessel sequence in , then , and thus, (15) follows directly from (17).
By observing the proof of [4, Proposition 2.1], we get
Lemma 2. Let , and . Then as in (7) is a MWBS in with Bessel bound if and only if
Lemma 3. Let , and . Suppose that as in (7) is an MWBS in with Bessel bound . Then
holds a.e. on .
Proof. Since as in (7) is a MWBS in with Bessel bound , by Lemma 2, we get.
Applying the Plancherel theorem and the Parseval identity, by a simple computationSimilarly, we haveFrom the above computation, we getBy the Fubini–Tonelli theorem, we haveandCollecting (23)–(25), we haveby the definition of for .
Suppose (19) does not hold, then there exists with such thatand thuson some with and . Define byin (26), then we getwhich contradicts with (20).
3. MWDF in Sobolev Spaces
This section is devoted to characterizing MWDF of the form (11) in . The following theorem provides us with a characterization by using a pair of equations.
Theorem 1. Given , let and , and . Define the multi-wavelet systems and as in (7) and (8), respectively. Suppose that and are two MWBSs in and , respectively. Then is a pair of MWDF in if and only if, for every .
Proof. By the definition, is a pair of MWDF in if and only if
for Applying Lemma 1 and the Plancherel theorem, we haveSo, (32) can be writtenClearly, (31) implies (34). Next, we will prove another direction.
Suppose (34) holds. Applying the Cauchy Schwarz inequality and Lemma 3, the seriesabsolutely converges a.e. on and belongs to therefore, almost all points on Lebesgue points. Let be such a point. For define f and g by
in (31), where Thenletting , we getFor , define and by in (34), where Thenletting , we getBy the arbitrariness of and , thus we obtain (31).
4. Conclusion
In this paper, we address multi-wavelet frames in Sobolev spaces setting, and provide a characterization of multi-wavelet dual frames in of the form (11) by using a pair of equations.
Data Availability
There is no data in this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11961072, 61861044, 61902339), and by the Doctoral Research Project of Yan’an University (Grant No. YDBK2017-21) and by the National Innovation and Entrepreneurship Training Program for Chinese College Students (Grant No. 201910719013).