A Class of Weak Dual Wavelet Frames for Reducing Subspaces of Sobolev Spaces
In recent years, dual wavelet frames derived from a pair of refinable functions have been widely studied by many researchers. However, the requirement of the Bessel property of wavelet systems is always required, which is too technical and artificial. In present paper, we will relax this restriction and only require the integer translation of the wavelet functions (or refinable functions) to form Bessel sequences. For this purpose, we introduce the notion of weak dual wavelet frames. And for generality, we work under the setting of reducing subspaces of Sobolev spaces, we characterize a pair of weak dual wavelet frames, and by using this characterization, we obtain a mixed oblique extension principle for such weak dual wavelet frames.
Let be a separable Hilbert space. An at most countable sequence in is called a frame for if there exist two constants such that where and are called lower and upper frame bounds; it is called a Bessel sequence in if the right-hand side inequality in (1) holds, where is called a Bessel bound. Given a frame for , a sequence is called a dual of if it is a frame such that
It is well-known that is also a dual of if is a dual of . So, in this case, we also say is a pair of dual frames. The fundamentals of frames can be found in [1–3]. We denote by , , and the set of integers, the set of positive integers, and the set of nonnegative integers, respectively. Let and a expansive matrix (an integer matrix with all its eigenvalues being greater than in modulus). The dilation operator and the shift operator with are, respectively, defined by for , and naturally extended to tempered distribution. In what follows, we denote by the transpose of , and by a set of representatives of distinct cosets of with .
The Fourier transform of a function is defined by and naturally extended to tempered distribution, where denotes the Euclidean inner product in . Similarly, the inverse Fourier transform of a function is defined by and extended to tempered distribution as usual. For a real number , we denote by the Sobolev space consisting of all distributions such that where denotes its Euclidean norm for . It is easy to check that is a Hilbert space under the inner product
In particular, by the Plancherel theorem. For with nonzero measure, we write
Then, . Obviously, for each , defines a continuous linear function on . Then, forms a pair of dual spaces; so does by taking . Write
It is well known that is dense in and is dense in for every .
Definition 1. Given a real number and a expansive matrix , a nonzero closed linear subspace of is called a reducing subspace if and for each , and where .
In particular, when , (12) is trivial and Definition 1 reduces to one in , which is characterized in Fourier domain as follows.
Proposition 2 (, (Theorem 1)). For a expansive matrix , is a reducing subspace of if and only if for some with nonzero measure satisfying .
Proposition 3 (, (Theorem 2.1)). Let be a real number and a expansive matrix. Then, is a reducing subspace of if and only if for some with nonzero measure satisfying .
So, to be specific, we denote a reducing subspace of by instead of . Given a distribution , we write for and . Let and be reducing subspaces of and , respectively, and for , , and finite subsets of and , respectively, the wavelet systems and are defined as
We say is a pair of dual wavelet frames for if (1) is a wavelet frame for and is a wavelet frame for (2)The identity
holds for and .
We say is a pair of weak dual wavelet frames for if (1) are Bessel sequences in and , respectively(2)There exist dense subsets of and of such that
where the series converges in the following sense: both and converge unconditionally (equivalently, both and converge), and
It is obvious that the convergence of series in (16) is weaker than that in (15). And a pair of dual wavelet frames must be a pair of weak dual wavelet frames, whereas the converse is not true. Also, observe that in the above definition of weak dual wavelet frames, need not belong to , and need not belong to .
Due to the great design freedom and the potential applications in signal denoising, image restoration, numerical analysis, etc., the study of wavelet frames for and Sobolev spaces has been attracting many researchers and seen great achievements (see [5–16] for details). In particular, Bownik in  obtained the following important characterization for homogeneous dual wavelet frames:
Proposition 4. Let and be Bessel sequences in . Then, is a pair of dual wavelet frames for if and only if where is defined by Definition 6 in Section 2.
Li and Zhang in  generalized Proposition 4 to Sobolev space pairs for nonhomogeneous dual wavelet frames:
Proposition 5. Given , let and be Bessel sequences in and , respectively. Then, is a pair of dual frames in if and only if
An important method to construct (dual) wavelet frames from refinable functions is extension principles. Ron and Shen in [15, 16] prosed the unitary extension principle (UEP) and the mixed extension principle (MEP). Subsequently, Daubechies et al. in  developed them in the form of the oblique extension principle (OEP) and the mixed oblique extension principle (MOEP). From then on, the study of the extension principles has interested many researchers [4, 5, 7, 8, 11, 18–20].
Observe that all above works, the wavelet systems (or the refinable functions) are required to be Bessel sequences. In order to achieve the Bessel property, some conditions have to be imposed on the wavelet systems (or the refinable functions) that are too technical and artificial. It is natural to ask what are expected from general refinable functions without too many restrictions. For this purpose, Jia and Li in  introduced the nation of weak wavelet biframes (weak dual wavelet frames). Starting from a pair of general refinable functions without smoothness restrictions, they obtained a construction of weak dual wavelet frames for reducing subspace of .
Inspired by all these works, in present paper, we investigate a class of weak dual wavelet frames for reducing subspaces of Sobolev spaces. In Section 2, we first give some necessary lemmas, and then, we give a Fourier-domain characterization of weak dual wavelet frames in associated with . In Section 3, by using the above characterization, we derive a mixed oblique extension principle for such weak dual wavelet frames.
Before proceeding, we introduce some necessary notations. We denote by the -dimensional torus and, for a Lebesgue measurable set in , by its Lebesgue measure and its characteristic function, respectively, by the Dirac sequence such that and for , and denote by the mapping from to defined by
For functions and on , we define if it is well defined, and the spectrum by
It is obvious that if and only if and for some . So is independent of . For simplicity, we use to replace .
2. The Characterization of Weak Dual Wavelet Frames
This section is devoted to characterizing weak dual wavelet frames in . Fist, we give some necessary lemmas for later use.
Definition 6. Let be a expansive matrix. Define a function by and set .
By a standard argument, we have the following three lemmas:
Lemma 8. For and , we have
Lemma 9. Let and be two complex sequences, and . Then,
Lemma 10 (, (Lemma 3.1)). Given , a expansive matrix , and , we have the following: (i) is a Bessel sequence in if and only if . In this case, is a Bessel bound(ii)If is a Bessel sequence in , then is a Bessel sequence in with the Bessel bound for , andfor , where .
Lemma 11. Let , , , and . Then, for , the -th Fourier coefficient of is . In particular, if is a Bessel sequence in .
Proof. Since and , we have , and thus,
by the Plancherel theorem. So the -th Fourier coefficient of is .
If is a Bessel sequence in , then is a Bessel sequence in by Lemma 10 (ii). It follows that , and thus, (27) holds.☐☐
Lemma 12 (, (Lemma 3.5)). Let be a bounded set in . Then, there exist finite sets and such that
Lemma 13. Given , let and be Bessel sequences in and , respectively. Then, for , where is defined by (11).
Proof. By Lemma 11, we have Write Then, by the Cauchy-Schwarz inequality, we have By a similar procedure, we also have If , then and with are Bessel sequences in . Also, observe that and are Bessel sequences in , and thus, by Lemma 10 (i). It follows that by the Fubini-Tonelli theorem. Therefore, we get By using the Cauchy-Schwarz inequality, we have which belongs to by Lemma 10 (i) since is also a Bessel sequence in if , and thus, Since , then there exists a bounded set in such that . By Lemma 12, there exist finite sets and such that Therefore, we have Write Then, for and , we have Also, observe that and . It follows that Therefore, we can change the order of summation and integration in (40). where Lemma 9 is used in the last equality. And thus, the conclusion follows by collecting (36), (38), and (44).☐☐
The following theorem characterizes a pair of weak dual wavelet frames for .
Theorem 14. Let and be reducing subspaces of and , respectively, and , and , finite subsets of and . Suppose and are both Bessel sequences in and , respectively. Then, is a pair of weak dual wavelet frames for if and only if
Proof. Since is dense in for every , then is a pair of weak dual wavelet frames for if and only if for , or equivalently for due to . By the Bessel assumptions, we know that the series and are absolutely convergent for since is dense both in and . By Lemma 13, (48) can be written as Obviously, (45) and (46) imply (49). Next, we prove the converse implication to complete the proof. Suppose (49) holds. For , the function and thus, almost every point in is a Lebesgue point of all function with . Let be such a point. For and , take and such that in (49), where . Then, we have