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.

1. Introduction

Let

be a separable Hilbert space. An at most countable sequence

is called a frame for

if there exist two constants 0 < C₁ ≤ C₂ < ∞ such that

(1)

where C₁ and C₂ are called lower and upper frame bounds; it is called a Bessel sequence in

if the right-hand side inequality in (1) holds, where C₂ is called a Bessel bound. Given a frame

for

, a sequence

is called a dual of

if it is a frame such that

(2)

It is well-known that

is also a dual of

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 ℤ, ℕ, and ℕ₀ the set of integers, the set of positive integers, and the set of nonnegative integers, respectively. Let d ∈ ℕ and M a d × d expansive matrix (an integer matrix with all its eigenvalues being greater than 1 in modulus). The dilation operator D and the shift operator T_k with k ∈ ℤ^d are, respectively, defined by

(3)

for f ∈ L²(ℝ^d), and naturally extended to tempered distribution. In what follows, we denote by M^∗ the transpose of M, and by

(4)

a set of representatives of distinct cosets of ℤ^d/M^∗ℤ^d with ε₀ = 0.

The Fourier transform of a function f ∈ L¹(ℝ^d)∩L²(ℝ^d) is defined by

(5)

and naturally extended to tempered distribution, where 〈·, ·〉 denotes the Euclidean inner product in ℝ^d. Similarly, the inverse Fourier transform of a function f ∈ L¹(ℝ^d)∩L²(ℝ^d) is defined by

(6)

and extended to tempered distribution as usual. For a real number s, we denote by H^s(ℝ^d) the Sobolev space consisting of all distributions f such that

(7)

where |ξ| denotes its Euclidean norm for ξ ∈ R^d. It is easy to check that H^s(ℝ^d) is a Hilbert space under the inner product

(8)

In particular, H⁰(ℝ^d) = L²(ℝ^d) by the Plancherel theorem. For Ω ⊂ ℝ^d with nonzero measure, we write

(9)

Then, FH^s(ℝ^d) = H^s(ℝ^d). Obviously, for each g ∈ FH^−s(Ω),

(10)

defines a continuous linear function on FH^s(Ω). Then, (FH^s(Ω), FH^−s(Ω)) forms a pair of dual spaces; so does (H^s(ℝ^d), H^−s(ℝ^d)) by taking Ω = ℝ^d. Write

(11)

It is well known that is dense in H^s(ℝ^d) and is dense in FH^s(Ω) for every s ∈ ℝ.

Definition 1. Given a real number s and a d × d expansive matrix M, a nonzero closed linear subspace X of H^s(ℝ^d) is called a reducing subspace if DX = X and T_kX = X for each k ∈ ℤ^d, and

(12)

where

In particular, when s = 0, (12) is trivial and Definition 1 reduces to one in [4], which is characterized in Fourier domain as follows.

Proposition 2 ([4], (Theorem 1)). For a d × d expansive matrix M, X is a reducing subspace of L²(ℝ^d) if and only if X = FL²(Ω) for some Ω ⊂ ℝ^d with nonzero measure satisfying Ω = M^∗Ω.

Proposition 3 ([4], (Theorem 2.1)). Let s be a real number and M a d × d expansive matrix. Then, X is a reducing subspace of H^s(ℝ^d) if and only if X = FH^s(Ω) for some Ω ⊂ ℝ^d with nonzero measure satisfying Ω = M^∗Ω.

So, to be specific, we denote a reducing subspace of H^s(ℝ^d) by FH^s(Ω) instead of X. Given a distribution f, we write

(13)

for j ∈ ℤ and k ∈ ℤ^d. Let FH^s(Ω) and FH^−s(Ω) be reducing subspaces of H^s(ℝ^d) and H^−s(ℝ^d), respectively, and for

, Ψ = {ψ_l : 1 ≤ l ≤ L}, and

finite subsets of FH^s(Ω) and FH^−s(Ω), respectively, the wavelet systems X^s(ψ₀, Ψ) and

are defined as

(14)

We say

is a pair of dual wavelet frames for (FH^s(Ω), FH^−s(Ω)) if

(1)
X^s(ψ₀, Ψ) is a wavelet frame for FH^s(Ω) and is a wavelet frame for FH^−s(Ω)
(2)
The identity

(15)

holds for f ∈ FH^s(Ω) and g ∈ FH^−s(Ω).

We say

is a pair of weak dual wavelet frames for (FH^s(Ω), FH^−s(Ω)) if

(1)
are Bessel sequences in H^s(ℝ^d) and H^−s(ℝ^d), respectively
(2)
There exist dense subsets V of FH^s(Ω) and of FH^−s(Ω) such that

(16)

where the series converges in the following sense: both

and

converge unconditionally (equivalently, both

and

converge), and

(17)

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, ψ_l, 0 ≤ l ≤ L need not belong to FH^s(Ω), and need not belong to FH^−s(Ω).

Due to the great design freedom and the potential applications in signal denoising, image restoration, numerical analysis, etc., the study of wavelet frames for L²(ℝ^d) and Sobolev spaces has been attracting many researchers and seen great achievements (see [5–16] for details). In particular, Bownik in [6] obtained the following important characterization for homogeneous dual wavelet frames:

Proposition 4. Let X(Ψ) and be Bessel sequences in L²(ℝ^d). Then, is a pair of dual wavelet frames for L²(ℝ^d) if and only if

(18)

where κ(k) is defined by Definition 6 in Section 2.

Li and Zhang in [17] generalized Proposition 4 to Sobolev space pairs (H^s(ℝ^d), H^−s(R^d)) for nonhomogeneous dual wavelet frames:

Proposition 5. Given s ∈ R, let X^s(ψ₀, Ψ) and be Bessel sequences in H^s(ℝ^d) and H^−s(ℝ^d), respectively. Then, is a pair of dual frames in (H^s(ℝ^d), H^−s(ℝ^d)) if and only if

(19)

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 [10] 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 [21] 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 FL²(Ω) of L²(ℝ^d).

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 (FH^s(Ω), FH^−s(Ω)) associated with (D∩FH^s(Ω), D∩FH^−s(Ω)). 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 T^d = [0, 1)^d the d-dimensional torus and, for a Lebesgue measurable set E in ℝ^d, by |E| its Lebesgue measure and χ_E its characteristic function, respectively, by δ the Dirac sequence such that δ_0,0 = 1 and δ_0,k = 0 for 0 ≠ k ∈ ℤ^d, and denote by τ the mapping from ℝ^d to

defined by

(20)

For functions f and g on ℤ^d, we define

(21)

if it is well defined, and the spectrum σ_t(f) by

(22)

It is obvious that ξ ∈ σ_t(f) if and only if ξ ∈ T^d and for some k ∈ Z^d. So σ_t(f) is independent of t. For simplicity, we use σ(f) to replace σ_t(f).

2. The Characterization of Weak Dual Wavelet Frames

This section is devoted to characterizing weak dual wavelet frames in (FH^s(Ω), FH^−s(Ω)). Fist, we give some necessary lemmas for later use.

Definition 6. Let M be a d × d expansive matrix. Define a function κ : ℤ^d⟶N₀ by

(23)

and set κ(0) = +∞.

By a standard argument, we have the following three lemmas:

Lemma 7.

Lemma 8. For ϕ ∈ H^s(ℝ^d) and , we have

(24)

Lemma 9. Let and be two complex sequences, and . Then,

(25)

Lemma 10 ([4], (Lemma 3.1)). Given s ∈ R, a d × d expansive matrix M, and ϕ ∈ H^s(R^d), we have the following:

(i)
{T_kϕ : k ∈ ℤ^d} is a Bessel sequence in H^s(ℝ^d) if and only if . In this case, is a Bessel bound
(ii)
If {T_kϕ : k ∈ Z^d} is a Bessel sequence in H^s(ℝ^d), then {ϕ_n,k : k ∈ ℤ^d} is a Bessel sequence in H^s(ℝ^d) with the Bessel bound for n ∈ Z, and

(26)

for f ∈ H^−s(ℝ^d), where

Lemma 11. Let s ∈ R, f ∈ H^s(ℝ^d), ψ ∈ H^−s(ℝ^d), and j ∈ Z. Then, for k ∈ Z^d, the k -th Fourier coefficient of is 〈f, ψ_j,k〉. In particular,

(27)

if {T_kψ : k ∈ ℤ^d} is a Bessel sequence in H^−s(ℝ^d).

Proof. Since f ∈ H^s(ℝ^d) and ψ ∈ H^−s(ℝ^d), we have , and thus,

(28)

by the Plancherel theorem. So the k-th Fourier coefficient of

is 〈f, ψ_j,k〉.

If {T_kψ : k ∈ ℤ^d} is a Bessel sequence in H^−s(ℝ^d), then {ψ_j,k : k ∈ ℤ^d} is a Bessel sequence in H^−s(ℝ^d) by Lemma 10 (ii). It follows that , and thus, (27) holds.☐☐

Lemma 12 ([4], (Lemma 3.5)). Let S be a bounded set in ℝ^d. Then, there exist finite sets F₁ ⊂ N₀ and F₂ ⊂ Z^d\{0} such that

(29)

Lemma 13. Given s ∈ ℝ, let {T_kψ_l : k ∈ ℤ^d, 0 ≤ l ≤ L} and be Bessel sequences in H^s(ℝ^d) and H^−s(ℝ^d), respectively. Then,

(30)

for

, where

is defined by (11).

Proof. By Lemma 11, we have

(31)

Write

(32)

Then, by the Cauchy-Schwarz inequality, we have

(33)

By a similar procedure, we also have

(34)

If g ∈ D, then {T_kg : k ∈ ℤ^d} and with are Bessel sequences in H^−s(ℝ^d). Also, observe that {T_kψ₀ : k ∈ ℤ^d} and {T_kψ_l : k ∈ ℤ^d, 1 ≤ l ≤ L} are Bessel sequences in H^s(ℝ^d), and thus, by Lemma 10 (i). It follows that

(35)

by the Fubini-Tonelli theorem. Therefore, we get

(36)

By using the Cauchy-Schwarz inequality, we have

(37)

which belongs to L^∞(R^d) by Lemma 10 (i) since {T_kf : k ∈ Z^d} is also a Bessel sequence in H^s(R^d) if f ∈ D, and thus,

(38)

Since f, g ∈ D, then there exists a bounded set S in R^d such that . By Lemma 12, there exist finite sets F₁ ⊂ N₀ and F₂ ⊂ Z^d\{0} such that

(39)

Therefore, we have

(40)

Write

(41)

Then, for j ∈ F₁ and k ∈ F₂, we have

(42)

Also, observe that and . It follows that

(43)

Therefore, we can change the order of summation and integration in (40).

(44)

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 (FH^s(Ω), FH^−s(Ω)).

Theorem 14. Let FH^s(Ω) and FH^−s(Ω) be reducing subspaces of H^s(R^d) and H^−s(R^d), respectively, and ψ₀ ∈ FH^s(Ω), and Ψ = {ψ_l : 1 ≤ l ≤ L}, finite subsets of FH^s(Ω) and FH^−s(Ω). Suppose {T_kψ_l : k ∈ Z^d, 0 ≤ l ≤ L} and are both Bessel sequences in H^s(R^d) and H^−s(R^d), respectively. Then, is a pair of weak dual wavelet frames for (FH^s(Ω), FH^−s(Ω)) if and only if

(45)

(46)

Proof. Since D∩FH^s(Ω) is dense in FH^s(Ω) for every s ∈ R, then is a pair of weak dual wavelet frames for (FH^s(Ω), FH^−s(Ω)) if and only if

(47)

for f ∈ D∩FH^s(Ω), g ∈ D∩FH^−s(Ω), or equivalently

(48)

for f, g ∈ D due to

. By the Bessel assumptions, we know that the series

and

are absolutely convergent for f, g ∈ D since D is dense both in H^s(R^d) and H^−s(R^d). By Lemma 13, (48) can be written as

(49)

Obviously, (45) and (46) imply (49). Next, we prove the converse implication to complete the proof. Suppose (49) holds. For 0 ≠ k ∈ Z^d, the function

(50)

and thus, almost every point in R^d is a Lebesgue point of all function

(51)

with 0 ≠ k ∈ Z^d. Let ξ₀ ∈ R^d be such a point. For 0 < ε < 1/2 and 0 ≠ k₀ ∈ Z^d, take f and g such that

in (49), where B(ξ₀, ε) = {ξ ∈ R^d : |ξ − ξ₀| < ε}. Then, we have

(52)

or equivalently,

(53)

since

if ξ and ξ + k₀ belong to Ω when ξ ∈ B(ξ₀, ε). Letting ε⟶0 in (53), we obtain

(54)

By the arbitrariness of ξ₀ ∈ R^d and 0 ≠ k₀ ∈ Z^d, then we have

(55)

And thus, (46) holds since the above function vanishes out of Ω. Now, we prove (45) holds.

Arbitrarily fix h ∈ L^∞(K), ∀ compact K ⊂ ℝ^d\{0}, then there exists a finite set {k₁, k₂, ⋯, k_m} ⊂ ℤ^d such that . Define for 1 ≤ i ≤ m. Then, we have , and thus, (45) holds for h if it holds for each h_i with 1 ≤ i ≤ m. Next, we prove (45) holds for each h_i with 1 ≤ i ≤ m. Take f and g in (49) such that

(56)

Then, we obtain

(57)

(45) therefore holds for h_i and holds for h. By the arbitrariness of h, we obtain (45). The proof is completed.☐☐

3. Constructing Weak Dual Wavelet Frames from a Pair of General Refinable Functions

This section is devoted to constructing weak dual wavelet frames for (FH^s(Ω), FH^−s(Ω)) from a pair of general refinable functions under the following assumptions:

Assumption 1. ψ₀ ∈ H^s(R^d) and are M-refinable functions with symbols in L^∞(T^d), i.e., there exists such that

(58)

Assumption 2. .

Assumption 3. for some δ > 0, where B(0, δ) denotes the δ-neighborhood of the origin 0.

Given L ∈ N, let

with 1 ≤ l ≤ L define Ψ = {ψ_l : 1 ≤ l ≤ L} and

(59)

Remark 15. By Lemma 10 (i), Assumption 2 is equivalent to the fact that {T_kψ₀ : k ∈ Z^d} and are Bessel sequences in H^s(R^d) and H^−s(R^d), respectively. It is easy to check that {T_kψ_l : k ∈ Z^d, 1 ≤ l ≤ L} and are also Bessel sequences in H^s(R^d) and H^−s(R^d), respectively.

Theorem 16. Let FH^s(Ω) and FH^−s(Ω) be reducing subspaces of H^s(R^d) and H^−s(R^d), respectively, ψ₀ ∈ FH^s(Ω) and two functions satisfy Assumptions 1–3, and let Ψ = {ψ_l : 1 ≤ l ≤ L} and be two finite subsets of FH^s(Ω) and FH^−s(Ω), respectively, defined by (59). Assume that there exists a function Θ ∈ L^∞(T^d) such that

(60)

(61)

Then, is a pair of weak dual wavelet frames for (FH^s(Ω), FH^−s(Ω)), where η is defined by .

Proof. First, we claim that the following equation holds:

(62)

a.e. on ℝ^d for n ∈ ℤ^d and 0 ≤ j < κ(n). Indeed, by Assumption 1 and (59) and the Z^d-periodicity of with 1 ≤ l ≤ L, we have

(63)

So (62) holds if

(64)

a.e. on ℝ^d with . Fix ξ with . Suppose for some and n_ξ ∈ ℤ^d. Then, by Lemma 8, and thus,

(65)

by (61). This is equivalent to (64) by the Z^d-periodicity of Θ, b_l, and

with 0 ≤ l ≤ L. Therefore, (62) holds.

By Theorem 14, to complete the proof, it is enough to prove that

(66)

for h ∈ L^∞(K), ∀ compact K ⊂ R^d\{0}, and

(67)

We first prove (67). For 0 ≠ n ∈ Z^d, by using (62), we have

(68)

By Lemma 7, we have for some n^′ ∈ Z^d and . It follows that

(69)

by (61), and (67) therefore holds.

Next, we prove (66) holds. By using (62), we have

(70)

where (61) was used in the last equality. And thus,

(71)

for h ∈ L^∞(K) with compact K ⊂ R^d\{0}. Since K is compact and K ⊂ R^d\{0}, then there exists J₀ > 0 such that

for J > J₀ + 1 and ξ ∈ K. And thus, there exists a constant M > 0 such that

(72)

by Assumption 3. This implies that

(73)

by Assumptions 1 and 3. Therefore, (66) holds by collecting (71) and (60).☐☐

Remark 17.

(i)
In the literature, the function Θ is related to the notion of mixed fundamental function, which plays a significant role in MOEP.
(ii)
In the above theorem, (60) is trivial and is a pair of weak dual wavelet frames for (FH^s(Ω), FH^−s(Ω)) if Θ(·) = 1 a.e. on T^d

4. Conclusion

In this paper, 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.

Conflicts of Interest

The author declares no conflicts of interest.

Authors’ Contributions

The author read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11961072 and 62041212), the Natural Science Basic Research Program of Shanxi (Grant Nos. 2020JM-547 and 2020JM-548), and the Doctoral Research Project of Yan’an University (Grant No. YDBK2017-21).

Open Research

Data Availability

There is no data in our paper.

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A Class of Weak Dual Wavelet Frames for Reducing Subspaces of Sobolev Spaces

Abstract

1. Introduction

2. The Characterization of Weak Dual Wavelet Frames

3. Constructing Weak Dual Wavelet Frames from a Pair of General Refinable Functions

4. Conclusion

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

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A Class of Weak Dual Wavelet Frames for Reducing Subspaces of Sobolev Spaces

Abstract

1. Introduction

2. The Characterization of Weak Dual Wavelet Frames

3. Constructing Weak Dual Wavelet Frames from a Pair of General Refinable Functions

4. Conclusion

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

References

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