# A Characterization of Multi-Wavelet Dual Frames in Sobolev Spaces

Authors: Jianping Zhang; Jiajia Li
Journal: Journal of Function Spaces
 (2020)

---

## 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.


---

## Body

## 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 L
2
R
d to Sobolev spaces pairs H
s
R
d
,
H
−
s
R
d 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 2
I
d 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 useZ, N, and N
0 to denote the set of integers, the set of positive integers, and the set of nonegative integers, respectively. Let d
∈
N, we use T
d
=
0
,
1
d to denote a d-dimensional torus, and, given a set E on R
d, E to denote the Lebesgue measure, and χ
E to denote the characteristic function on E. Throughout this paper, we write δ as the Dirac sequences so that δ
0
,
0
=
1, δ
0
,
k
=
0 for 0
≠
k
∈
Z
d. For a function f in L
1
R
d, it’s Fourier transform f
^ is defined by f
^
·
=
∫
R
d
f
x
e
−
2
π
i
x
,
·
d
x, and is naturally extended to tempered distributions, where ·
,
· denotes the Euclidean inner product on R
d.Givens
∈
R, the Sobolev space H
s
R
d consists of all distributions f such that(1)
f
H
s
R
d
2
=
∫
R
d
f
^
ξ
2
1
+
ξ
2
2
s
d
ξ
<
∞
,where·
2 denotes the Euclidean norm on R
d. Note that H
s
R
d is a separable Hilbert space under the following definition of inner product:(2)
f
,
g
H
s
R
d
=
∫
R
d
f
^
ξ
g
^
ξ
¯
1
+
ξ
2
2
s
d
ξ
,
f
,
g
∈
H
s
R
d
.Furthermore, for everyg
∈
H
−
s
R
d,(3)
f
,
g
=
∫
R
d
f
^
ξ
g
^
ξ
¯
d
ξ
,
f
∈
H
s
R
dgives a linear continuous functional inH
s
R
d. The Sobolev space H
s
R
d is the dual space of H
−
s
R
d, and vice versa.Forf
,
g
:
R
d
↦
C, we define(4)
f
,
g
·
=
∑
k
∈
Z
d
f
·
+
k
g
·
+
k
¯
.We useA
∗ to denote the conjugate transpose of a d
×
d order matrix A. It is called an expansive matrix if the modulus of all its eigenvalues is greater than 1. In a vast body of the literature, A is required to be an expansive integer matrix so that A
=
2
I
d (I
d is the identity matrix). Moreover, for the sake of convenience, we always write(5)
m
=
det
A
1
/
d
,and write(6)
f
j
,
k
·
=
m
j
d
/
2
f
A
j
·
−
k
a
n
d
f
j
,
k
s
·
=
m
−
j
s
f
j
,
k
·
=
m
j
d
2
−
s
f
A
j
·
−
k
,
j
∈
Z
,
k
∈
Z
dfor a tempered distributionf.Givenr
∈
N, let Φ
=
ϕ
1
,
ϕ
2
,
⋯
,
ϕ
r
T
∈
H
s
R
d
r and Ψ
l
=
ψ
1
l
,
ψ
2
l
,
⋯
,
ψ
r
l
T
∈
H
s
R
d
r
l
=
1
L, Φ
̃
=
ϕ
̃
1
,
ϕ
̃
2
,
⋯
,
ϕ
̃
r
T
∈
H
−
s
R
d
r andΨ
̃
l
=
ψ
̃
1
l
,
ψ
̃
2
l
,
⋯
,
ψ
̃
r
l
T
∈
H
−
s
R
d
r
l
=
1
L, we denote by X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L and X
−
s
Φ
̃
;
Ψ
̃
1
,
Ψ
̃
2
,
⋯
,
Ψ
̃
L the following two multi-wavelet systems in H
s
R
d and H
−
s
R
d:(7)
X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L
=
ϕ
n
;
0
,
k
:
n
=
1
,
2
,
⋯
,
r
;
k
∈
Z
d
∪
ψ
n
;
j
,
k
l
,
s
:
n
=
1
,
2
,
⋯
,
r
;
j
∈
N
0
,
k
∈
Z
d
,
l
=
1
,
2
,
⋯
,
Land(8)
X
−
s
Φ
̃
;
Ψ
̃
1
,
Ψ
̃
2
,
⋯
,
ψ
̃
L
=
ϕ
̃
n
;
0
,
k
:
n
=
1
,
2
,
⋯
,
r
;
k
∈
Z
d
∪
ψ
̃
n
;
j
,
k
l
,
s
:
n
=
1
,
2
,
⋯
,
r
;
j
∈
N
0
,
k
∈
Z
d
,
l
=
1
,
2
,
⋯
,
L
.If there existA
,
B
>
0 such that(9)
A
f
H
s
R
d
2
≤
∑
n
=
1
r
∑
k
∈
Z
d
f
,
ϕ
n
;
0
,
k
H
s
R
d
2
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
f
,
ψ
n
;
j
,
k
l
,
s
H
s
R
d
2
≤
B
f
H
s
R
d
2
,
∀
f
∈
H
s
R
d
,then we sayX
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L is a multi-wavelet frame ( MWF) in H
s
R
d, where A and B are called upper and lower frame bounds, respectively; If the inequality on the right side of (9) holds, then we say X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L is a multi-wavelet Bessel sequence (MWBS) in H
s
R
d, where B is called a Bessel bound. In addition, if the multi-wavelet systems X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L and X
−
s
Φ
̃
;
Ψ
̃
1
,
Ψ
̃
2
,
⋯
,
Ψ
̃
L are two Bessel sequences in H
s
R
d and H
−
s
R
d, respectively, and the identity(10)
f
,
g
=
∑
n
=
1
r
∑
k
∈
Z
d
f
,
ϕ
̃
n
;
0
,
k
ϕ
n
;
0
,
k
,
g
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
f
,
ψ
̃
n
;
j
,
k
l
,
−
s
ψ
n
;
j
,
k
l
,
s
,
g
,
f
∈
H
s
R
d
,
g
∈
H
−
s
R
dholds, then we say(11)
X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L
,
X
−
s
Φ
̃
;
Ψ
̃
1
,
Ψ
̃
2
,
⋯
,
Ψ
̃
Lis a pair of multi-wavelet dual frames (MWDF) inH
s
R
d
,
H
−
s
R
d.IfX
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L
,
X
−
s
Φ
̃
;
Ψ
̃
1
,
Ψ
̃
2
,
⋯
,
Ψ
̃
L is a pair of MWDF in H
s
R
d
,
H
−
s
R
d, then it follows from (10) that(12)
f
=
∑
n
=
1
r
∑
k
∈
Z
d
f
,
ϕ
̃
n
;
0
,
k
ϕ
n
;
0
,
k
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
f
,
ψ
̃
n
;
j
,
k
l
,
−
s
ψ
n
;
j
,
k
l
,
s
,
f
∈
H
s
R
dand(13)
g
=
∑
n
=
1
r
∑
k
∈
Z
d
g
,
ϕ
n
;
0
,
k
ϕ
̃
n
;
0
,
k
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
g
,
ψ
n
;
j
,
k
l
,
s
ψ
̃
n
;
j
,
k
l
,
−
s
,
g
∈
H
−
s
R
d
,where the series in (12) and (13) converge absolutely in H
s
R
d and H
−
s
R
d, respectively.In the following section, we provide some necessary lemmas used latter. In Section3, we provide a characterization of MWDF of the form (11) in H
s
R
d
,
H
−
s
R
d 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.
LetA be a d
×
d expansive matrix. Define a function κ
:
Z
d
→
Z by(14)
κ
n
=
s
u
p
j
≥
0
:
A
∗
−
j
n
∈
Z
d
,
and setκ
0
=
+
∞.Lemma 1.
Lets
∈
R, f
∈
H
s
R
d
,
ψ
∈
H
−
s
R
d and j
∈
Z. Then, for k
∈
Z
d, the kth Fourier coefficient of det
A
j
2
f
^
A
∗
j
·
,
ψ
^
·
ξ is f
,
ψ
j
,
k. In particular,(15)
det
A
j
2
f
^
A
∗
j
·
,
ψ
^
·
ξ
=
∑
k
∈
Z
d
f
,
ψ
j
,
k
e
2
π
i
k
,
ξ
,
ifψ
j
,
k
:
k
∈
Z
d is a Bessel sequence in H
−
s
R
d.Proof.
Sincef
∈
H
s
R
d, ψ
∈
H
−
s
R
d, we have f
^
A
∗
j
·
ψ
^
·
¯
∈
L
1
R
d, and thus
(16)
∫
T
d
det
A
j
/
2
f
^
A
∗
j
·
,
ψ
^
·
ξ
e
−
2
π
i
k
,
ξ
d
ξ
=
d
e
t
A
j
2
∫
T
d
∑
l
∈
Z
d
f
^
A
∗
j
ξ
+
l
ψ
^
ξ
+
l
¯
e
−
2
π
i
k
,
ξ
d
ξ
=
d
e
t
A
j
2
∫
R
d
f
^
A
∗
j
ξ
ψ
^
ξ
¯
e
−
2
π
i
k
,
ξ
d
ξ
=
det
A
−
j
2
∫
R
d
f
^
ξ
ψ
^
A
∗
−
j
ξ
¯
e
−
2
π
i
k
,
A
∗
−
j
ξ
d
ξ
=
∫
R
d
f
^
ξ
ψ
j
,
k
·
∧
ξ
¯
d
ξ
.
By the Plancherel theorem, we get(17)
∫
T
d
d
e
t
A
j
/
2
f
^
A
∗
j
·
,
ψ
^
·
ξ
e
−
2
π
i
k
,
ξ
d
ξ
=
f
,
ψ
j
,
k
.
Ifψ
j
,
k
:
k
∈
Z
d is a Bessel sequence in H
−
s
R
d, then f
,
ψ
j
,
k
k
∈
Z
d
∈
ℓ
2
Z
d, and thus, (15) follows directly from (17).By observing the proof of [4, Proposition 2.1], we getLemma 2.
Lets
∈
R, Φ
=
ϕ
1
,
ϕ
2
,
⋯
,
ϕ
r
T
∈
H
s
R
d
r and Ψ
l
=
ψ
1
l
,
ψ
2
l
,
⋯
,
ψ
r
l
T
∈
H
s
R
d
r
l
=
1
L. Then X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L as in (7) is a MWBS in H
s
R
d with Bessel bound B if and only if
(18)
∑
n
=
1
r
∑
k
∈
Z
d
g
,
ϕ
n
;
0
,
k
2
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
g
,
ψ
n
;
j
,
k
l
,
s
2
≤
B
g
H
−
s
R
d
2
f
o
r
g
∈
H
−
s
R
d
.Lemma 3.
Lets
∈
R, Φ
=
ϕ
1
,
ϕ
2
,
⋯
,
ϕ
r
T
∈
H
s
R
d
r and Ψ
l
=
ψ
1
l
,
ψ
2
l
,
⋯
,
ψ
r
l
T
∈
H
s
R
d
r
l
=
1
L. Suppose that X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L as in (7) is an MWBS in H
s
R
d with Bessel bound B. Then
(19)
∑
n
=
1
r
ϕ
^
n
·
2
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
m
−
2
j
s
ψ
^
n
l
A
∗
−
j
·
2
≤
B
1
+
·
2
2
−
s
,holds a.e. on R
d.Proof.
SinceX
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L as in (7) is a MWBS in H
s
R
d with Bessel bound B, by Lemma 2, we get.
(20)
∑
n
=
1
r
∑
k
∈
Z
d
g
,
ϕ
n
;
0
,
k
2
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
g
,
ψ
n
;
j
,
k
l
,
s
2
≤
B
g
H
−
s
R
d
2
f
o
r
g
∈
H
−
s
R
d
.
Applying the Plancherel theorem and the Parseval identity, by a simple computation(21)
∑
k
∈
Z
d
g
,
ϕ
n
·
−
k
2
=
∑
k
∈
Z
d
∫
R
d
g
^
ξ
ϕ
^
n
ξ
¯
e
2
π
i
k
,
ξ
d
ξ
2
=
∑
k
∈
Z
d
∑
k
′
∈
Z
d
∫
T
d
g
^
ξ
+
k
′
ϕ
^
n
ξ
+
k
′
¯
e
2
π
i
k
,
ξ
d
ξ
2
=
∫
T
d
∑
k
′
∈
Z
d
g
^
ξ
+
k
′
ϕ
^
n
ξ
+
k
′
¯
2
d
ξ
.
Similarly, we have(22)
∑
k
∈
Z
d
g
,
ψ
n
;
j
,
k
l
,
s
2
=
m
−
j
d
+
2
s
∑
k
∈
Z
d
∫
R
d
g
^
ξ
ψ
^
n
l
A
∗
−
j
ξ
¯
e
2
π
i
k
,
A
∗
−
j
ξ
d
ξ
2
=
m
j
d
−
2
s
∑
k
∈
Z
d
∑
k
′
∈
Z
d
∫
T
d
g
^
A
∗
j
ξ
+
k
′
ψ
^
n
l
ξ
+
k
′
¯
e
2
π
i
k
,
ξ
d
ξ
2
=
m
j
d
−
2
s
∫
T
d
∑
k
′
∈
Z
d
g
^
A
∗
j
ξ
+
k
′
ψ
^
n
l
ξ
+
k
′
¯
2
d
ξ
.
From the above computation, we get(23)
∑
n
=
1
r
∑
k
∈
Z
d
g
,
ϕ
n
;
0
,
k
2
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
g
,
ψ
n
;
j
,
k
l
,
s
2
=
∑
n
=
1
r
∫
T
d
∑
k
∈
Z
d
g
^
ξ
+
k
ϕ
^
n
ξ
+
k
¯
2
d
ξ
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
m
j
d
−
2
s
∫
T
d
∑
k
∈
Z
d
g
^
A
∗
j
ξ
+
k
ψ
^
n
l
ξ
+
k
¯
2
d
ξ
=
∑
n
=
1
r
∫
T
d
∑
k
∈
Z
d
g
^
ξ
+
k
ϕ
^
n
ξ
+
k
¯
∑
k
∈
Z
d
ϕ
^
n
ξ
+
k
g
^
ξ
+
k
¯
d
ξ
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
m
j
d
−
2
s
∫
T
d
∑
k
∈
Z
d
g
^
A
∗
j
ξ
+
k
ψ
^
n
l
ξ
+
k
¯
∑
k
∈
Z
d
ψ
^
n
l
ξ
+
k
g
^
A
∗
j
ξ
+
k
¯
d
ξ
.
By the Fubini–Tonelli theorem, we have(24)
∑
n
=
1
r
∫
T
d
∑
k
∈
Z
d
g
^
ξ
+
k
ϕ
^
n
ξ
+
k
¯
∑
k
∈
Z
d
ϕ
^
n
ξ
+
k
g
^
ξ
+
k
¯
d
ξ
=
∑
n
=
1
r
∫
R
d
ϕ
^
n
ξ
g
^
ξ
¯
∑
k
∈
Z
d
g
^
ξ
+
k
ϕ
^
n
ξ
+
k
¯
d
ξ
,
and(25)
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
m
j
d
−
2
s
∫
T
d
∑
k
∈
Z
d
g
^
A
∗
j
ξ
+
k
ψ
^
n
l
ξ
+
k
¯
∑
k
∈
Z
d
ψ
^
n
l
ξ
+
k
g
^
A
∗
j
ξ
+
k
¯
d
ξ
=
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
m
j
d
−
2
s
∫
R
d
ψ
^
n
l
ξ
g
^
A
∗
j
ξ
¯
∑
k
∈
Z
d
g
^
A
∗
j
ξ
+
k
ψ
^
n
l
ξ
+
k
¯
d
ξ
=
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
m
−
2
j
s
∫
R
d
ψ
^
n
l
A
∗
−
j
ξ
g
^
ξ
¯
∑
k
∈
Z
d
g
^
ξ
+
A
∗
j
k
ψ
^
n
l
A
∗
−
j
ξ
+
k
¯
d
ξ
.
Collecting (23)–(25), we have(26)
∑
n
=
1
r
∑
k
∈
Z
d
g
,
ϕ
n
;
0
,
k
2
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
g
,
ψ
n
;
j
,
k
l
,
s
2
=
∑
n
=
1
r
∫
R
d
ϕ
^
n
ξ
g
^
ξ
¯
∑
k
∈
Z
d
g
^
ξ
+
k
ϕ
^
n
ξ
+
k
¯
d
ξ
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
m
−
2
j
s
∫
R
d
ψ
^
n
l
A
∗
−
j
ξ
g
^
ξ
¯
∑
k
∈
Z
d
g
^
ξ
+
A
∗
j
k
ψ
^
n
l
A
∗
−
j
ξ
+
k
¯
d
ξ
=
∫
R
d
g
^
ξ
2
∑
n
=
1
r
ϕ
^
n
ξ
2
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
m
−
2
j
s
ψ
^
n
l
A
∗
−
j
ξ
2
d
ξ
+
∫
R
d
g
^
ξ
¯
∑
0
≠
k
∈
Z
d
g
^
ξ
+
k
∑
n
=
1
r
ϕ
^
n
ξ
ϕ
^
n
ξ
+
k
¯
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
κ
k
m
−
2
j
s
ψ
^
n
l
A
∗
−
j
ξ
ψ
^
n
l
A
∗
−
j
ξ
+
k
¯
d
ξ
by the definition ofκ for 0
≠
k
∈
Z
d.
Suppose (19) does not hold, then there exists E
⊂
R
d with E
>
0 such that(27)
∑
n
=
1
r
ϕ
^
n
·
2
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
m
−
2
j
s
ψ
^
n
l
A
∗
−
j
·
2
>
B
1
+
·
2
2
−
s
o
n
E
,
and thus(28)
∑
n
=
1
r
ϕ
^
n
·
2
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
m
−
2
j
s
ψ
^
n
l
A
∗
−
j
·
2
>
B
1
+
·
2
2
−
s
,
on someE
′
=
E
∩
0
,
1
d
+
k
0 with E
′
>
0 and k
0
∈
Z
d. Define g by(29)
g
^
·
=
1
+
·
2
2
s
2
χ
E
′
,
in (26), then we get(30)
∑
n
=
1
r
∑
k
∈
Z
d
g
,
ϕ
n
;
0
,
k
2
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
g
,
ψ
n
;
j
,
k
l
,
s
2
>
B
E
′
=
B
g
H
−
s
R
d
2
,
which contradicts with (20).


## 3. MWDF in Sobolev Spaces

This section is devoted to characterizing MWDF of the form (11) in H
s
R
d
,
H
−
s
R
d. The following theorem provides us with a characterization by using a pair of equations.Theorem 1.
Givens
∈
R, let Φ
=
ϕ
1
,
ϕ
2
,
⋯
,
ϕ
r
T
∈
H
s
R
d
r and Ψ
l
=
ψ
1
l
,
ψ
2
l
,
⋯
,
ψ
r
l
T
∈
H
s
R
d
r
l
=
1
L, Φ
̃
=
ϕ
̃
1
,
ϕ
̃
2
,
⋯
,
ϕ
̃
r
T
∈
H
−
s
R
d
r and Ψ
̃
l
=
ψ
̃
1
l
,
ψ
̃
2
l
,
⋯
,
ψ
̃
r
l
T
∈
H
−
s
R
d
r
l
=
1
L. Define the multi-wavelet systems X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L and X
−
s
Φ
̃
;
Ψ
̃
1
,
Ψ
̃
2
,
⋯
,
Ψ
̃
L as in (7) and (8), respectively. Suppose that X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L and X
−
s
Φ
̃
;
Ψ
̃
1
,
Ψ
̃
2
,
⋯
,
Ψ
̃
L are two MWBSs in H
s
R
d and H
−
s
R
d, respectively. Then X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L
,
X
−
s
Φ
̃
;
Ψ
̃
1
,
Ψ
̃
2
,
⋯
,
Ψ
̃
L is a pair of MWDF in H
s
R
d
,
H
−
s
R
d if and only if, for every k
∈
Z
d.
(31)
∑
n
=
1
r
ϕ
^
n
∙
ϕ
̃
^
n
∙
+
k
¯
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
κ
k
ψ
^
n
l
A
∗
−
j
∙
ψ
̃
^
n
l
A
∗
−
j
∙
+
k
¯
=
δ
0
,
k
,
a
.
e
.
o
n
R
d
.Proof.
By the definition,X
s
Φ
;
Ψ
1
,
Ψ
2
,
⋯
,
Ψ
L
,
X
−
s
Φ
̃
;
Ψ
1
̃
,
Ψ
2
̃
,
⋯
,
Ψ
L
̃ is a pair of MWDF in H
s
R
d
,
H
−
s
R
d if and only if
(32)
∑
n
=
1
r
∑
k
∈
Z
d
f
,
ϕ
̃
n
;
0
,
k
ϕ
n
;
0
,
k
,
g
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
f
,
ψ
̃
n
;
j
,
k
l
,
−
s
ψ
n
;
j
,
k
l
,
s
,
g
=
f
,
g
forf
∈
H
s
R
d
,
g
∈
H
−
s
R
d
. Applying Lemma 1 and the Plancherel theorem, we have(33)
∑
n
=
1
r
∑
k
∈
Z
d
f
,
ϕ
̃
n
;
0
,
k
ϕ
n
;
0
,
k
,
g
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
f
,
ψ
̃
n
;
j
,
k
l
,
−
s
ψ
n
;
j
,
k
l
,
s
,
g
=
∑
n
=
1
r
∫
T
d
∑
k
∈
Z
d
f
^
ξ
+
k
ϕ
̃
^
n
ξ
+
k
¯
∑
k
∈
Z
d
ϕ
^
n
ξ
+
k
g
^
n
ξ
+
k
¯
d
ξ
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
d
e
t
A
j
∫
T
d
∑
k
∈
Z
d
f
^
A
∗
j
ξ
+
k
ψ
̃
^
n
l
ξ
+
k
¯
∑
k
∈
Z
d
ψ
^
n
l
ξ
+
k
g
^
n
A
∗
j
ξ
+
k
¯
d
ξ
=
∑
n
=
1
r
∫
R
d
∑
k
∈
Z
d
f
^
ξ
+
k
ϕ
̃
^
n
ξ
+
k
¯
ϕ
^
n
ξ
g
^
ξ
¯
d
ξ
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
d
e
t
A
j
∫
R
d
∑
k
∈
Z
d
f
^
A
∗
j
ξ
+
k
ψ
̃
^
n
l
ξ
+
k
¯
ψ
̃
^
n
l
ξ
g
^
A
∗
j
ξ
¯
d
ξ
=
∫
R
d
f
^
ξ
g
^
ξ
¯
∑
n
=
1
r
ϕ
^
n
ξ
ϕ
̃
^
¯
n
ξ
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
ψ
^
n
l
A
∗
−
j
ξ
ψ
̃
^
n
l
A
∗
−
j
ξ
¯
d
ξ
+
∫
R
d
g
^
ξ
¯
∑
n
=
1
r
∑
0
≠
k
∈
Z
d
f
^
ξ
+
k
ϕ
n
^
ξ
ϕ
n
̃
^
ξ
+
k
¯
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
∑
k
∈
Z
d
f
^
ξ
+
A
∗
j
k
ψ
^
n
l
A
∗
−
j
ξ
ψ
̃
^
n
l
A
∗
−
j
ξ
+
k
¯
d
ξ
=
∫
R
d
f
^
ξ
g
^
ξ
¯
∑
n
=
1
r
ϕ
^
n
ξ
ϕ
̃
^
¯
n
ξ
¯
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
ψ
^
n
l
A
∗
−
j
ξ
ψ
̃
^
n
l
A
∗
−
j
ξ
¯
d
ξ
+
∫
R
d
g
^
ξ
¯
∑
0
≠
k
∈
Z
d
f
^
ξ
+
k
∑
n
=
1
r
ϕ
^
n
ξ
ϕ
̃
^
¯
n
ξ
+
k
¯
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
ψ
^
n
l
A
∗
−
j
ξ
ψ
̃
^
n
l
A
∗
−
j
ξ
+
k
¯
d
ξ
.
So, (32) can be written(34)
∫
R
d
f
^
ξ
g
^
ξ
¯
∑
n
=
1
r
ϕ
^
n
ξ
ϕ
̃
^
¯
n
ξ
−
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
ψ
^
n
l
A
∗
−
j
ξ
ψ
̃
^
n
l
A
∗
−
j
ξ
¯
d
ξ
+
∫
R
d
g
^
ξ
¯
∑
0
≠
k
∈
Z
d
f
^
ξ
+
k
∑
n
=
1
r
ϕ
^
n
ξ
ϕ
̃
^
¯
n
ξ
+
k
−
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
ψ
^
n
l
A
∗
−
j
ξ
ψ
̃
^
n
l
A
∗
−
j
ξ
+
k
¯
d
ξ
=
∫
R
d
f
^
ξ
g
^
ξ
¯
d
ξ
.
Clearly, (31) implies (34). Next, we will prove another direction.
Suppose (34) holds. Applying the Cauchy Schwarz inequality and Lemma 3, the series(35)
∑
n
=
1
r
ϕ
^
n
∙
ϕ
̃
^
¯
n
∙
+
k
¯
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
κ
k
ψ
^
n
l
A
∗
−
j
∙
ψ
̃
^
n
l
A
∗
−
j
.
+
k
¯
,
k
∈
Z
d
absolutely converges a.e. onR
d and belongs to L
∞
R
d therefore, almost all points on R
d Lebesgue points. Let ξ
0
∈
R
d be such a point. For 0
<
∈
<
1
/
2 define f and g by
(36)
f
^
·
=
1
+
·
2
2
−
s
2
χ
B
ξ
0
,
ϵ
B
ξ
0
,
ϵ
and
g
^
·
=
1
+
·
2
2
s
2
χ
B
ξ
0
,
ϵ
B
ξ
0
,
ϵ
.
in (31), where B
ξ
0
,
ϵ
=
ξ
∈
R
d
:
ξ
−
ξ
0
<
ϵ
. Then(37)
1
B
ξ
0
,
ϵ
∫
B
ξ
0
,
ϵ
∑
n
=
1
r
ϕ
^
n
ξ
ϕ
̃
^
¯
n
ξ
¯
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
ψ
^
n
l
A
∗
−
j
ξ
ψ
̃
^
n
l
A
∗
−
j
ξ
¯
d
ξ
=
1
B
ξ
0
,
ϵ
∫
B
ξ
0
,
ϵ
d
ξ
,
lettingϵ
→
0, we get(38)
∑
n
=
1
r
ϕ
^
n
ξ
0
ϕ
̃
^
¯
n
ξ
0
¯
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
∞
ψ
^
n
l
A
∗
−
j
ξ
0
ψ
̃
^
n
l
A
∗
−
j
ξ
0
¯
=
1
.
For0
≠
k
0
∈
Z
d, define f and g by   (39)
f
^
·
+
k
0
=
1
+
·
2
2
−
s
2
χ
B
ξ
0
,
ϵ
B
ξ
0
,
ϵ
and
g
^
·
¯
=
1
+
·
2
2
s
2
χ
B
ξ
0
,
ϵ
B
ξ
0
,
ϵ
.
in (34), where 0
<
ϵ
<
1
/
2
. Then(40)
1
B
ξ
0
,
ϵ
∫
B
ξ
0
,
ϵ
1
∑
n
=
1
r
ϕ
^
n
ξ
ϕ
̃
^
¯
n
ξ
+
k
0
¯
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
κ
k
0
ψ
^
n
l
A
∗
−
j
ξ
ψ
̃
^
n
l
A
∗
−
j
ξ
+
k
0
¯
d
ξ
=
0
,
lettingϵ
→
0, we get(41)
∑
n
=
1
r
ϕ
^
n
ξ
0
ϕ
̃
^
¯
n
ξ
0
+
k
0
¯
+
∑
n
=
1
r
∑
l
=
1
L
∑
j
=
0
κ
k
0
ψ
^
n
l
A
∗
−
j
ξ
0
ψ
̃
^
n
l
A
∗
−
j
ξ
0
+
k
0
¯
=
0
.
By the arbitrariness ofξ
0
∈
R
d and k
0
∈
Z
d, 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 inH
s
R
d
,
H
−
s
R
d of the form (11) by using a pair of equations.



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*Source: 3148946-2020-01-03.xml*