The Nice Characters of Orthogonal Bidimensional Vector Wavelet Packs and Orthogonal Binary Framelets - Advanced Research on Computer Education, Simulation and Modeling - page 74

and construction of vector-valued wavelets. However, vector-valued wavelets [5] and
multiwavelets are different. Hence, studying vector-valued wavelets is useful in
multiwavelet theory and representations of signals. It is known that the majority of
information is multi-dimensional information. Thus, it is significant and necessary to
generalize the concept of multivariate wavelet packets to the case of multiple vector-
valued multivariate wavelets. Based on some ideas from [3], the goal of this paper is
to discuss the properties of multiple vector-valued multivariate wavelet packets.
Let R and C be all real and all complex numbers, respectively. Z and N denote,
respectively, all integers and all positive integers. Set as well Z

+ =

{0} U Nas N , , ∈

2

a ≥ 2 a elements

as By algebra theory, it is obviously follows that there are

2

dd d − ,, , a L Z = {( zz zz Z , ) : , ∈ }

2 2

in such that Z = U ( daZ + ) ;

01

2

+ 12 12 +

1

d ∈Ω

0

2 2

( d Z d Z + ) I ( + ) = φ , where Ω= {,, , dd d − L } denotes the aggregate of all

1 2 0 0 1

2

a 1

2 2

the different representative elements in the quotient group Z /( aZ ) and order

dd denote two arbitrary distinct

d = { 0 } where { 0 } is the null element of

2
Z + and

12

0

Ω

elements in .Let Ω=Ω − { 0 } and ΩΩ , to be two index sets. Define,

0

0 0

s
LRC , we denote the set of all vector-valued functions

2 (,

By )

LRC y hy hy hy hy LR l

2

( ,

2 s

):{() ( ()), (), , ()) : () = h = L

T

∈ =

2

( ),

2

1,2, ,} L s , where

1 2 u l

T means the transpose of a vector. For any

2 2 s

h ∈ LRC (, )

its integration is defin-

∫ ∫ ∫ ∫

ed as h () ydy = ( h ydy h ydy () , () , , L h ydy () ) , and the Fourier tra- 2 2

1

2

2

2

s

R R R R

h () y

nsform of is defined by

ˆ ():

∫

h ω = h ()exp{ y ⋅ − i y 〈 〉 , } ω dy , (1) 2

R

〈 〉 denotes the inner product of y and y , ω ω

where . For multiple vector-valued

<Λ> h ,

functions denotes their symbol inner product, i.e.,

h h , (2)
where ∗ means the transpose and the complex conjugate.

*

〈 , Λ 〉 : =

R yyy

() () Λ

2

2 2 s

Definition 1. A sequenc {() h y

∈ ⊂

L R C (, } is called an orthogonal set, if

n

2

nZ

2

〈 〉 hh

nv nv s I nvZ

, = δ , , ∈

, (3)

,

I stands for the ss

× δ

where identity matrix and , is generalized Kronecker

nv

δ = 1

as nv

= δ = 0

symbol, i.e., and , otherwise.

nv , nv ,

h

Definition 2. A sequence of vector-valued functions {( } y

∈ ⊂⊂

W

LRC

2 2

(,

s

)

is

n

2

nZ

called a Riesz basis in W , if it satisfies

Λ∈ ( yW

, there exists a unique ss

× { n nZ P

(i) For any matrix sequence

2

∈

such that