The Biorthogonal Features of the Ternary Vector-Valued Wavelet Wraps with Filter Functions and Pseudoframes - Advanced Research on Computer Education, Simulation and Modeling - page 68

2 3

Definition 1. We say that a pair of vector-valued functions Hx Hx (), () ∈ L R C ( ,

v

)

are biorthogonal, if their translations satisfy

3

[( , ( HH n ⋅ ⋅−= ] δ I , nZ ∈ , (2)

0 , nv

I denotes the vv

× δ

where is the Kronecker symbol.

Definition 2. A sequence of vector-valued functions

indentity matrix and

0 , n

2 3 v

is

{( } Hx

∈ ⊂⊂

U LRC (, )

n

3

nZ

called a Riesz basis of U if itsatisfies (i) for any

Gx () ∈ U

, there exists a unique

2 3 vv ×

vv × {} Q

∈ ∈ ℓ

( ) Z

matrix sequence such that

n

s

nZ

∑

3

Gx () = QH x (), x R ∈ ,

(3)

nn

3

nZ ∈

1

v

∑∑

2 3 vv × 3 uu × 2

ℓ () Z = {: Q Z → = C , Q | qn ( ) | ) <+∞ ,

where

2

(ii) there

ls ,

2

4

ls , = ∈ 1 nZ

0 <≤<+∞ CC {}

n

M

exist two constants such that, for any matrix sequence ,

1 2

3

nZ ∈

the following equality follows.i.e.,

∑

CM {}|| ≤ MHx C M ( || ≤ {},

(4)

1 n n n 2 n

* *

3

nZ ∈

In what follows, we introduce the notion of vector- valued multiresolution analysis
and give the definition of biorthogonal vector-valued wavelets of space

v
LR C .

2 3

(, )

Definition 3. A vector-valued multiresolution analysis of the space

v
LR C is a

2 3

(, )

ℓ nested sequence of closed subspaces {} Z

Y YY

⊂∀∈ 1 ,

Z ;

such that (i) (ii)

ℓℓ ∈ ℓ ℓ

∩ ℓ ∪ ℓ

Y = {0}

Z Y

and is dense in

v
LR C , where 0 denotes an zero vector of

2 3

(, )

∈ Z ℓ ∈ ℓ

v

ℓ

R ; Φ∈⇔Φ

ℓ

( xY ( Mx )

∈∀∈

Y , Z

space (iii) ; (iv) there exists

ℓ 1

Fx () ∈ Y

,called a vector-valued scaling function, such that its translates

0

3
nZ

n Fx

(): =− Fx n ( ), ∈ Y

0 .

forms a Riesz basis of subspace

Fx () ∈⊂ Y Y

, by Definition 3 and (4) there exists a finitely supported se-
quence of constant vv

Since

0 1

×

matrice such that {}

Ω ℓ

2

( Z

3 vv ×

n

3

nZ

∑

(5)

Fx ()

∈ =Ω

FMx n ( − ).

n

3

nZ

Equation (6) is called a refinement equation. Define

∑

3

⋅ () γ = ⋅ exp{ − ⋅ in γ }, γ ∈ R .

m Ω Ω (6)

n

3

nZ ∈

s

, which is 2

π Z Fx . Thus, (6) becomes

3

()

where ( ) Ω γ fun., is called a symbol of

ˆ ˆ

FM ( γ ) =Ω () (), γ F γ γ ∈ R . (7)

Xj Z

∈

Y in Y + 1 .

Let be the direct complementary subspace of Assume that

,

ψ () xLRC ∈

23

( ,

v

), μ ∈Γ

there exist 63 vector-valued functions such that their

μ

X

tran- slations and dilations form a Riesz basis of i.e.,

j ,