# Construction of Multiwavelets on an Interval

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## Abstract

**:**

## 1. Introduction and Overview

## 2. Review of Wavelet Theory

#### 2.1. Multiresolution Approximation

- (i)
- ${V}_{n}\subset {V}_{n+1}$ for all $n\in \mathbb{Z}$;
- (ii)
- $f\left(x\right)\in {V}_{n}\u27faf\left(2x\right)\in {V}_{n+1}$ for all $n\in \mathbb{Z}$;
- (iii)
- $f\left(x\right)\in {V}_{n}\u27f9f(x-{2}^{-n}k)\in {V}_{n}$ for all $n,k\in \mathbb{Z}$;
- (iv)
- ${\bigcap}_{n\in \mathbb{Z}}{V}_{n}=\left\{0\right\}$;
- (v)
- $\overline{{\bigcup}_{n\in Z}{V}_{n}}={L}^{2}\left(\mathbb{R}\right)$;
- (vi)
- there exists a function vector$$\varphi \left(x\right)=\left(\begin{array}{c}{\varphi}_{1}\left(x\right)\\ \vdots \\ {\varphi}_{r}\left(x\right)\end{array}\right),\phantom{\rule{2.em}{0ex}}{\varphi}_{i}\in {L}^{2}\left(\mathbb{R}\right)$$

#### 2.2. Discrete Wavelet Transform

#### 2.3. Approximation Order

## 3. Wavelets on an Interval

- •
- recursion relations; or
- •
- linear combinations of shifts of the underlying scaling functions; or
- •
- linear algebra techniques.

#### 3.1. Basic Assumptions and Notation

- •
- The underlying multiwavelet is orthogonal, continuous, with multiplicity r and approximation order $p\ge 1$, and has recursion coefficients ${H}_{0},\dots ,{H}_{N}$, ${G}_{0},\dots ,{G}_{N}$. This means the support of $\varphi $ and ψ is contained in the interval $[0,N]$.
- •
- The interval I is $[0,M]$ with M large enough so that the left and right endpoint functions do not interfere with each other.
- •
- The boundary functions have support on $[0,N-1]$ and $[M-N+1,M]$, respectively (that is, smaller support than the interior functions).

#### 3.2. Recursion Relations

#### 3.3. Linear Combinations

#### 3.4. Linear Algebra

#### 3.5. Uniqueness Results

**Lemma 3.1**If ${T}_{0}$, ${T}_{1}$ are square matrices of size $2r\times 2r$ that satisfy relations (11), then

**Theorem 3.2**If ${\Delta}_{3}$ is orthogonal and has the structure given in (14), then ${L}_{0}$, ${L}_{1}$, ${R}_{0}$, ${R}_{1}$ must have sizes $2{\rho}_{1}\times {\rho}_{1}$, $2{\rho}_{1}\times 2r$, $2{\rho}_{0}\times 2r$, and $2{\rho}_{0}\times {\rho}_{0}$, respectively.

**Theorem 3.3**Assume that ${\Delta}_{3}$, ${\tilde{\Delta}}_{3}$ are two orthogonal matrices of form (14). Then there exist orthogonal matrices ${Q}_{L}$, ${Q}_{R}$ so that

## 4. Madych Approach for Scalar Wavelets

**Definition 4.1**A multiscaling function based on four recursion coefficients satisfies Condition M if

## 5. A New Approach to Multiwavelet Endpoint Modification

## 6. Imposing Regularity Conditions

#### 6.1. Approximation Order for Boundary Functions

#### 6.2. Simplifying the Problem

#### 6.3. Deriving the Algorithm

## 7. Examples

#### 7.1. Example 1: Daubechies ${D}_{2}$

#### 7.2. Example 2: CL(3) Multiwavelet

#### 7.3. Example 3: DGHM Multiwavelet

**Figure 3.**Boundary functions for DGHM. The single left boundary function has approximation order 1. Three right boundary functions provide approximation order 2.

## 8. Summary

## References

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**MDPI and ACS Style**

Altürk, A.; Keinert, F.
Construction of Multiwavelets on an Interval. *Axioms* **2013**, *2*, 122-141.
https://doi.org/10.3390/axioms2020122

**AMA Style**

Altürk A, Keinert F.
Construction of Multiwavelets on an Interval. *Axioms*. 2013; 2(2):122-141.
https://doi.org/10.3390/axioms2020122

**Chicago/Turabian Style**

Altürk, Ahmet, and Fritz Keinert.
2013. "Construction of Multiwavelets on an Interval" *Axioms* 2, no. 2: 122-141.
https://doi.org/10.3390/axioms2020122