4th-order spline wavelets on a bounded interval

Applied Mathematics and Mechanics (English Edition, Vol. 21, No. 4, Apt 2000)

Published by Shanghai University, Shanghai, China

Article ID: 0253-4827(2000)04-0437-10

4TH-ORDER SPLINE WAVELETS ON A BOUNDED INTERVAL* Duan Jiwei ( ~ j ) l ,

Peter Kai-kwong Lee (~j~J~mS~)~-

( 1. Department of Civil Engineering, Zhejiang University of Technology, Hangzhou 310014, P R China; 2. Department of Civil Engineering, The University of Hong Kong, Hong Kong, P R China) (Communicated by Wu Qiguang) Abstract: The 4th-order spline wavelets on a bounded interval are constructed by the 4 thorder truncated B- spline functions. These wavelets consist of inner and boundary wavelets. They are bases of wavelet space with finite dimensions. Any function on an interval will be expanded as the sum of finite items of the scaling functions and wavelets. It plays an important role for numerical analysis of partial differential equations, signal processes, and other similar problems.

Key words: B-spline; wavelet; bounded interval CLC numbers: 0242.29; O241.82 Document cede: A

Introduction Wavelet analysis has experienced an enormous development in recent years. One of the important fields is the numerical analysis of differential equations. The advantage of adopting wavelets has been reported [1-4] Classical approaches to wavelet construction deal with multiresolution analysis ( M R A ) on the whole real axis, but recently, interests have been developed to construct wavelets on interval. Any function on an interval can easily be expanded into wavelet series as long as these wavelets are obtained. Some techniques have been developed in dealing with wavelets on an interval TM 6] One simple technique is to extend a function, given on the interval [ 0, 1 ] , to the real line by setting its values outside [ 0, 1 ] to zeros. This will introduce discontinuities at the end points but can be solved by increasing the number of wavelets. Another method is to periodize a given function. By using the scale and wavelet function in a periodic MR_A, a function can be expressed approximately into an interval similar to a function in Fourier expansion. The focus of the present study is to construct wavelets on the interval. Since the scale and wavelet space are finite dimensional at a given level, any function on an interval can be represented by a t-mite wavelet series for the interval. The wavelet developed can play an important role in partial differential

Received date: 1997-09-22; Revised date: 1998-09-i0

Biography: Duan Jiwei (1964 ~ ), Associate Professor, Doctor 437

438

I)uan Jiwei and Peter Kai-kwong Lee

equation, signal processing and similar problems. A method for the construction of spline wavelets on the interval is proposed. Once constructed, it becomes easy for wavelets to be applied in problems involving computational mechanics.

1

M u l t i r e s o l u t i o n A n a l y s i s a n d W a v e l e t s o n [0, 1]

D e f i n i t i o n 1 ( M u l f i r e s o l u t i o n analysis on [0, 1]) A f u n c t i o n f ( x ) E L2[0, 1] is said to be able to generate a multiresolution analysis (MRA) if it can generate a nested sequence of closed subspaces V~~ t] that satisfies 1)

V[oo, t] C V[~ 11 C "";

2) 3)

eIoSL2[0,l](.UaVJ~ j;,,0

) = L2[0, 1];

~

4) f ( x ) E V~~ 1] 0, sup Bj < ~ , such that kE l~dta

for all { ct } E f-. If a function ~ ( x ) generates an MRA, then ~ ( x ) is called a scaling function. The different integer translation and the dyadic dilation of ~ (x) will give a new family of the functions, namely ~ii(x) = 2J/"q)(2Jx - i) which are the base in V5~ 1] Definition 2 {Wavelets on [0, 1]) A function r E L'-[0, 1] is called a wavelet if it generates the complement arc orthogonal subspaces W5~ 1] of an MRA, i . e . ,

nz o, ll = in which Cj,(x) = 2JP-r

- i) E WS~

E Z+), E Z+).

Dilation and translation of a wavelet also generates a family of wavelet similar to Definition 1. If the inner product is given by J0 with J0 = m i n { j E Z . :2 i I> 2 m - 1 } .

(21)

With the wavelets obtained for level j 0 , scaling functions and wavelets for the j level when j I> J0 are given [7] . When m = 4 , j0 becomes 3. Using the Mathematics software, wavelets can be presented in the following figures:

0.010

O.OLO

0.005

0.005

0

t

0.8

- 0.005

o

-o.oo5!

0.010

-0.010

- 0.015

-0.015

-

0:2

(b)

Ca) Fig.2

The inner wavelets

1)

the

a)

the translate integer, i = 0

b)

the translate integer, i = 1

2)

the 0-boundary wavelets ( j = Jo = 3)

a)

the translate integer, i = m -

inner wavelets ( j = Yo = 3)

1 =-3

b)

the translate integer, i = - 2

c)

the translate integer, i = -

3)

the I-boundary wavelets ( j = jo = 3)

a)

the translate integer, i = 2

b)

the translate integer, i = 3

c)

the translate integer, i = 4

1

445

4th-Order Spline Wavelets on a Bounded Interval

A

0.015 0.010 0.005 0 - 0.005 -

000

0.5

v0 V

0.010

015

1 0.005

016 - 0.005 -0.010

- 0.015

-I 0.015 -

(a)

0.03 O.02 0.01 0

••

0.015 0.010 0.005

0 ~ " 2

-0.01 -0.02

(a)

0~

014

0

015

I_ 0.005 - 0.010 - 0.015

- 0.03

h)

(b) 0.03

0.010

o:/o.9t

O.005 0 -

0:005

-

0.010

.

,

.

,

0.5

0.02_

\

0.6

-0.015

(c)

(c) Fig.3

5

The 0-boundary wavelets

Fig.4

The 1-boundary wavelets

Conclusion The 4th-order truncated spline has been used successfully for consu-ucting wavelets on the

interval. These wavelets may have applications in differential equations, signal processing and other similar p r o b l e m s . It is considered that these wavelets can help to o v e r c o m e the difficult treatment o f any function on the interval.

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Chen Mingquayer, Hwang Chyi, Shih Yenping. A wavelet-Gale"rkin method for solving population

[2]

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[3]

2921 ~ 2944. Williams John R, Amaratunga Kevin. Introduction to wavelets in engineering[J]. International

446

Duan Jiwei and Peter Kai-kwong Lee

Journal for Numerical Method in Engineering, 1994,37 (14) : 2365 ~ 2388. [4]

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Daubechies Ingrid. Two recent results on wavelets: wavelet bases for the interval, and biothogonal wavelets diagonalizing the derivative operator[ A ] . In: Larry L Schumaker, Glenn Webb Eds. Recent Advances in Wavelet Analysis, Academic Press, Incorporeted, 1993.

[6]

Quak Ewald, Weyrich Norman. Wavelets on the interval[A]. In: Singh S P E d . Approximation Theory Wavelets and Applications [ C ] . 1995. Quak Ewald, Weyrich Norman. Decomposition and reconstruction algorithms for spline wavelets on a bounded interval[J]. Applied and Computational Harmonic Analysis, 1994,1:217 ~ 231. Guan Liitai. Truncated B-spline-wavelets on a bounded interval and its vanishing moment property [J] . Acta Scientiarum Natrualium Universitatis Sunyatseni, 1996, 35(3) :28 ~ 33. (in Chinese)

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