Definition of Terms

 

Compactly Supported Wavelets

The class of compactly supported wavelet bases was introduced by I. Daubechies in 1988 [2]. They are an orthonormal basis for functions in. A ``wavelet system'' consists of the function, referred to as the scaling function and the functionreferred to as the wavelet function.

By convention we define the translates ofas

 

The scaling relation that definesis

 

The scaling relation that definesis

 

From the scaling relation (2) above one sees thatis equal to a sum of scaled and shifted versions of itself. The scaling factor in this equation is 2 and hence we refer toandof this form as a multiplier 2 system. One can generalize wavelet systems to an arbitrary nonnegative integer multiplier. For higher order multipliers there are multiple functionswith different sets ofcoefficients. For the purposes of this paper, however, we restrict ourselves to the multiplier 2 case with real valued's.

The wavelet expansion of a functionis of the form

The indices k and j represent translation and scaling respectively.

 

 

Iffor, then f(x) has an alternative expansion in terms of dilated scaling functions only.

 

This simple but important relation can be found by repeated application of the scaling relation (see [6]. Therefore a finite wavelet expansion of f(x) can be written solely in terms of translated scaling functions. For ease of notation we remove the J from our equations and assume that the's are all at the finest level of resolution.

The scaling functions we have used are those of Daubechies [2]. In her work, Daubechies found and exploited the link between vanishing moments of the waveletand regularity of the waveletand scaling function. We say that the wavelethas K vanishing moments if

A necessary and sufficient condition for this to hold is that integer translates of the scaling functionperfectly interpolate polynomials of degree up to K; that is, for eachthere exist constantssuch that

Daubechies [2] introduced scaling functions satisfying this property and distinguished by having the shortest possible support. The scaling function DN (where N is an even integer) will have supportand (N/2 -1) vanishing wavelet moments. In [2] and with a refined analysis in [3] Daubechies showed that there existssuch that DN hascontinuous derivatives; for small N,.

Derivatives and Connection Coefficients

In order to solve PDE's we will need to evaluate derivatives of f(x) in terms of. We first define the following shorthand for differentiation of a function:

 

In general a superscript will represent differentiation. From equation (6) we derive the Galerkin approximation of a derivative of f(x) as

 

Nowcan be approximated in terms ofas,

where

The coefficientis a 2-term connection coefficient which in its most general form is defined as

There is an analogous connection coefficient for the functionbut by use of equation (6) there is no necessity for calculating these values. It is, however, possible to derive a connection coefficient which is an integral of products of differentiated's and's from just the scaling function connection coefficients (demonstrated below). For the rest of this paper the reference to connection coefficient will always imply scaling function connection coefficient.

In general, a connection coefficient is a coefficient in the Galerkin expansion (approximation) of a function of the form

 

The resulting connection coefficient is

In the most general case we allowto be differentiated which gives rise to the n-term connection coefficient: