## On the regularity of refinable functions

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http://hdl.handle.net/10019.1/2881##### Collections

##### Abstract

This work studies the regularity (or smoothness) of continuous finitely supported refinable
functions which are mainly encountered in multiresolution analysis, iterative interpolation
processes, signal analysis, etc. Here, we present various kinds of sufficient conditions on
a given mask to guarantee the regularity class of the corresponding refinable function.
First, we introduce and analyze the cardinal B-splines Nm, m ∈ N. In particular, we
show that these functions are refinable and belong to the smoothness class Cm−2(R). As
a generalization of the cardinal B-splines, we proceed to discuss refinable functions with
positive mask coefficients. A standard result on the existence of a refinable function in
the case of positive masks is quoted. Following [13], we extend the regularity result in
[25], and we provide an example which illustrates the fact that the associated symbol to
a given positive mask need not be a Hurwitz polynomial for its corresponding refinable
function to be in a specified smoothness class. Furthermore, we apply our regularity result
to an integral equation.
An important tool for our work is Fourier analysis, from which we state some standard
results and give the proof of a non-standard result. Next, we study the H¨older regularity
of refinable functions, whose associated mask coefficients are not necessarily positive, by
estimating the rate of decay of their Fourier transforms. After showing the embedding of
certain Sobolev spaces into a H¨older regularity space, we proceed to discuss sufficient conditions
for a given refinable function to be in such a H¨older space. We specifically express
the minimum H¨older regularity of refinable functions as a function of the spectral radius
of an associated transfer operator acting on a finite dimensional space of trigonometric
polynomials.
We apply our Fourier-based regularity results to the Daubechies and Dubuc-Deslauriers
refinable functions, as well as to a one-parameter family of refinable functions, and then
compare our regularity estimates with those obtained by means of a subdivision-based
result from [28]. Moreover, we provide graphical examples to illustrate the theory developed.

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