| Summary: | Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. === We study refinable functions where the dilation factor is not always assumed to be 2. In
our investigation, the role of convolutions and refinable step functions is emphasized as a
framework for understanding various previously published results. Of particular importance
is a class of polynomial factors, which was first introduced for dilation factor 2 by
Berg and Plonka and which we generalise to general integer dilation factors.
We obtain results on the existence of refinable functions corresponding to certain reduced
masks which generalise similar results for dilation factor 2, where our proofs do not
rely on Fourier methods as those in the existing literature do.
We also consider subdivision for general integer dilation factors. In this regard, we extend
previous results of De Villiers on refinable function existence and subdivision convergence
in the case of positive masks from dilation factor 2 to general integer dilation factors.
We also obtain results on the preservation of subdivision convergence, as well as on the
convergence rate of the subdivision algorithm, when generalised Berg-Plonka polynomial
factors are added to the mask symbol.
We obtain sufficient conditions for the occurrence of polynomial sections in refinable
functions and construct families of related refinable functions.
We also obtain results on the regularity of a refinable function in terms of the mask
symbol factorisation. In this regard, we obtain much more general sufficient conditions
than those previously published, while for dilation factor 2, we obtain a characterisation of
refinable functions with a given number of continuous derivatives.
We also study the phenomenon of subsequence convergence in subdivision, which explains
some of the behaviour that we observed in non-convergent subdivision processes
during numerical experimentation. Here we are able to establish different sets of sufficient
conditions for this to occur, with some results similar to standard subdivision convergence,
e.g. that the limit function is refinable. These results provide generalisations of the corresponding
results for subdivision, since subsequence convergence is a generalisation of
subdivision convergence. The nature of this phenomenon is such that the standard subdivision
algorithm can be extended in a trivial manner to allow it to work in instances where
it previously failed.
Lastly, we show how, for masks of length 3, explicit formulas for refinable functions can
be used to calculate the exact values of the refinable function at rational points.
Various examples with accompanying figures are given throughout the text to illustrate
our results.
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