On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergence

[摘要] We study refinable functions where the dilation factor is not always assumed to be 2. Inour investigation, the role of convolutions and refinable step functions is emphasized as aframework for understanding various previously published results. Of particular importanceis a class of polynomial factors, which was first introduced for dilation factor 2 byBerg and Plonka and which we generalise to general integer dilation factors.We obtain results on the existence of refinable functions corresponding to certain reducedmasks which generalise similar results for dilation factor 2, where our proofs do notrely on Fourier methods as those in the existing literature do.We also consider subdivision for general integer dilation factors. In this regard, we extendprevious results of De Villiers on refinable function existence and subdivision convergencein 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 theconvergence rate of the subdivision algorithm, when generalised Berg-Plonka polynomialfactors are added to the mask symbol.We obtain sufficient conditions for the occurrence of polynomial sections in refinablefunctions and construct families of related refinable functions.We also obtain results on the regularity of a refinable function in terms of the masksymbol factorisation. In this regard, we obtain much more general sufficient conditionsthan those previously published, while for dilation factor 2, we obtain a characterisation ofrefinable functions with a given number of continuous derivatives.We also study the phenomenon of subsequence convergence in subdivision, which explainssome of the behaviour that we observed in non-convergent subdivision processesduring numerical experimentation. Here we are able to establish different sets of sufficientconditions 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 correspondingresults for subdivision, since subsequence convergence is a generalisation ofsubdivision convergence. The nature of this phenomenon is such that the standard subdivisionalgorithm can be extended in a trivial manner to allow it to work in instances whereit previously failed.Lastly, we show how, for masks of length 3, explicit formulas for refinable functions canbe used to calculate the exact values of the refinable function at rational points.Various examples with accompanying figures are given throughout the text to illustrateour results.

[发布机构] Stellenbosch University