Regularity of Refinable Functions via Non-shift-invariant Tight Frames
Alberto Viscardi (University of Milano-Bicocca)
given at NuHAG seminar (16.05.18 11:00)
The study of the smoothness of refineable functions has been investigated
deeply in the last 25 years using Fourier analysis, wavelet analysis, difference operators
and joint spectral radius techniques. However most of these techniques
works only in the shift-invariant setting, i.e. when the refineable functions considered
are just the integer shifts of a single function. It is well known (e.g. )
that analyzing the decay of wavelet coefficients is an efficient way to approximate
the Hoelder-Zygmund regularity of a function. However in the non-shift-invariant
case, e.g. when the refineable functions are entwined with particular meshes,
we are not able to compute the wavelet coefficients of the considered functions
with respect to any shift-invariant wavelet system.
In the semi-regular setting we can construct appropriate non-shift-invariant wavelet tight frames [1, 3].
In this talk we prove that, under suitable
assumptions, these frames characterize the Hoelder-Zygmund regularity similarly
to orthogonal wavelets.
 C. K. Chui, W. He, and J. Stoeckler, Nonstationary tight wavelet frames.
II. Unbounded intervals, Appl. Comput. Harmon. Anal., 18 (2005), pp. 25-66.
 Y. Meyer, Wavelets and operators, vol. 37 of Cambridge Studies in Advanced
Mathematics, Cambridge University Press, Cambridge, 1992. Translated from
the 1990 French original by D. H. Salinger.
 A. Viscardi, Semi-regular Dubuc-Deslauriers wavelet tight frames, J. Comput.
Appl. Math., (submitted).