**ABSTRACT:**

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. [2])

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.

[1] C. K. Chui, W. He, and J. Stoeckler, Nonstationary tight wavelet frames.

II. Unbounded intervals, Appl. Comput. Harmon. Anal., 18 (2005), pp. 25-66.

[2] 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.

[3] A. Viscardi, Semi-regular Dubuc-Deslauriers wavelet tight frames, J. Comput.

Appl. Math., (submitted).