Dimension functions, scaling sequences, and wavelet sets

Arambašić Ljiljana¹ ; Damir Bakić¹ ; Rajna Rajić²

¹ Department of Mathematics University of Zagreb Bijenička c. 30 10000 Zagreb, Croatia
² Faculty of Mining, Geology and Petroleum Engineering University of Zagreb Pierottijeva 6 10000 Zagreb, Croatia

Studia Mathematica, Tome 198 (2010) no. 1, pp. 1-32

Résumé

The paper is a continuation of our study of dimension functions of orthonormal wavelets on the real line with dyadic dilations. The main result of Section 2 is Theorem 2.8 which provides an explicit reconstruction of the underlying generalized multiresolution analysis for any MSF wavelet. In Section 3 we reobtain a result of Bownik, Rzeszotnik and Speegle which states that for each dimension function $D$ there exists an MSF wavelet whose dimension function coincides with $D$. Our method provides a completely new explicit construction of an admissible generalized multiresolution analysis (and, a posteriori, of a wavelet) from an arbitrary dimension function. Several examples are included.

Détail

DOI : 10.4064/sm198-1-1

Keywords: paper continuation study dimension functions orthonormal wavelets real line dyadic dilations main result section nbsp theorem which provides explicit reconstruction underlying generalized multiresolution analysis msf wavelet section reobtain result bownik rzeszotnik speegle which states each dimension function there exists msf wavelet whose dimension function coincides method provides completely explicit construction admissible generalized multiresolution analysis posteriori wavelet arbitrary dimension function several examples included