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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm198-1-1,
author = {Aramba\v si\'c Ljiljana and Damir Baki\'c and Rajna Raji\'c},
title = {Dimension functions, scaling sequences, and wavelet sets},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {1-32},
zbl = {1204.42050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm198-1-1}
}
Arambašić Ljiljana; Damir Bakić; Rajna Rajić. Dimension functions, scaling sequences, and wavelet sets. Studia Mathematica, Tome 196 (2010) pp. 1-32. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm198-1-1/