The Journal of Geometric Analysis

, Volume 5, Issue 2, pp 181–236

Solution of two problems on wavelets

  • Pascal Auscher
Article

DOI: 10.1007/BF02921675

Cite this article as:
Auscher, P. J Geom Anal (1995) 5: 181. doi:10.1007/BF02921675

Abstract

We solve two problems on wavelets. The first is the nonexistence of a regular wavelet that generates a wavelet basis for the Hardy space ℍ2(ℝ). The second is the existence, given any regular wavelet basis for\(\mathbb{H}^2 (\mathbb{R})\), of aMulti-Resolution Analysis generating the wavelet. Moreover, we construct a regular scaling function for this Multi-Resolution Analysis. The needed regularity conditions are very mild and our proofs apply to both the orthonormal and biorthogonal situations. Extensions to more general cases in dimension 1 and higher are given. In particular, we show in dimension larger than 2 that a regular wavelet basis for\(\mathbb{L}^2 (\mathbb{R}^n )\) arises from a Multi-Resolution Analysis that is regular modulo the action of a unitary operator, which is whenn = 2 a Calderón-Zygmund operator of convolution type.

Math Subject Classification

42C15 42B20 55R10 55P15 

Key Words and Phrases

Wavelet basis Multi-Resolution Analysis Hardy space Calderón-Zygmund operator vector bundle index theory simple connectivity 

Copyright information

© Mathematica Josephina, Inc. 1995

Authors and Affiliations

  • Pascal Auscher
    • 1
  1. 1.IRMARUniversité de Rennes IRennes CedexFrance