On the correspondence of Haar-type wavelet series in one and several dimensions

Abstract

In this article we show that, given a Haar-type wavelet series { f_j } in with respect to a dilation matrix A and tile Q, we can construct another Haar-type wavelet series in ℝ and tile in such a way that the stochastic processes defined by { f_j } and on the probability spaces and , respectively, have the same finite-dimensional distributions. Thus, many properties of { f_j } can be deduced from . This is a technique that, to our knowledge, has not been used in the literature. We show the power of this method by extending some known results for Haar series in ℝ to Haar-type wavelet series in with respect to a dilation matrix A, using simple proofs.

Keywords:

• Haar-type wavelets
• equivalent stochastic processes
• representation problem
• convergence theorems

AMS Subject Classifications::

• 42C40
• 60G46

Acknowledgements

The authors would like to thank Richard F. Gundy for reading the manuscript and for his valuable comments and suggestions.