## Inequivalence of Wavelet Systems in $L_1({\Bbb R}^d)$ and ${\rm BV}({\Bbb R}^d)$

### Volume 53 / 2005

Bulletin Polish Acad. Sci. Math. 53 (2005), 25-37
MSC: 42C40,46E30, 26A45, 26B30.
DOI: 10.4064/ba53-1-4

#### Abstract

Theorems stating sufficient conditions for the inequivalence of the $d$-variate Haar wavelet system and another wavelet system in the spaces $L_1(\mathbb R^d)$ and $\mathop{\rm BV}(\mathbb R^d)$ are proved. These results are used to show that the Strömberg wavelet system and the system of continuous Daubechies wavelets with minimal supports are not equivalent to the Haar system in these spaces. A theorem stating that some systems of smooth Daubechies wavelets are not equivalent to the Haar system in $L_1(\mathbb R^d)$ is also shown.