Construction of Wavelet and Gabor's Parseval Frames

Abstract

A new way to build wavelet and Gabor's Parseval frames for $L^2(\R^d)$ is shown in this paper. In the first case the construction is done using an expansive matrix $B$, together with only one function $h\in L^2(\R^d)$.\ In the second one, we work with a function $g\in L^2(\R^d)$ and two invertible matrixes $B$ and $C$, with the condition that $C^t\mathbb{Z}^d\subset\mathbb{Z}^d$. The only requirement for $h$ and $g$ is that they have to be supported in a set $Q$, such that the measure of $Q$ is finite and positive. $Q$ has diameter lower than $1$, and its border has null measurement. In addition, $\{B^jQ\}_{j\in \mathbb{Z}}$ $(\{T_{Bj}Q\}_{j\in \mathbb{Z}^d})$ is a covering of $\R^d\backslash\{0\}$ $(\R^d)$, and $\{h(B^j)\}_{j\in\mathbb{Z}}$
$(\{T_{Bj}g\}_{j\in\mathbb{Z}^d})$ is a Riesz Partition of unity for $L^2(\R^d)$. Then, it is possible to obtain the Parseval frames with good localization properties, after adding conditions to $ h (g)$. At the end, we show two examples of building of wavelet Parseval frames and Gabor's Parseval frames with a good decay, as required.