Smooth Parseval frames for $L^2(\mathbb{R})$ and generalizations to $L^2(\mathbb{R}^d)$

TL;DR

This paper constructs smooth Parseval frame wavelets in $L^2( ^d)$, addressing the challenges of smoothing wavelet set wavelets in higher dimensions and exploring their properties and limitations.

Contribution

It introduces a new construction of wavelet sets in higher dimensions that can be smoothed and analyzes the limitations of convolutional smoothing for wavelet set wavelets.

Findings

Smoothing wavelet set wavelets in higher dimensions is non-trivial.

A new method for constructing smoothed wavelet sets in $ ^d$ is presented.

Certain functions cannot be obtained through convolutional smoothing of wavelet set wavelets.

Abstract

Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in $L^2(\mathbb{R}^d)$ which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over $\mathbb{R}^d$, $d >1$. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Han's proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in $\hat{\mathbb{R}}^d$ which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.