New Tight Wavelet Frame Constructions Sharing Responsibility

TL;DR

This paper introduces new methods for constructing tight wavelet frames that distribute the design responsibility between refinable functions and mother wavelets, offering solutions to existing limitations and the SOS problem.

Contribution

The paper proposes novel tight wavelet frame construction methods that share responsibility between refinable functions and mother wavelets, improving flexibility and addressing the SOS problem.

Findings

New construction methods demonstrated with examples.

Methods effectively distribute construction responsibility.

Addresses limitations of existing TWF construction approaches.

Abstract

Tight wavelet frames (TWFs) in \(L^2(\mathbb{R}^n)\) are versatile, and are practically useful due to their perfect reconstruction property. Nevertheless, existing TWF construction methods exhibit limitations, including a lack of specific methods for generating mother wavelets in extension-based construction, and the necessity to address the sum of squares (SOS) problem even when specific methods for generating mother wavelets are provided in SOS-based construction. Many TWF constructions begin with a given refinable function. However, this approach places the entire burden on finding suitable mother wavelets. In this paper, we introduce TWF construction methods that spread the burden between both types of functions: refinable functions and mother wavelets. These construction methods offer an alternative approach to addressing the SOS problem. We present examples to illustrate our construction methods.