New Tight Wavelet Frame Constructions Sharing Responsibility

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.