Tight wavelet frames (TWFs) in $L^2(\mathbb{R}^n)$ are versatile and practical structures that provide the 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. It is a common practice for current TWF constructions to 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 circumvent the SOS problem while providing specific techniques for generating mother wavelets. We present examples to illustrate our construction methods.

翻译：在$L^2(\mathbb{R}^n)$空间中的紧小波框架（TWFs）是具有完美重构特性的多功能实用结构。然而，现有TWF构建方法存在若干局限性：在基于扩展的构建中缺乏生成母小波的具体方法；而在基于平方和（SOS）的构建中，即使提供了生成母小波的具体方法，仍需解决SOS问题。当前TWF构建通常从给定可细化函数出发，但这种方法将全部负担置于寻找合适母小波的过程。本文提出在可细化函数与母小波两类函数间分担责任的TWF构建方法。这些构建方法提供了规避SOS问题的替代途径，同时给出了生成母小波的具体技术。我们通过示例阐明所提出的构建方法。