Internal structure of the multiresolution analyses defined by the unitary extension principle (Q999271)

This paper investigates the internal structure of the multiresolution analysis of \(L^2(\mathbb{R}^d)\) associated with a wavelet tight frame constructed by the unitary extension principle. Let \(V_0\) denote the closed linear span of the integer shifts of the scaling refinable function and \(V_1\) be the dyadic dilation of \(V_0\). Let \(W_0\) denote the closed linear span of the integer shifts of all the wavelet functions. This paper characterizes the conditions such that \(V_1=W_0\), \(V_1\) is the sum of \(V_0\) and \(W_0\), and \(V_1\) is a direct sum of \(V_0\) and \(W_0\), respectively. A more detailed analysis for trigonometric filters is also provided.