Explicit Structures Of Compactly Supported M-band Minimum-energy Tight Frames

We know that except of the Haar wavelet, the characters of compact support, symmetric\anti-symmetry and orthogonal don't exit in a simple wavelet function at the same time, but it is very disadvantageous in the process of dealing with signals. To solve this question, we study compactly supported M-band minimum-energy tight framesψ= {ψ~1,...,ψ~N} associated with M-band refinable function with compact support. First, we discuss the relations between M-scale symbol of refinable function and M-scale symbols of frame functions, then give a precise existence criterion ofψin terms of an inequality condition on the Laurent polynomial symbols of the refinable function, finally give two constructive proofs of compactly supported M-band minimum-energy tight frames, then we compute the expression of minimum-energy tight frames associated with B-splines function and give the figures of symbols of refinable function and frame functions.