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Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix. (English) Zbl 1021.42020
For any d×d dilation matrix M it is proved how to construct compactly supported tight wavelet frames and orthonormal wavelet bases having exponential decay; the bases have the form ψ j,k =|detM| j/2 ψ(M j ·-k),j,k d for some functions ψL 2 ( d ); they are derived from refinable functions ϕ, in the sense that they have the form ψ=|detM| k d b k ϕ(M·-k) for some sequence {b k } k d . One of the main results is as follows. Given any positive integer r, there exists a collection Ψ of at most (3/2) d |detM| functions in C r ( d ), derived from a refinable function with compact support, such that Ψ has vanishing moments of order r and generates a tight wavelet frame for L 2 ( d ).
42C40Wavelets and other special systems