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Semiorthogonal Multiresolution Analysis Frames in Higher Dimensions

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Abstract

Benedetto and Li proposed the theory of frame multiresolution analysis (FMRA) in one dimension. This paper generalizes Benedetto and Li’s theory of FMRA to higher dimensions with arbitrary integral expansive matrix dilations, and gives two necessary and sufficient conditions to characterize (semiorthogonal) multiresolution analysis frames for L 2(ℝn). One of the two conditions is put on the frame scaling function and the low-pass and high-pass filters only. Multiresolution analysis Parseval frames are also characterized. The theory is implemented to a bidimensional example with the nonseparable quincunx dilation \(\scriptsize\bigl(\begin{array}{c@{\ }c}1&-1\\1&1\end{array}\bigr)\) , along with its potential application in subband signal processing.

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  1. Department of Mathematics, Tianjin Polytechnic University, 63 Chenglin Street, Hedong District, Tianjin, 300160, China

    Xiaojiang Yu

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  1. Xiaojiang Yu
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Yu, X. Semiorthogonal Multiresolution Analysis Frames in Higher Dimensions. Acta Appl Math 111, 257–286 (2010). https://doi.org/10.1007/s10440-009-9544-z

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