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Inhomogeneous refinement equations

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Abstract

Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications also include an inhomogeneous forcing term F(t). We develop here a part of the existence theory for the inhomogeneous refinement equation

$$\phi (t) = \sum\limits_{k \in \mathbb{Z}} {a(k)\phi (2t - k) + F(t)}$$

where a (k) is a finite sequence and F is a compactly supported distribution on ℝ.

The existence of compactly supported distributional solutions to an inhomogeneous refinement equation is characterized in terms of conditions on the pair (a, F).

To have Lp solutions from F ∈ Lp(ℝ), we construct by the cascade algorithm a sequence of functions φ0 ∈ Lp(ℝ) from a compactly supported initial function ℝ as

$$\phi _n (t) = \sum\limits_{k \in \mathbb{Z}} {a(k)\phi _{n - 1} (2t - k) + F(t)}$$

A necessary and sufficient condition for the sequence {φn} to converge in Lp(ℝ)(1 ≤ p ≤ ∞) is given by the p-norm joint spectral radius of two matrices derived from the mask a. A convexity property of the p-norm joint spectral radius (1 ≤ p ≤ ∞) is presented.

Finally, the general theory is applied to some examples and multiple refinable functions.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA

    Gilbert Strang

  2. Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

    Ding-Xuan Zhou

Authors
  1. Gilbert Strang
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  2. Ding-Xuan Zhou
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Additional information

Communicated by John J. Benedetto

Acknowledgements and Notes. Research supported in part by Research Grants Council and City University of Hong Kong under Grants #9040281, 9030562, 7000741.

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Strang, G., Zhou, DX. Inhomogeneous refinement equations. The Journal of Fourier Analysis and Applications 4, 733–747 (1998). https://doi.org/10.1007/BF02479677

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  • DOI: https://doi.org/10.1007/BF02479677

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