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L p –Solutions of Vector Refinement Equations with General Dilation Matrix

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Abstract

The purpose of this paper is to investigate the solutions of refinement equations of the form

$$ \varphi (x) = {\sum\limits_{\alpha \in \mathbb{Z}^{s} } {a(\alpha )\varphi {\left( {Mx - \alpha } \right)},\;\;\;\;x \in \mathbb{R}^{s} } }\;, $$

where the vector of functions ϕ = (ϕ1, . . . ,ϕr)T is in (L p (ℝs))r, 0 < p ≤ ∞, a(α), α ∈ ℤs¸ is a finitely supported sequence of r× r matrices called the refinement mask, and M is an s × s integer matrix such that limn→ ∞ M –n = 0. In this article, we characterize the existence of an L p –solution of the refinement equation for 0 < p ≤ ∞. Our characterizations are based on the p–norm joint spectral radius.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China

    Song Li & Ruei Fang Hu

  2. Department of Mathematics, Shaoxing University, Shaoxing 312000, P. R. China

    Xiang Qing Wang

Authors
  1. Song Li
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  2. Ruei Fang Hu
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  3. Xiang Qing Wang
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Correspondence to Song Li.

Additional information

This project is supported by NSF of China under Grant No. 10071071 and No. 10471123

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Li, S., Hu, R.F. & Wang, X.Q. L p –Solutions of Vector Refinement Equations with General Dilation Matrix. Acta Math Sinica 22, 51–60 (2006). https://doi.org/10.1007/s10114-004-0465-5

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  • DOI: https://doi.org/10.1007/s10114-004-0465-5

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