Higher-Dimensional Wavelet Systems

Howard L. Resnikoff () and Raymond O. Wells ()
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Howard L. Resnikoff: Future WAVE Inc.
Raymond O. Wells: Rice University, Department of Mathematics

Chapter 7 in Wavelet Analysis, 1998, pp 165-187 from Springer

Abstract: Abstract In this chapter, we construct compactly supported (orthonormal) systems of wavelets for L2(R d ), d > 1. Each system is of the form 7.1 $$\left\{ {{\psi ^s}\left( {{M^k}x - p} \right):1 \le s \le m - 1;k \in Z;p \in \Lambda } \right\},$$ where Λ is a rank d lattice subgroup of R d , M is a strictly expansive linear transformation with MΛ ⊂ Λ, and m = |detM|. The fundamental wavelets, ψ s , 1 ≤ s ≤ m −1 are defined by functional equations having the form 7.2 $${\psi ^s}\left( x \right) = \sum\limits_{p \in \Lambda } {a_p^s\varphi \left( {Mx - p} \right),\,\,1 \le s \le m - 1,} $$ where a p s are complex-valued functions having finite support on Λ and satisfy certain algebraic conditions, and the scaling function ϕ satisfies the scaling equation $$\varphi \left( x \right) = \sum\limits_{p \in \Lambda } {a_p^0\varphi \left( {Mx - p} \right).}$$

Keywords: Discrete Fourier Transform; Scaling Function; Trigonometric Polynomial; Wavelet Function; Fundamental Domain (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1007/978-1-4612-0593-7_7

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