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Refinable functions for dilation families

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Abstract

We consider a family of d × d matrices W e indexed by e ∈ E where (E, μ) is a probability space and some natural conditions for the family (W e ) e ∈ E are satisfied. The aim of this paper is to develop a theory of continuous, compactly supported functions \(\varphi: {{\mathbb R}}^d \to {\mathbb{C}}\) which satisfy a refinement equation of the form

$$ \varphi (x) = \int_E \sum\limits_{\alpha \in {{\mathbb Z}}^d} a_e(\alpha)\varphi\left(W_e x - \alpha\right) d\mu(e) $$

for a family of filters \(a_e : {{\mathbb Z}}^d \to {\mathbb{C}}\) also indexed by e ∈ E. One of the main results is an explicit construction of such functions for any reasonable family (W e ) e ∈ E . We apply these facts to construct scaling functions for a number of affine systems with composite dilation, most notably for shearlet systems.

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  1. ETH Zürich, Zurich, Switzerland

    Philipp Grohs

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  1. Philipp Grohs
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Correspondence to Philipp Grohs.

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Communicated by Tomas Sauer.

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Grohs, P. Refinable functions for dilation families. Adv Comput Math 38, 531–561 (2013). https://doi.org/10.1007/s10444-011-9248-6

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  • DOI: https://doi.org/10.1007/s10444-011-9248-6

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