research-article

Optimal Hölder-Zygmund exponent of semi-regular refinable functions

Authors:

Costanza Conti,

Joachim Stöckler,

Alberto ViscardiAuthors Info & Claims

Volume 251, Issue C

https://doi.org/10.1016/j.jat.2019.105340

Published: 01 March 2020 Publication History

Abstract

The regularity of refinable functions has been investigated deeply in the past 25 years using Fourier analysis, wavelet analysis, restricted and joint spectral radii techniques. However the shift-invariance of the underlying regular setting is crucial for these approaches. We propose an efficient method based on wavelet tight frame decomposition techniques for estimating Hölder-Zygmund regularity of univariate semi-regular refinable functions generated, e.g., by subdivision schemes defined on semi-regular meshes t = − h ℓ N ∪ { 0 } ∪ h r N, h ℓ, h r ∈ ( 0, ∞ ). To ensure the optimality of this method, we provide a new characterization of Hölder-Zygmund spaces based on suitable irregular wavelet tight frames. Furthermore, we present proper tools for computing the corresponding frame coefficients in the semi-regular setting. We also propose a new numerical approach for estimating the optimal Hölder-Zygmund exponent of refinable functions which is more efficient than the linear regression method. We illustrate our results with several examples of known and new semi-regular subdivision schemes with a potential use in blending curve design.

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Index Terms

Optimal Hölder-Zygmund exponent of semi-regular refinable functions
1. Mathematics of computing
  1. Mathematical analysis
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

Index terms have been assigned to the content through auto-classification.

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Published In

cover image Journal of Approximation Theory

Journal of Approximation Theory Volume 251, Issue C

Mar 2020

244 pages

ISSN:0021-9045

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Elsevier Inc.

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Academic Press, Inc.

United States

Publication History

Published: 01 March 2020

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