2018/07/28 by Charina, Maria, Conti, Costanza, Romani, Lucia +2
#42C15 #42C40 #65D17 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1807.10909
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_ℓℕ∪\0\∪ hrℕ, h_ℓ,hr ∈ (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.