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Numerical Algorithms

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12-03-2020 | Original Paper

On a family of non-oscillatory subdivision schemes having regularity C^r with r > 1

Authors: Sergio Amat, Juan Ruiz, Juan C. Trillo, Dionisio F. Yáñez

Published in: Numerical Algorithms | Issue 2/2020

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Abstract

In this paper, the properties of a new family of nonlinear dyadic subdivision schemes are presented and studied depending on the conditions imposed to the mean used to rewrite the linear scheme upon which the new scheme is based. The convergence, stability, and order of approximation of the schemes of the family are analyzed in general. Also, the elimination of the Gibbs oscillations close to discontinuities is proved in particular cases. It is proved that these schemes converge towards limit functions that are Hölder continuous with exponent larger than 1. The results are illustrated with several examples.

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$L^{\infty }$stability of the limit function: Let S be a linear uniformly convergent subdivision scheme and let ϕ be its limit function defined by $\phi =S^{\infty } \delta $with $\delta _{n}=0 \quad \forall n \in \mathbb {N}\backslash \left \{0\right \}$and δ₀ = 1. The limit function ϕ is said to be $L^{\infty }$stable if:

$$ \exists A,B>0 \text{ s.t. } \forall f \in l^{\infty}(\mathbb{Z}), A||f||_{\infty} \leq ||{\sum}_{n \in \mathbb{Z}} f_{n}\phi(.-n)||_{L^{\infty}} \leq B ||f||_{\infty}, $$

where $ ||f||_{\infty }=sup_{n\in \mathbb {Z}}\left \{ |f_{n}|\right \}$.

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Title: On a family of non-oscillatory subdivision schemes having regularity Cr with r > 1
Authors: Sergio Amat
Juan Ruiz
Juan C. Trillo
Dionisio F. Yáñez
Publication date: 12-03-2020
Publisher: Springer US
Published in: Numerical Algorithms / Issue 2/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI: https://doi.org/10.1007/s11075-019-00826-3

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