Abstract
A nonlinear quaternary 4-point approximating subdivision scheme is presented. It is based on a nonlinear perturbation of the quaternary subdivision scheme studied in Mustafa and Khan (Abstr Appl Anal 2009:14, 2009). The convergence of the scheme and the regularity of the limit function are analyzed. It is shown that the Gibbs phenomenon, classical in linear schemes, is eliminated. The stability, that in the nonlinear case is not a consequence of the convergence, is also established. Up to our knowledge, this is the first subdivision scheme of regularity larger than three, avoiding Gibbs oscillations and for which the stability is rigorously obtained. All these properties are very important for real applications.
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Notes
For \(0<\alpha <1\), \(f \in C^\alpha (\mathbb R )\) iff \(f\) is bounded and \(\exists C>0\) such that \(\forall x,y \in \mathbb R , |f(x)-f(y)|\le C|x-y|^\alpha \). For \(\alpha >1\), \(f \in C^\alpha (\mathbb R )\) iff \(f^{(\left[ \alpha \right] )}\) is bounded and \(f^{(\left[ \alpha \right] )} \in C^{(\alpha -\left[ \alpha \right] )}\) where \(\left[ \alpha \right] \) is the integer part of \(\alpha \).
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S. Amat was supported in part by the Spanish grants MICINN-FEDER MTM2010-17508 and 08662/PI/08.
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Amat, S., Liandrat, J. On a nonlinear 4-point quaternary approximating subdivision scheme eliminating the Gibbs phenomenon. SeMA 62, 15–25 (2013). https://doi.org/10.1007/s40324-013-0006-1
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DOI: https://doi.org/10.1007/s40324-013-0006-1
Keywords
- Quaternary subdivision scheme
- Nonlinear harmonic mean
- Convergence
- Regularity
- Stability
- Gibbs oscillations