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Lacunary Interpolation by Fractal Splines with Variable Scaling Parameters

Part of:Approximations and expansionsClassical measure theory

Published online by Cambridge University Press:  20 February 2017

P. Viswanathan,
A.K.B. Chand and
K.R. Tyada
P. Viswanathan*
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India
A.K.B. Chand*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai, India
K.R. Tyada*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai, India
*
*Corresponding author. Email addresses:amritaviswa@gmail.com (P. Viswanathan), chand@iitm.ac.in (A.K.B. Chand), kurmaths86@gmail.com (K.R. Tyada)
*Corresponding author. Email addresses:amritaviswa@gmail.com (P. Viswanathan), chand@iitm.ac.in (A.K.B. Chand), kurmaths86@gmail.com (K.R. Tyada)
*Corresponding author. Email addresses:amritaviswa@gmail.com (P. Viswanathan), chand@iitm.ac.in (A.K.B. Chand), kurmaths86@gmail.com (K.R. Tyada)

Abstract

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For a prescribed set of lacunary data with equally spaced knot sequence in the unit interval, we show the existence of a family of fractal splines satisfying for v = 0, 1, … ,N and suitable boundary conditions. To this end, the unique quintic spline introduced by A. Meir and A. Sharma [SIAM J. Numer. Anal. 10(3) 1973, pp. 433-442] is generalized by using fractal functions with variable scaling parameters. The presence of scaling parameters that add extra “degrees of freedom”, self-referentiality of the interpolant, and “fractality” of the third derivative of the interpolant are additional features in the fractal version, which may be advantageous in applications. If the lacunary data is generated from a function Φ satisfying certain smoothness condition, then for suitable choices of scaling factors, the corresponding fractal spline satisfies , as the number of partition points increases.

Keywords

lacunary interpolationfractal interpolation functionvariable scaling parametersMeir-Sharma quintic splineconvergence

MSC classification

Secondary:28A80: Fractals41A05: Interpolation41A15: Spline approximation41A25: Rate of convergence, degree of approximation

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 1 , February 2017, pp. 65 - 83
Copyright
Copyright © Global-Science Press 2017 

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