Rocky Mountain Journal of Mathematics

Constrained shape preserving rational cubic fractal interpolation functions

A.K.B. Chand and K.R. Tyada

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In this paper, we discuss the construction of $\mathcal {C}^1$-rational cubic fractal interpolation function (RCFIF) and its application in preserving the constrained nature of a given data set. The $\mathcal {C}^1$-RCFIF is the fractal design of the traditional rational cubic interpolant of the form ${p_i(\theta )}/{q_i(\theta )}$, where $p_i(\theta )$ and $q_i(\theta )$ are cubic and quadratic polynomials with three tension parameters. We present the error estimate of the approximation of RCFIF with the original function in $\mathcal {C}^k[x_1,x_n]$, $k=1,3$. When the data set is constrained between two piecewise straight lines, we derive the sufficient conditions on the IFS parameters of the RCFIF so that it lies between those two lines. Numerical examples are given to support the theoretical results.

Article information

Rocky Mountain J. Math., Volume 48, Number 1 (2018), 75-105.

First available in Project Euclid: 28 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A80: Fractals [See also 37Fxx] 37C25: Fixed points, periodic points, fixed-point index theory 41A30: Approximation by other special function classes 41A55: Approximate quadratures 42A15: Trigonometric interpolation

Iterated function systems fractal interpolation convergence analysis bounding Cauchy remainder Peano-kernel theorem constrained data interpolation positivity


Chand, A.K.B.; Tyada, K.R. Constrained shape preserving rational cubic fractal interpolation functions. Rocky Mountain J. Math. 48 (2018), no. 1, 75--105. doi:10.1216/RMJ-2018-48-1-75.

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