Positivity Preserving Rational Cubic Trigonometric Fractal Interpolation Functions

In this paper, we propose a family of l(1)-rational cubic trigonometric fractal interpolation function (RCTFIF) to preserve positivity inherent in a set of data. The proposed RCTFIF is a generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form p(i)(theta)/q(i)(theta), where p(i) (theta) and q(i) (theta) are cubic trigonometric polynomials. The RCTFIF involves a scaling factor and four shape parameters in each subinterval. The convergence of the RCTFIF towards the original function is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the l(1)-RCTFIF so that the proposed RCTFIF preserves the positivity property of the given positive data set. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval, and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The effects of the scaling factors and shape parameters on the RCTFIF and its first derivative are illustrated graphically.