Uniform algebraic hyperbolic spline quasi‐interpolant based on mean integral values

Abstract

In the present work, a novel spline quasi-interpolation operator reproducing both constant polynomials and algebraic hyperbolic functions is presented. The quasi-interpolant to a given function is defined from the integrals on every interval of the function to be approximated. Compared to the other existing methods, this operator does not need any additional end conditions and it is easy to be implemented without solving any system of equations. The approximation properties of the operator are theoretically analyzed and some numerical tests are presented to illustrate its performance.

C2 Cubic Algebraic Hyperbolic Spline Interpolating Scheme by Means of Integral Values

“…Our goal now is to compare the method introduced herein with the methods that use only algebraic splines, as well as those that combine the features of algebraic and hyperbolic functions. To this end, we consider the approaches presented by A. Boujraf et al in [29], D. Barrera et al in [21], J. Wu and X. Zhang in [40] and S. Eddargani et al in [34], so we implement the approaches described in [29,34,40] to be able to execute any test function, because those involved in the cited references are simple examples and one of them (exponential function) is reproduced by our approach.…”

“…These classes of splines are also known as cycloidal spaces, and they have become the subjects of a considerable amount of research [2,[20][21][22][23][24][25][26]. The algebraic hyperbolic spaces spanned by the functions 1, x, .…”

“…. ., x n−2 , cosh(x), sinh(x), for x ∈ R, have been widely considered in the literature; see [21] and references quoted. They yield the tension splines, which are extremely useful for avoiding undesirable oscillations in the interpolation curves [27].…”

“…An integro quartic spline scheme has been constructed in [33]. The authors in [21,34] provided some integro spline schemes for the case of non-polynomial splines. More recent work on the integro spline approximation is given in [35,36] In this work, a new class of integro spline approximant is introduced.…”

On nonpolynomial monotonicity‐preservingC1spline interpolation

“…From these local AT Hermite interpolants a  1 AT Hermite spline interpolant is produced, and the slopes m i could be chosen as in Reference 23 by minimizing the functional I 1 defined in (1). A straightforward calculation shows that…”