On algebraic trigonometric integro splines

Abstract

In this paper, we present a new kind of quadratic approximation operator reproducing of both algebraic and trigonometric functions. It is called integro quadratic splines interpolant, which agree with the given integral values of a univariate real-valued function over the same intervals, rather than the functional values at the knots. Efficient approximations of fractional integrals and fractional Caputo derivatives based on this interpolant, are constructed and well studied. The general approximation error is studied too, and the super convergence property is also derived when the interval is equally partitioned. Numerical examples illustrate that our method is very effective and our quadratic algebraic trigonometric integro spline has higher approximation ability than others. K E Y W O R D Salgebraic trigonometric splines, error bound, fractional caputo derivatives, fractional integrals, integro spline quasi-interpolant

A C1 composite spline Hermite interpolant on the sphere

“…This subsection aims to derive the error bounds for algebraic trigonometric spline interpolant. For the sake of simplicity, we proceed by the same strategy described in Eddargani et al 21 and Lyche et al 22 To this end, suppose that g ∈ C 3 ( J ) and let be a differential operator. Its null space is Γ 3 , that is, , for all f ∈ Γ 3 .…”

“…It is easy to verify that this function satisfies the (C 1 )-(C 4 ) conditions. We give in Table 2 a comparison between the interpolant  𝑓 (21) and the ones presented in Schumaker and Traas 6 and Lamnii et al, 7 based on quadratic algebraic composite splines and the trigonometric composite splines of order 3. The proposed scheme is simple and computationally attractive and shows a very high precision comparing with other existing numerical methods (see Schumaker and Traas 6 and Lamnii et al 7 ).…”

“…Theorem 3. If the function f lies in  , then the associated Hermite interpolant  k,𝓁 𝑓 , given in (21), lies also in  . Moreover, we have…”

“…In order to illustrate the performance of the interpolant built above, we suggest to give two examples where, in the first one, we consider a function f on a rectangular domain, and we use the interpolant (21) to find the approximate S  𝑓 of the closed-surface S f associated with f. Then we give the error bounds of the interpolant  𝑓 as a function of k and 𝓁. While in the second one, we will deal with real data by considering a set of data points forming a 3D closed surfaces.…”

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

“…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.…”