Effective and accurate high-degree spline interpolation is still a challenging task in today’s applications. Higher degree spline interpolation is not so commonly used, because it requires the knowledge of higher order derivatives at the nodes of a function on a given mesh.

In this article, our goal is to demonstrate the continuity of the piecewise polynomials and their derivatives at the connecting points, obtained with a method initially developed by Beaudoin (1998, 2003) and Beauchemin (2003). This new method, involving the discrete Fourier transform (DFT/FFT), leads to higher degree spline interpolation for equally spaced data on an interval $[0,T]$. To do this, we analyze the singularities that may occur when solving the system of equations that enables the construction of splines of any degree. We also note an important difference between the odd-degree splines and even-degree splines. These results prove that Beaudoin and Beauchemin’s method leads to spline interpolation of any degree and that this new method could eventually be used to improve the accuracy of spline interpolation in traditional problems.

A new minimization principle for the Poisson equation using two variables – the solution and the gradient of the solution – is introduced. This principle allows us to use any conforming finite element spaces for both variables, where the finite element spaces do not need to satisfy the so-called inf–sup condition. A numerical example demonstrates the superiority of this approach.

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.

In this paper, we prove that for $\ell \,=\,1$ or 2 the rate of best $\ell $ - monotone polynomial approximation in the ${{L}_{p}}$ norm $\left( 1\,\le \,p\,\le \,\infty \right)$ weighted by the Jacobi weight ${{w}_{\alpha ,\,\beta }}\left( x \right)\,:=\,{{\left( 1\,+\,x \right)}^{\alpha }}{{\left( 1\,-\,x \right)}^{\beta }}$ with $\alpha ,\,\beta \,>\,-1/p$ if $p\,<\,\infty $ , or $\alpha ,\,\beta \,\ge \,0$ if $p\,=\,\infty $ , is bounded by an appropriate $\left( \ell \,+\,1 \right)$ -st modulus of smoothness with the same weight, and that this rate cannot be bounded by the $\left( \ell \,+\,2 \right)$ -nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.

For the approximation in Lp-norm, we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots. For p = 1, ∞, we obtain its values. By these results we know that for the Sobolev classes, the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths. At the same time, the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.

This paper proposes a new family of symmetric $4$-point ternary non-stationary subdivision schemes  that can generate the limit curves of $C^3$ continuity. The continuity of this scheme is higher than the existing 4-point ternary approximating schemes. The proposed scheme has been developed using trigonometric B-spline basis functions and analyzed using the theory of asymptotic equivalence. It has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines as well. Some graphical and numerical examples are being considered, by choosing an appropriate tension parameter $0<\alpha <\pi /3 $, to show the usefulness of the proposed scheme. Moreover, the Hölder regularity and the reproduction property are also being calculated.

In this paper we derive local error estimates for radial basis function interpolation on the unit sphere . More precisely, we consider radial basis function interpolation based on data on a (global or local) point set for functions in the Sobolev space with norm , where s>1. The zonal positive definite continuous kernel ϕ, which defines the radial basis function, is chosen such that its native space can be identified with . Under these assumptions we derive a local estimate for the uniform error on a spherical cap S(z;r): the radial basis function interpolant ΛXf of satisfies , where h=hX,S(z;r) is the local mesh norm of the point set X with respect to the spherical cap S(z;r). Our proof is intrinsic to the sphere, and makes use of the Videnskii inequality. A numerical test illustrates the theoretical result.

Positive, copositive, onesided and intertwining (co-onesided) polynomial and spline approximations of functions are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained.

Using a new B-spline basis due to Dahmen, Micchelli and Seidel, we construct a univariate spline approximation operator of Bernstein-Schoenberg type. We show that it shares all the shape preserving properties of the usual Bernstein-Schoenberg operator and we derive a Voronovskaya type asymptotic error estimate.

The problem of linear independence of the integer translates of μ * B, where μ is a compactly supported distribution and B is an exponential box spline, is considered in this paper. The main result relates the linear independence issue with the distribution of the zeros of the Fourier-Laplace transform, of μ on certain linear manifolds associated with B. The proof of our result makes an essential use of the necessary and sufficient condition derived in [12]. Several applications to specific situations are discussed. Particularly, it is shown that if the support of μ is small enough then linear independence is guaranteed provided that does not vanish at a certain finite set of critical points associated with B. Also, the results here provide a new proof of the linear independence condition for the translates of B itself.

It is proved that a regular piecewise n-convex function differs from an n-convex function only by a polynomial spline of degree n - 1. The argument is given in terms of Peano and de la Vallée Poussin derivatives.