On the best choice of nodes at interpolation of functions by even Hermitian splines

We have found exact values of deviation of even Hermitian splines on some classes of functions and pointed out the best choice of nodes at approximation of concrete functions by these splines.

On the reconstruction via Urysohn-Chlodovsky operators

The main first goal of this work is to introduce an Urysohn type Chlodovsky operators defined on positive real axis by using the Urysohn type interpolation of the given function f and bounded on every finite subinterval. The basis used in this construction are the Fréchet and Prenter Density Theorems together with Urysohn type operator values instead of the rational sampling values of the function. Afterwards, we will state some convergence results, which are generalization and extension of the theory of classical interpolation of functions to operators.

A continuant and an estimate of the remainder of the interpolating continued C-fraction

The problem of the interpolation of functions of a real variable by interpolating continued $C$-fraction is investigated. The relationship between the continued fraction and the continuant was used. The properties of the continuant are established. The formula for the remainder of the interpolating continued $C$-fraction proved. The remainder expressed in terms of derivatives of the functional continent. An estimate of the remainder was obtained. The main result of this paper is contained in the following Theorem 5:Let \(\mathcal{R}\subset \mathbb{R} \) be a compact, \(f \in \mathbf{C}^{(n+1)}(\mathcal{R})\) andthe interpolating continued $C$-fraction~($C$-ICF) of the form$$D_n(x)=\frac{P_n(x)}{Q_n(x)}=a_0+\bfrac{K}{k=1}{n}\frac{a_k(x-x_{k-1})}{1}, \ a_k \in \mathbb{R}, \; k=\overline{0,n},$$be constructed by the values the function \(f\) at nodes $X=\{x_i : x_i \in \mathcal{R}, x_i\neq x_j, i\neq j, i,j=\overline{0,n}\}.$If the partial numerators of $C$-ICF satisfy the condition of the Paydon--Wall type, that is\(0<a^* \ {\rm diam}\, \mathcal{R} \leq p\), then$\displaystyle|f(x)-D_n(x)|\leq \frac{f^*\prod\limits_{k=0}^n |x-x_k|}{(n+1)!\, \Omega_n(t)} \Big( \kappa_{n+1}(p)+\sum_{k=1}^r \tbinom{n+1}{k} (a^*)^k \sum_{i_1=1}^{n+1-2k} \kappa_{i_1}(p)\times$$\displaystyle\times \sum_{i_2=i_1+2}^{n+3-3k} \kappa_{i_2-i_1-1}(p)\dots\sum_{i_{k-1}=i_{k-2}+2}^{n-3} \kappa_{i_{k-1}-i_{k-2}-1}(p)\sum_{i_k=i_{k-1}+2}^{n-1} \kappa_{i_k-i_{k-1}-1}(p)\, \kappa_{n-i_{k}}(p)\Big),$ where $\displaystyle f^*=\max\limits_{0\leq m \leq r}\max\limits_{x \in \mathcal{R}} |f^{(n+1-m)}(x)|,$$\displaystyle \kappa_n(p)=\cfrac{(1\!+\!\sqrt{1+4p})^n\!-\!(1\!-\!\sqrt{1+4p})^n}{2^n\, \sqrt{1+4p}},$\ $a^*=\max\limits_{2\leqslant i \leqslant n}|a_i|,$\ $p=t(1-t),\;t\in(0;\tfrac{1}{2}], \; r=\big[\tfrac{n}{2}\big].$

Orthogonal structure on a quadratic curve

Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. We discuss applications to the Fourier extension problem, interpolation of functions with singularities or near singularities and the solution of Schrödinger’s equation with nondifferentiable or nearly nondifferentiable potentials.

The order of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions

We show that the error of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions comprising $$$N$$$ elements has the order $$$N^{-2}$$$ as $$$N \rightarrow \infty$$$.