arXiv subjects
Publications and source records attributed to Vladimir S. Chelyshkov.
We introduce sequences of functions orthogonal on a finite interval: proper orthogonal rational functions, orthogonal exponential functions, orthogonal logarithmic functions, and transmuted orthogonal polynomials
We introduce an algorithm of joint approximation of a function and its first derivative by alternative orthogonal polynomials on the interval [0,1].The algorithm exhibits properties of shape preserving approximation for the function. A weak formulation of approximation is presented. An example on shape preserving extrapolation is given. The weak form is reduced for approximation on a discrete set of abscissas. Also, we introduce a new system of orthogonal functions with nice properties - structured orthogonal polynomials - and show that the system can be employed for a different kind of joint approximation of a function and its first derivative and may have property of shape preserving approximation. In addition, we show that structured orthogonal polynomials generate wavelet functions We complement these results with definition of structured semi-orthogonal polynomials and introduce wavelet basis functions.
System of alternatively orthogonalized rational functions of Jacobi type on the half line $[1, \infty)$ is defined and its properties are established. Three subsystems of proper and mixed systems of rational functions with nice properties are presented.
Sequences of orthogonal polynomials that are alternative to the Jacobi polynomials on the interval $[0,1]$ are defined and their properties are established. An $(α,β)$-parameterized system of orthogonal polynomials of the exponential function on the semi-axis $[0,\infty)$ is presented. Two subsystems of the alternative Jacobi polynomials, as well as orthogonal exponential polynomials are described. Two parameterized systems of discretely almost orthogonal functions on the interval $[0,1]$ are introduced.
A new approach for integration of the initial value problem for ordinary differential equations is suggested. The algorithm is based on approximation of the solution by a system of functions that contains orthogonal exponential polynomials.