scholarly journals Rational interpolation of the function |x|α by an extended system of Chebyshev – Markov nodes

2020 ◽  
Vol 55 (4) ◽  
pp. 391-405
Author(s):  
E. A. Rovba ◽  
V. Yu. Medvedeva
Keyword(s):  
Rational Functions ◽  
Rational Interpolation ◽  
Asymptotic Equality ◽  
Fixed Number ◽  
Theoretical Research ◽  
Extended System ◽  
Uniform Approximations ◽  
Upper Estimate ◽  
Interpolation Functions ◽  
Interpolation Nodes

In this paper, we study the approximations of a function |x|α, α > 0 by interpolation rational Lagrange functions on a segment [–1,1]. The zeros of the even Chebyshev – Markov rational functions and a point x = 0 are chosen as the interpolation nodes. An integral representation of an interpolation remainder and an upper bound for the considered uniform approximations are obtained. Based on them, a detailed study is made:a) the polynomial case. Here, the authors come to the famous asymptotic equality of M. N. Hanzburg;b) at a fixed number of geometrically different poles, the upper estimate is obtained for the corresponding uniform approximations, which improves the well-known result of K. N. Lungu;c) when approximating by general Lagrange rational interpolation functions, the estimate of uniform approximations is found and it is shown that at the ends of the segment [–1,1] it can be improved.The results can be applied in theoretical research and numerical methods. 

scholarly journals On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions

2020 ◽  
pp. 6-27
Author(s):  
Pavel G. Patseika ◽  
Yauheni A. Rouba ◽  
Kanstantin A. Smatrytski
Keyword(s):  
Integral Operator ◽  
Uniform Approximation ◽  
Asymptotic Expression ◽  
Classical Problem ◽  
Rational Functions ◽  
Fixed Number ◽  
Approximation Properties ◽  
Elementary Functions ◽  
Rational Chebyshev ◽  
Rational Integral

The purpose of this paper is to construct an integral rational Fourier operator based on the system of Chebyshev – Markov rational functions and to study its approximation properties on classes of Markov functions. In the introduction the main results of well-known works on approximations of Markov functions are present. Rational approximation of such functions is a well-known classical problem. It was studied by A. A. Gonchar, T. Ganelius, J.-E. Andersson, A. A. Pekarskii, G. Stahl and other authors. In the main part an integral operator of the Fourier – Chebyshev type with respect to the rational Chebyshev – Markov functions, which is a rational function of order no higher than n is introduced, and approximation of Markov functions is studied. If the measure satisfies the following conditions: suppμ = [1, a], a > 1, dμ(t) = ϕ(t)dt and ϕ(t) ἆ (t − 1)α on [1, a] the estimates of pointwise and uniform approximation and the asymptotic expression of the majorant of uniform approximation are established. In the case of a fixed number of geometrically distinct poles in the extended complex plane, values of optimal parameters that provide the highest rate of decreasing of this majorant are found, as well as asymptotically accurate estimates of the best uniform approximation by this method in the case of an even number of geometrically distinct poles of the approximating function. In the final part we present asymptotic estimates of approximation of some elementary functions, which can be presented by Markov functions.

SENSITIVITY ANALYSIS FOR HIDDEN VARIABLE FRACTAL INTERPOLATION FUNCTIONS AND THEIR MOMENTS

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 161-170
Author(s):  
HONG-YONG WANG
Keyword(s):  
Sensitivity Analysis ◽  
Iterated Function System ◽  
Fractal Interpolation ◽  
Hidden Variable ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
The Difference ◽  
The Moment ◽  
Upper Estimate ◽  
Interpolation Functions

The sensitivity analysis for a class of hidden variable fractal interpolation functions (HVFIFs) and their moments is made in the work. Based on a vector valued iterated function system (IFS) determined, we introduce a perturbed IFS and investigate the relations between the two HVFIFs generated by the IFS determined and its perturbed IFS, respectively. An explicit expression for the difference between the two HVFIFs is presented, from which, we show that the HVFIFs are not sensitive to a small perturbation in IFSs. Furthermore, we compute the moment integrals of the HVFIFs and discuss the error of moments of the two HVFIFs. An upper estimate for the error is obtained.

Spectral Estimates for Towers of Noncompact Quotients

1999 ◽  
Vol 51 (2) ◽  
pp. 266-293 ◽  
Author(s):  
Anton Deitmar ◽  
Werner Hoffman
Keyword(s):  
Analogous Result ◽  
Congruence Subgroup ◽  
Kernel Method ◽  
Linear Algebraic Group ◽  
Fixed Number ◽  
Principal Congruence ◽  
Congruence Subgroups ◽  
Spectral Estimates ◽  
The Family ◽  
Upper Estimate

AbstractWe prove a uniform upper estimate on the number of cuspidal eigenvalues of the Γ-automorphic Laplacian below a given bound when Γ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each Γ in the family is assumed to contain a principal congruence subgroup whose index in Γ does not exceed a fixed number. The bound we prove depends linearly on the covolume of Γ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice Γ.

Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products

2001 ◽  
Vol 21 (2) ◽  
pp. 563-603 ◽  
Author(s):  
HIROKI SUMI
Keyword(s):  
Hausdorff Dimension ◽  
Riemann Sphere ◽  
Julia Set ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Empty Interior ◽  
Skew Products ◽  
Wandering Domain ◽  
Upper Estimate ◽  
Interior Points

We consider dynamics of sub-hyperbolic and semi-hyperbolic semigroups of rational functions on the Riemann sphere and will show some no wandering domain theorems. The Julia set of a rational semigroup in general may have non-empty interior points. We give a sufficient condition that the Julia set has no interior points. From some information about forward and backward dynamics of the semigroup, we consider when the area of the Julia set is equal to zero or an upper estimate of the Hausdorff dimension of the Julia set.

scholarly journals Rational interpolation to |x| at the Chebyshev nodes

1997 ◽  
Vol 56 (1) ◽  
pp. 81-86 ◽  
Author(s):  
Lev Brutman ◽  
Eli Passow
Keyword(s):  
Polynomial Interpolation ◽  
Rational Interpolation ◽  
Chebyshev Nodes ◽  
Rational Interpolants ◽  
Equidistant Points ◽  
Special Case ◽  
Interpolation Nodes

Recently the authors considered Newman-type rational interpolation to |x| induced by arbitrary sets of interpolation nodes and showed that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to |x|. In the present paper we consider the special case of the Chebyshev nodes which are known to be very efficient for polynomial interpolation. It is shown that, in contrast to the polynomial case, the approximation of |x| induced by rational interpolation at the Chebyshev nodes has the same order as rational interpolation at equidistant points.

Sign in / Sign up

Export Citation Format

Share Document