scholarly journals On rational Abel – Poisson means on a segment and approximations of Markov functions

2021 ◽  
pp. 6-24
Author(s):  
Pavel G. Patseika ◽  
Yauheni A. Rouba
Keyword(s):  
Integral Operator ◽  
Integral Representation ◽  
Asymptotic Expression ◽  
Fixed Number ◽  
Asymptotic Estimates ◽  
Markov System ◽  
Uniform Approximations ◽  
Optimal Values ◽  
Fourier Type ◽  
Rational Integral

Approximations on the segment [−1, 1] of Markov functions by Abel – Poisson sums of a rational integral operator of Fourier type associated with the Chebyshev – Markov system of algebraic fractions in the case of a fixed number of geometrically different poles are investigated. An integral representation of approximations and an estimate of uniform approximations are found. Approximations of Markov functions in the case when the measure µ satisfies the conditions suppµ = [1, a], a > 1, dµ(t) = φ(t)dt and φ(t) ≍ (t − 1)α on [1, a], a are studied and estimates of pointwise and uniform approximations and the asymptotic expression of the majorant of uniform approximations are obtained. The optimal values of the parameters at which the majorant has the highest rate of decrease are found. As a corollary, asymptotic estimates of approximations on the segment [−1, 1] are given by the method of rational approximation of some elementary Markov functions under study.

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.

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 Random Walks Reaching Against all Odds the other Side of the Quarter Plane

2013 ◽  
Vol 50 (1) ◽  
pp. 85-102 ◽  
Author(s):  
Johan S. H. van Leeuwaarden ◽  
Kilian Raschel
Keyword(s):  
Random Walk ◽  
Integral Representation ◽  
Nearest Neighbor ◽  
Asymptotic Expression ◽  
Exclusion Process ◽  
Vertical Axis ◽  
The Other ◽  
Asymmetric Exclusion Process ◽  
Quarter Plane ◽  
Asymptotic Evaluation

For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state (i0,j0), we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive a certain integral representation for the probability of this event, and an asymptotic expression for the case when i0 becomes large, a situation in which the event becomes highly unlikely. The integral representation follows from the solution of a boundary value problem and involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model, and the asymmetric exclusion process.

Asymptotic approximation for the dispersion relation of a hot magnetized plasma

1987 ◽  
Vol 38 (2) ◽  
pp. 275-286 ◽  
Author(s):  
A. Bravo-Ortega ◽  
D. G. Swanson ◽  
A. H. Glasser
Keyword(s):  
Dispersion Relation ◽  
Integral Representation ◽  
Asymptotic Approximation ◽  
Asymptotic Expression ◽  
Typical Case ◽  
Relative Accuracy ◽  
Magnetized Plasma ◽  
Dielectric Tensor

An asymptotic expression for the dielectric tensor e of a hot magnetized plasma is obtained employing the steepest descents method, via the transformation of the components of ε into their integral representation. The electrostatic Bernstein dispersion relation for oblique and perpendicular propagation is discussed under this treatment. It is shown that with this procedure the computation of the dispersion relation is up to 20 times faster when it is compared with the original expression, and the relative accuracy is usually as good as O·l% for a typical case.

scholarly journals Fejer means of rational Fourier – Chebyshev series and approximation of function |x|s

2019 ◽  
pp. 18-34
Author(s):  
Pavel G. Patseika ◽  
Yauheni A. Rouba
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Chebyshev Series ◽  
Approximation Properties ◽  
Fejér Means ◽  
Markov System ◽  
Asymptotic Expressions ◽  
Uniform Approximations ◽  
Optimal Value

Approximation properties of Fejer means of Fourier series by Chebyshev – Markov system of algebraic fractions and approximation by Fejer means of function |x|s, 0 < s < 2, on the interval [−1,1], are studied. One orthogonal system of Chebyshev – Markov algebraic fractions is considers, and Fejer means of the corresponding rational Fourier – Chebyshev series is introduce. The order of approximations of the sequence of Fejer means of continuous functions on a segment in terms of the continuity module and sufficient conditions on the parameter providing uniform convergence are established. A estimates of the pointwise and uniform approximation of the function |x|s, 0 < s < 2, on the interval [−1,1], the asymptotic expressions under n→∞ of majorant of uniform approximations, and the optimal value of the parameter, which provides the highest rate of approximation of the studied functions are sums of rational use of Fourier – Chebyshev are found.

Application of synthetic gravity model with exponential-power gravity function for calculation of passenger traffic splitting by different modes of public transport

2020 ◽  
pp. 3-9
Author(s):  
Galina Adolfovna Timofeeva ◽  
◽  
Olga Nikolaevna Ie ◽  
Keyword(s):  
Gravity Model ◽  
Public Transport ◽  
Statistical Data ◽  
Fixed Number ◽  
Traffic Splitting ◽  
Matrix Recovery ◽  
Passenger Traffic ◽  
Optimal Values ◽  
Exponential Power ◽  
Matrix Calculation

The paper considers a task of a correspondence matrix recovery for transport with a fixed number of routes according to the statistical data on incoming and departing passenger traffic. A classic gravity model was generalized for several modes of transport that resulted in combination of a correspondence matrix calculation and a calculation of passenger traffic splitting by different modes of transport. The authors propose an approach to determine the correspondence matrix for several modes of transport on the basis of synthetic gravity model with exponential-power gravity function. The approach described is realized on a specific example of analysis of public transport in Ekaterinburg. With the use of numerical methods the authors have solved a task of multidimensional optimization and found optimal values of parameters for gravity function. The paper also presents that the exponential-power gravity function is the most suitable for the model.

Boundedness of pseudo-differential operator associated with fractional Hankel transform

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Akhilesh Prasad ◽  
V. Singh
Keyword(s):  
Differential Operator ◽  
Integral Operator ◽  
Integral Representation ◽  
Integral Operators ◽  
Hankel Transform ◽  
Pseudo Differential Operator ◽  
Fractional Hankel Transform ◽  
Boundedness Result

AbstractA pseudo-differential operator (p.d.o.) associated with the fractional Hankel transform involving the symbol a(x, ξ) is defined. An integral representation of p.d.o. and boundedness result of the composition of operators Δμr and A μ,α are obtained. A generalized integral operator A μ,aα corresponding to p.d.o. is also defined and the properties of the product of two generalized integral operators corresponding to p.d.o. are studied.

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