首页|Complete Asymptotics of Approximations by Certain Singular Integrals in Mathematical Modeling
Complete Asymptotics of Approximations by Certain Singular Integrals in Mathematical Modeling
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When solving some types of problems of applied character, nowadays the most efficient are the methods of the theory of approximation of functions. In a modern stage of development of the theory of approximation of functions, one mostly deals with either an approximation of individual functions or whole function classes by preset subsets of functions that turn to be in a certain sense more convenient to deal with in calculations in comparison with the functions that should be approximated. In practice, a set of algebraic polynomials or a set of trigonometric polynomials of a given order often play the role of such a subset. As a result, a new type of problems appeared, that further was called the extremal problems of the theory of approximation. In its turn, among all of the extremal problems of the theory of approximation, the most interesting from the mathematical modeling point of view are the so-called Kolmogorov-Nikol'skii problems. Their essence is the determination of asymptotic equalities for the values of the approximation of functions of certain classes by specific methods of summation of the Fourier series. Here we consider a problem of approximation of 2π-periodic functions of the Lipshitz class by certain singular integrals. The most vivid examples of such integrals are the so-called generalized Poisson integrals. As a result of studying, we wrote down complete asymptotic expansions in powers of 1/δ, δ →∞, of the least upper borders of deviations of functions of the Lipshitz class from their generalized Poisson integrals. The obtained result allows us to write down not only the main term of the asymptotic expansion but also write down its second, third terms, etc., using the Riemann ζ-function, which, respectively, much simplifies the problem of algorithmization when solving the stated applied problem. Moreover, the generalized Poisson integrals are the solutions of partial differential equations, and they are connected directly with the methods of solving integral, difference-differential, and integral-differential games, that are related to the game problems of dynamics.
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