asymptotics

About Asymptotics of Chebyshev Polynomials Orthogonal on an Uniform Net

Authors: Sultanov E. S.

In this article asymptotic properties of the Chebyshev polynomials Tn(x,N) (0 ≤ n ≤ N − 1) orthogonal on an uniform net ΩN = {0,1,...,N − 1} with the constant weight µ(x) = 2 N (discrete analog of the Legendre polynomials) by n = O(N 1 2 ), N → ∞ were researched. The asymptotic formula that is relating polynomials Tn(x,N) with Legendre polynomials Pn(t) for x = N 2 (1 + t) − 1 2 was determined.

Development of Asymptotic Methods for the Analysis of Dispersion Relations for a Viscoelastic Solid Cylinder

Authors: Wilde M. V., Sergeeva N. V.

Propagation of time-harmonic waves in a viscoelastic solid cylinder is considered. Vibrations of the cylinder are described by three-dimensional viscoelasticity equations in  cylindrical coordinates. The stress-free surface boundary conditions are imposed. Viscoelastic properties are described by integral operators with a fractional-exponential kernel. For the case of a rational singularity parameter the method of asymptotic analysis of dispersion relations is proposed, which is based on the generalized power series expansion.

Dirac System with Undifferentiable Potential and Antiperiodic Boundary Conditions

Authors: Kornev V. V., Khromov A. P.

The object of the paper is Dirac system with antiperiodic boundary conditions and complex-valued conditions potential. A new method

is suggested for investigating spectral properties of this boundary problem. The method is based on the formulas of the transform

operators type. It is rather elementary and simple. Using this method asymptotic behaviour of eigenvalues is specificated and it is

proved that eigen and associated functions form Riesz basis with brackets in the space of quadratic summerable two-dimensional