Mathematics
An inverse spectral problem for discrete operators of a triangular structure in topological spaces is studied. A constructive procedure for the solution of the inverse problem is provided. Necessary and sufficient conditions for its solvability are obtained.
The new notion of affine system of Walsh type is introduced and studied. We proved results about orthogonalization and completion of affine systems of Walsh type with preservation of structure of affine systems.
The paper deals with extremal properties of diagonal quadratic Hermite – Pad’e approximants of type I for exponential system {eλjz}2j =0 with arbitrary real λ0, λ1, λ2. Proved theorems complement known results of P. Borwein, F. Wielonsky.
We provide two solutions to the Fagnano problem on finding a three-link billiard traectory in a triangle.
The analysis of the solutions of Clairaut equation with an arbitrary number of independent variables is completed. It is assumed that the function of the derivatives, which is part of the equation is multi-homogeneous. This means that the set of function arguments can be represented as the union of subsets, and the function is homogeneous on each of these subsets.
Provides analytical expressions representing all the roots of a random algebraic equation of n-th degree through the coefficients of the initial equation. These formulas consist of two relations infinite Toeplitz determinants, the diagonal elements of which are the coefficients of algebraic equations. For finding complex roots additionally used the method of summation of divergent continued fractions.
Let kλ(x) be a measurable essentially bounded 2π-periodic function (kernel), where λ > 1.
In this paper we introduce the discrete series with the «sticking»-property of the periodic ({sinx sinkx} system) and non-periodic (using the system of the second kind of Chebyshev polynomials Uk(x)) cases. It is shown that series of the system {sinx sinkx}
have an advantage over cosine Fourier series because they have better approximation properties near the bounds of the [0, π] segment. Similarly discrete series of the system Uk(x) near the bound of the [−1, 1] approximates given function significantly
In present paper there were introduced two-dimensional special series of the system {sinx sinkx}. It’s shown that these series have the advantage over two-dimensional cosine Fourier series, because they have better approximation properties near the bounds of the square [0, 1]2. It’s given convergence speed estimate of special series partial sums to functions f(x, y) from the space of even 2π-periodic continuous functions.
The paper deals with generalization of Lebesgue and Jessen –Marcinkiewicz – Zygmund theorems of the differentiation of multiple integrals for the intermediate case of regularity of the system of sets. The application of the result to the Fourier-Haar series and to orthorecursive expansions with respect to system of indicators of multi-dimensional intervals is considered.
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