Mathematics
We prove that the local field of positive characteristic is a vector space over a finite field.
A new approach to the construction of two-dimensional Haar and Vilenkin considered. To the obtained systems fast algorithms Fourier – Haar and Fourier – Vilenkin developed. Comparative analysis of algorithms developed in the problemstill image compression performed.
We consider one type of convergence of double trigonometric Fourier series intermediate between convergence over squares and λ-convergence for λ> 1. We construct an example of continuous functions of two variables, Fourier series of which diverges in this sense, almost everywhere.
The well-known Tsar’kov’s characterisation of finite-dimensional Banach spaces in which every bounded Chebyshev set (bounded P-acyclic set) is convex is extended to the asymmetrical setting.
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.
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