Haar system

Haar Series on Compact Zero-Dimensional Abelian Group

Authors: Lukomskii S. F.

In this article we construct a Haar system on compact zerodimensional abelian group as shifts and powers of some characters system. We find conditions under which a Fourier-Haar series of continuous function converge uniformly. We find groups for which Haar functions generated from one function by the operation of shifts, powers and dilations.

Solution of Cauchy Problem for Equation First Order Via Haar Functions

Authors: Lukomskii D. S., Lukomskii S. F., Terekhin P. A.

In this article we consider a Cauchy problem for the first order differential equation and are looking for its numerical solution. For this aim we represent the derivative of the solution as Haar decomposition. We also obtain estimates of approximate solution. The method is computationally simple and applications are demonstrated through illustrative examples. These examples show that in some cases the error of the proposed method is much less, than in second order Runge – Kutta method.

The Intermediate Case of Regularity in the Problem of Differentiation of Multiple Integrals

Authors: Fufaev D. V.

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.