Table of contents

In this paper it is proved that for a certain class of systems (systems of type (X)) one may construct a series

(1)

having the following properties: 1) uniformly on the interval . 2) For any measurable function on the interval and for any number , one can find a partial series

from (1) which converges to almost everywhere on the set where is finite, and converges to in measure on . 3) If, in addition, the functions () and are piecewise continuous and , then

       for all        and    

It is shown that systems of type (X) include, for example, trigonometric systems, the systems of Haar and Walsh, indexed in their original or a different order, any basis of the space , and others. Bibliography: 19 items.

A complete solution is found of the interpolation problem for constructing entire functions given their values on a geometric progression

where is a complex number, . Bibliography: 12 items.

This paper treats the convergence of the Fourier series of a summable function in a system of rational functions on the unit circle, from the inside of the unit circle, and from the outside of the unit circle, for various distributions of the poles of the system. Bibliography: 8 items.

We consider exceptional sets occurring in the solution of problems of uniqueness and approximation of analytic functions, as well as in problems of convergence of Fourier series and of removal of singularities of analytic and polyharmonic functions. In the formulation of the theorems the smallness of exceptional sets is characterized by a special set function, the so-called (p, l)-capacity. Bibliography: 38 items.

We study partial differential equations and systems all of whose solutions are analytic, and obtain a priori estimates on the analytic continuations of these solutions to the complex plane. Using these estimates, we obtain conditions which are necessary for the analyticity of all solutions of linear differential equations and systems with analytic coefficients. Bibliography: 20 items.

We give an axiomatization for homology and cohomology theory in the categories and of countable locally finite polyhedra and of locally compact metrizable spaces, respectively, with proper mappings; in the category of metrizable compacta and continuous mappings; and (for cohomology) in the category of locally compact metrizable spaces and arbitrary continuous mappings. In we determine the kernel of the natural homomorphism over compact for a -additive cohomology (in particular, for Aleksandrov-Čech cohomology). Finally, we analyze the axioms of Skljarenko (Math. USSR Sb. 14(1971), 199-218; MR 44 #4738). Bibliography: 6 items.

The following is proved. Theorem.For a compactum of codimension greater than or equal to three lying in Euclidean space there exists an arbitrarily close approximation by a locally homotopically unknotted (1-ULC) imbedding. A series of corollaries about approximation of imbeddings of manifolds and polyhedra is derived. A problem about Menger universal compacta is solved. The article contains the complete proof of previously announced results stated in the references. Bibliography: 17 items.