Spaces of piecewise-continuous almost-periodic functions and of almost-periodic sets on the line. I

In connection with the study of piecewise-continuous almost-periodic functions we introduce the notion of a countable almost-periodic number set. We investigate various properties of it: In particular, we prove that the space of almost-periodic sets is closed with respect to the operation of free union.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1613–1619, December, 1991.     

Zygmund functions on the real line and quasiconformal deformations

By a complex approach, we obtain some results concerning Zygmund functions on the real line, which corresponds to some classical results about Zygmund functions on the unit circle.     

Classes of functions defined on the real line and their approximation by entire functions. I

We introduce classes of functions defined on the real line which in the periodic case become the classes defined by multipliers (·) and translations with respect to the argument equal to . We study the simplest properties of these classes needed for the study of their approximation characteristics which will be discussed in the second part of the paper.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 102–112, January, 1990.     

Approximation of locally integrable functions on the real line

We introduce the notion of generalized-derivatives for functions locally integrable on the real axis and investigate problems of approximation of the classes of functions determined by these derivatives with the use of entire functions of exponential type. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1549–1561, November, 1999.     

Multiply periodic real functions and their period sets

Multiply periodic real functions are rather pathological; continuous ones have to be constant and, as will be shown, measurable ones must be constant a.e. Beyond this, some other properties of these functions are discussed, and examples are given. Since the set of periods of a real function is a subgroup of (R,+), it is possible to apply algebraic methods to get a better survey. It will be shown that the homogeneous sets defined by Borel are just cosets of these subgroups; this leads to a transfer of theorems about homogeneous sets to results about period sets (and, at the same time, subgroups of (R,+)).     

Factorization of almost-periodic matrix-valued functions

Translated from Matematicheskie Zametki, Vol. 45, No. 6, pp. 74–82, June, 1989.     

Orthogonal rational functions and quadrature on the real half line

In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Padé approximants for the Stieltjes transform of the measure. Convergence of both the quadrature formula and the multipoint Padé approximants is discussed.     

Special functions as subordinated semigroups on the real line

We give examples of convolution semigroups on the positive half-line and on the real line. Such semigroups are expressed in terms of special functions which arise in classical differential equations.     

Size direction games over the real line. II

Variants of two basic infinite games of perfect information are studied. A notion of continuous strategy for the playerS (Size) is shown to be related to a notion of convergence norm for sequences of reals. With each such norm, a variant of each of the basic games is associated in which the size player has to see that each play obeys the norm. Restriction to choose only rational numbers is also imposed onS. Some games are completely solved, and in this caseS has a winning strategy iff his set includes a perfect subset, andD has a winning strategy iffS's set is at most denumerable. Some other games, in whichS has to choose only rationals and obey a norm, induce a hierarchy structure on the class of nowhere dense perfect sets, that is embedded cofinally in the lattice of infinite sequences of integers modulo finite differences.     

Classes of functions defined on the real axis and their approximations by entire functions. II

Approximation properties of functions (defined on the real axis) of the classes introduced in the first part of the paper are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 210–222, February, 1990.     

Product sets in the plane,sets of the formA+B on the real line and Hausdorff measures

Research supported by the Hungarian National Foundation for Scientific Research Grant No. 2114.     

Decomposing the real line into Borel sets closed under addition

We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in and even in the theory if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition (0, ∞), and so on.