CONTENTS &
ABSTRACTS

In
English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics

** **

Volume 51 No. 3
September 2002

On the
summability of Fourier expansions in Banach spaces;
131–136

Ants Aasma

**Abstract.** Let *X* be a Banach space with an orthogonal system of
projections. Let _{} be the method of
Zygmund, _{} the triangular method
of summation, generated by the differentiable function _{} and _{} be _{}- and _{}-means of Fourier expansions of _{} respectively. The
author of this paper has proved the theorem (see *Facta Univ. Nið. Ser. Math.
Inform*., 1997, **12**, 233–238) that gives sufficient conditions for _{} if it is assumed that
_{} for the same _{} and _{} on _{} In the present paper
this theorem is applied in the cases, where _{} is either the method
of Riesz, Jackson–de La Vallée Poussin, Bohman–Korovkin, Zhuk or
Favard.

**Key words:** Fourier expansions, summability methods,
approximation order.

Some summability
methods *b*-equivalent to the Cesàro methods; 137–148

Olga Meronen and Anne Tali

**Abstract. **The paper deals with summability methods which are equivalent for
summing bounded sequences (*b*-equivalent). It is well known that the
Cesàro methods _{} and the Abel method *A*
are *b*-equivalent. More generally, different authors have proved that
generalized Nörlund methods (*N*, *a*, *b*) and Abel-type power
series methods _{} are *b*-equivalent
under certain conditions on these methods. It turns out that quite often these
conditions imply the *b*-equivalence of the methods (*N*, *a*, *b*)
and _{} to _{} as well. The idea of
this paper is to investigate the *b*-equivalence of the methods (*N*,
*a*, *b*), _{} and _{}

Key words: Summability methods,
generalized Nörlund methods, Cesàro methods, power series methods, *b*-equivalence
of methods.

Two-dimensional
quadrature for functions with a point singularity; 149–159

Enn Tamme

**Abstract.** Numerical integration over the unit square of functions having
a weak singularity at a vertex is considered. The cubature formula resulting
from using in both directions a one-dimensional composite quadrature formula on
graded grid is studied. The dependence of the error of the cubature rule on
nonuniformity of the grid is investigated and the conditions for the grid under
which the method has the maximal possible convergence rate are found.
Theoretical results are verified by numerical examples in the case of the
Gaussian quadrature.

**Key words:** Two-dimensional
quadrature, point singularity, graded grid.

Upgrading the maximin principle for nonzero-sum
games; 160–169

Ants Tauts

**Abstract.** A modification of the maximin principle in the nonzero-sum
game where players do not wish to harm one another but only benefit themselves
is considered. First a player excludes such strategies of other players that
are less favourable for them than any other choice. Assuming that other players
have carried out the same procedure, in the new situation the line of reasoning
similar to the first step will be repeated. So a gradual detailing process will
start that with stabilization will give the final result.

**Key words:** characteristic
function, nonzero-sum games, strategies, coalitions.

A method to solve ordinary differential equations;
170–178

Mario Castagnino,
Luis Lara, and Roberto Aquilano

**Abstract.** Algebraic transformations for nonlinear and nonhomogeneous
ordinary differential equations are introduced that yield, step by step,
quasilinear forms and analytic solutions on small subintervals. Usually in the
relevant physical example these analytical solutions allow us to obtain the
time evolution operator, for any initial conditions.

**Key words:** differential
equations, Lyapunov stability, Bessel’s equation.

General solution of a system of differential
equations modelling a class of exactly-solvable potentials. Part II: extended
results; 179–193

Axel Schulze-Halberg

**Abstract.** The complete, closed-form solution of a system of coupled
differential equations introduced by Ge et al. (*Phys. Rev. A*, 2000, **62**,
052110–052117) and representing a set of potentials for which shift operators
can be constructed is given. The general solution obtained can be used to
perform a systematic search for new exactly-solvable potentials. This note is
an extension of the paper published in *Proc. Estonian Acad. Sci. Phys. Math*.
(2001, **50**, 1, 42–48).

**Key words:** exactly-solvable
potential, Schrödinger equation, shift operator.

IN MEMORIAM

Juhan Ross (1925–2002); 194–195