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.



Juhan Ross (1925–2002); 194–195