CONTENTS &
ABSTRACTS

In
English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics

** **

Volume 51 No. 4
December 2002

On sampling series based on some
combinations of sinc functions; 203–220

Andi Kivinukk and Gert Tamberg

**Abstract.** The generalized sampling series given in *Jahresber.
Deutsch. Math.-Verein*., 1988, **90**, 1–70 are studied. We introduce
some new generalized sampling series which are defined by certain combinations
of the sinc functions and find the order of approximation by those series.
Sampling series of this kind are motivated by some summation methods of
trigonometric Fourier series. Our discussion is based on the Rogosinski-type
sampling series.

**Key words: **generalized sampling series, Rogosinski-type sampling series, sinc
function, operator norms, order of approximation.

Quadratic
spline collocation method for weakly singular integral equations on graded
grids; 221–237

Rene Pallav

**Abstract.** The spline collocation method for weakly singular Fredholm
integral equations of the second kind is studied. Using smooth quadratic
splines, we derive the rate of uniform convergence of this method on graded
grids.

**Key words:** weakly singular integral equation, collocation method,
quadratic splines, graded grids.

Removing or lowering the orders of input
shifts in discrete-time generalized state-space systems with Mathematica; 238–254

Ülle Kotta and Maris Tõnso

**Abstract.** The problem of lowering the orders of input forward shifts in
generalized~state-space systems by generalized state transformations is studied
using the language of differential forms. Necessary and sufficient conditions
are given for the local existence of such transformations. These conditions are
formulated in terms of the integrability of certain subspaces of one-forms
classified according to their relative degree. The sufficiency part of the
proof gives a constructive procedure (up to finding the integrating factors and
integration of the set of one-forms) for finding these generalized state
transformations. In a particular case, these conditions show when it is
possible to transform a generalized state-space representation into the
classical state equations. A set of functions developed in Mathematica 4.0 is described, allowing us to
test if the generalized state equation is transformable into the classical
state-space form, and also to find the transformation for simple examples. The
application of the developed functions is demonstrated on four examples
obtained via identification.

**Key words:** nonlinear systems, discrete-time systems, generalized
dynamics, generalized state transformations, linear algebraic approach,
symbolic computation, Mathematica.

A
technique to study the mechanisms of pressure broadening of spectral holes;
255–268

Kristjan Leiger and Jaak Kikas

**Abstract.** An experimental procedure to distinguish between the two
possible mechanisms (dispersive and diffusive) of pressure broadening of
spectral holes in nonisobaric experiments is proposed. The method makes use of
nonlinear saturation properties of the hole burning process. The proposed
four-step procedure consists of burning a hole, applying hydrostatic pressure
to the system, additional burning at the hole minimum at this second pressure,
and then applying more pressure. By model calculations it is demonstrated that,
under certain conditions, the dispersive mechanism yields a double-minimum
hole, while the diffusive mechanism does not. Experimental observability of
such an effect is discussed.

**Key words: **hole burning,
pressure tuning, inhomogeneous broadening.

CHRONICLE

Harald Keres 90; 269

Contents of volume 51; 270–271