CONTENTS &
ABSTRACTS

In English. Summaries in Estonian

Proceedings of the
Estonian Academy of Sciences.

Physics *
Mathematics

** **

Volume 51 No. 1
March 2002

Dual
pairs of sequence spaces. II;
3–17

Johann Boos and Toivo Leiger

**Abstract.** The authors proceed their
investigation of dual pairs _{} where _{} is a sequence space, _{} is a *K*-space on which a sum _{} is defined in the sense of Ruckle, and _{} is the space of all corresponding factor
sequences. Here, the particular case is considered that the sum _{} has the representation _{} where _{} is a directed set of indices _{} and _{} is a finite sequence for each _{} On the basis of this representation the *S*-sections
of any sequence _{} and both, their convergence (*AK*(*S*))
and boundedness (*AB*(*S*)) in *K*-spaces _{} are studied. Further, inclusion theorems due
to Bennett and Kalton are proved in this more general situation. Following an
idea of Schaefer to consider “section convergence barrels”, the notion of *AK*(*S*)-barrelled
*K*-spaces is introduced which leads to the result that a Mackey *K*-space
_{} containing all finite sequences is *AK*(*S*)-barrelled
if and only if _{} The paper covers some results concerning the
Köthe–Toeplitz duals and related section properties, for example, the _{}(*T*)-dual
and the *STK*-property (considered by Buntinas and Meyers).

**Key words:** topological sequence
spaces, Köthe–Toeplitz duals, section convergence, sum space, solid (normal)
topology, inclusion theorems.

On the
convexity theorem of M. Riesz;
18–34

Veera Pavlova and Anne Tali

**Abstract.** We consider the
well-known convexity theorem proved by M. Riesz in 1923, which gives
certain convexity conditions for Riesz summability methods _{} Later on different
authors have extended this theorem by modifying the convexity conditions and
the definition of methods _{} Our aim is to extend
the convexity theorem of M. Riesz to a wider class of summability methods
and, afterwards, apply it to the estimation of the speed of summability.

**Key words: **integral summability methods,
Riesz methods, integral Nörlund methods, convexity theorem of M. Riesz.

On the
group structure and parabolic points of the Hecke group *H*(*l*); 35–46

Nihal Yilmaz Özgür and I. Naci Cangül

**Abstract.** We consider the group structure of the Hecke
groups _{} which is isomorphic
to the free product of two cyclic groups of orders 2 and infinity and compute
all parabolic points of _{}

**Key words:** Hecke group, fundamental region,
parabolic point.

Quadratic
spline collocation method for weakly singular integral equations; 47–60

Raul Kangro, Rene Pallav, and Arvet Pedas

**Abstract.** The quadratic spline
collocation method for Fredholm integral equations of the second kind with
weakly singular kernels is studied. The rate of uniform convergence of this
method on quasi-uniform grids is derived.

**Key words:** weakly singular
integral equation, quadratic splines, collocation method, quasi-uniform grid.

SHORT
COMMUNICATIONS

Nanotechnology
should not neglect frequency dimension; 61–64

Karl K. Rebane

**Abstract.** In considering the
packing density of working units of a device the frequency dimension (pixel _{} should be also taken
into account, in addition to the conventional space-only domain characterized
by the pixel _{} Persistent spectral
hole burning space-and-time domain holography is a spectacular example of
usefulness of the frequency dimension in very high density data storage and
ultrafast processing.

**Key words:** nanotechnology, frequency dimension, persistent spectral hole
burning, optical data storage and processing, space-and-time domain holography.