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.




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.