an integrability condition for cosine series. No condition superior to that has been given so far. In this paper we identify the atomic structure of the Hardy type space that can be associated with this condition. As a consequence, we conclude that Telyakovskiȋ's condition is equivalent to certain Sidon type inequalities. Then on the basis of this equivalence we show how the atomic technique can be used to extend Telyakovskiȋ's condition to several systems, including Walsh series and integrals, in a uniform way.

Introduction. In this paper we deal with the connection (in different spaces) among the Vilenkin--Fourier sums, the modulus of continuity and the Lebesgueconstants (with respect to the Vilenkin-system). We give two sided estimates for an expression containing these quantities. The corresponding problem for the trigonometric system was considered by Lebesgue [7] and Oskolkov [8].

The aim of this paper is twofold. First we want to show how a duality relation provides a vehicle to deduce strong summability and approximation properties of Fourier series from some basic inequalities, called Sidon type inequalities. This way the technicalities concerning several strong summability and approximation problems can be reduced to proving such inequalities. On the other hand, we will isolate two properties that induce the sharpest version of these inequalities for a number of orthonormal systems, find their counterparts in terms of strong approximation, and show some of their consequences. We note that these results are known to be the best possible for the trigonometric system. Academic Press

In this paper we develop an adaptive transformdomain technique based on rational function systems. It is of general importance in several areas of signal theory, including filter design, transfer function approximation, system identification, control theory etc. The construction of the proposed method is discussed in the framework of a general mathematical model called variable projection. First we generalize this method by adding dimension type free parameters. Then we deal with the optimization problem of the free parameters. To this order, based on the well-known particle swarm optimization (PSO) algorithm, we develop the multi-dimensional hyperbolic PSO algorithm. It is designed especially for the rational transforms in question. As a result, the system along with its dimension is dynamically optimized during the process. The main motivation was to increase the adaptivity while keeping the computational complexity manageable. We note that the proposed method is of general nature. As a case study the problem of electrocardiogram (ECG) signal compression is discussed. By means of comparison tests performed on the PhysioNet MIT-BIH Arrhythmia database we demonstrate that our method outperforms other transformation techniques.

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