Regular matrix methods that improve and accelerate the convergence of sequences and series are studied. Some problems related to the speed of convergence of sequences and series with respect to matrix methods are discussed. Several theorems on the improvement and acceleration of the convergence are proved. The results obtained are used to increase the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in certain Banach spaces.

Abstract. In this paper sufficient conditions for a matrix M = (m nk ) (m nk are Cesàro numbers A s n−k , s ∈ C if k ≤ n and m nk = 0 if k > n) to be a transform from the summability domain of the Cesàro method C α into the summability domain of another Cesàro method C β , where α, β ∈ C\{−1, −2, . . .}, are found.

In this work some new nonclassical convergence acceleration concepts are described and compared with the classical convergence acceleration concept. It is shown that these concepts allow to compare the speeds of convergence for a larger set of sequences than the classical convergence acceleration concept. For the acceleration of convergence of sequences regular matrix methods are used. As an application the obtained results can be used for accelerating the convergence of Fourier expansions and for increasing the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in Banach space.

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Abstract. The present paper continues the study of acceleration of convergence started in the paper [A. Aasma, Proc. Estonian Acad. Sci. Phys. Math., 2006, 55, 4, 195-209]. The new, non-classical convergence acceleration concept, called strong µ-acceleration of convergence (µ is a positive monotonically increasing sequence), is introduced. It is shown that this concept allows to compare the speeds of convergence for a larger set of sequences than the classical convergence acceleration concept. Regular matrix methods are used to accelerate the convergence of sequences.