ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 40, 41, 42, 46 Aasma, Ants, Tallinn Pedagogical University, Estonia Summability with speed and comparison of orders of approximation of Fourier expansions by different matrix methods of summability A sequence x = (x_{k}) is said to be bounded with speed λ = (λ_{k}), 0 < λ_{k} ↗, if ∃ lim_{k} x_{k} = ξ, and l_{k} = λ_{k}(x_{k} − ξ) = O(1). If ∃ lim_{k} l_{k}, then x is said to be convergent with speed λ. Let m^{λ} and c^{λ} be the sets of λ-bounded and λ-convergent sequences, respectively and A = (α_{nk}) be a matrix of real numbers. Let m_{A}^{λ} = {x |(A_{n}x) ∈ m^{λ}} and c_{A}^{λ} = {x |(A_{n}x) ∈ c^{λ}} where A_{n} x = ∑_{k} α_{nk}x_{k}. We find necessary and sufficient conditions for a matrix M to transform m_{A}^{λ} into m_{B}^{μ} and c_{A}^{λ} into c_{B}^{μ} for a normal A, a triangular matrix B and μ = (μ_{k}), 0 < μ_{k} ↗. As an application we shall find the relation between the orders of approximation of Fourier expansions by the method of Zygmund Z^{r} and the method M^{φ} = (φ(\frac{k}{n+1})), generated by the continuous function φ(t) on [0, ∞] with φ(0) = 1 and φ(t) = 0, t ≥ 1.