On the strong $(L)$ summability of Fourier series

Further result on the strong summability of Fourier series

This article deals with some special cases which are extension of the strong summability of Fourier series with constant factor. We obtain a new equivalent form of inequalities A 2π 0 φ(e iθ) r dθ ≤ 2π 0 1 0 (1 − ρ) φ (z) 2 dρ r/2 dθ ≤ B 2π 0 φ(e iθ) r dθ, (1) 2π 0 1 0 (1 − ρ) q−1 φ (z) q dρ r/q dθ ≤ C 2π 0 φ(e iθ) r dθ, (2) D 2π 0 φ(e iθ) r dθ ≤ 2π 0 1 0 (1 − ρ) p−1 φ (z) p dρ r/p dθ.

Almost Everywhere Strong Summability of Two-dimensional Walsh-fourier Series

A BMO-estimation of two-dimensional Walsh-Fourier series is proved from which an almost everywhere exponential summability of quadratic partial sums of double Walsh-Fourier series is derived.

Logarithmic Summability of Fourier Series

A set of regular summations logarithmic methods is introduced. This set includes Riesz and Nörlund logarithmic methods as limit cases. The application to logarithmic summability of Fourier series of continuous and integrable functions are given. The kernels of these logarithmic methods for trigonometric system are estimated.

On the Summability of the Differentiated Fourier Series

On the Summability of Fourier Series. Fifth Note.

and evaluating the generalized sum of (1) as the lim rm(x; f). The matrix ?2 will be subjected to the restrictions (i-iv) below. These restrictions are of such general nature, however, that they are satisfied by a great majority of the definitions of summability known in the literature. Our first condition on 21 is significant only in the case where 91 is of infinite reference. It is automatica...