On the Strong Approximation by (C, α)‐Means of Fourier Series

Abstract

The degree of pointwise approximat.ion in the strong sense of 2nperiodic functions from LP (p = (1 + a)-1, a = . -i/2) is examined. An answer to the modified version of LEINDLER'S problem [ 4 ] is given.

Pointwise $$\left( H,\Phi \right) $$H,Φ Strong Approximation by Fourier Series of $$L^{1}$$L1 Integrable Functions

“…It is also clear, as it shown Gogoladze [4] and Rodin [14], that H Φ n f (x) −means, with Φ (u) = exp u − 1 also tend to 0 almost everywhere (as n → ∞). The estimates of H Φ n f (x) −means, for f ∈ L p (1 < p < ∞) was obtained in [10] but in the case f ∈ L 1 the estimates of H Φ n f (x) − means with Φ (u) = u q was obtained in [9]. Here we present estimation of the H Φ n f (x) − mean for f ∈ L 1 by G x f π n+1 1,Ψ as approximation version of the Totik type (see [15,16]) generalization of the mentioned results of J. Marcinkiewicz and A. Zygmund.…”

Pointwise (H, Φ) Strong Approximation by Fourier Series of LΨ Integrable Functions

“…It is also clear, as it shown L. Gogoladze [3] and W. A. Rodin [13], that H Φ n f (x) −means, with Φ (u) = exp u − 1 also tend to 0 almost everywhere (as n → ∞). The estimates of H Φ n f (x) −means, for f ∈ L p (1 < p < ∞) was obtained in [9] but in the case f ∈ L 1 the estimates of H Φ n f (x) − means with Φ (u) = u q was obtained in [8,12]. Finally, the estimation of the…”