Strong Approximation via Sidon Type Inequalities

Abstract

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

BMO-Estimation and Almost Everywhere Exponential Summability of Quadratic Partial Sums of Double Fourier Series

“…Similarly we can get the same bound for S (2) nn (x, y, f ), which together with (16) completes the proof of lemma.…”

Strong approximation by Marcinkiewicz means of two-dimensional Walsh–Kaczmarz–Fourier series

“…Proof of Theorem 2. a) It is easily seen that if ϕ ∈ Ψ, then e ϕ − 1 ∈ Ψ. Besides, (5) implies the existence of a number A such that…”

“…Leindler has also published a monograph [22].The results on strong summation and approximation of trigonometric Fourier series have been extended for several other orthogonal systems. For instance, concerning the Walsh system see Schipp [24,25,26], Fridli, Schipp [4,5], Fridli [3], Rodin [23], Goginava, Gogoladze [13,12], Gát, Goginava, Karagulyan [6,7], Goginava, Gogoladze, Karagulyan [14] and concerning the Ciesielski system see Weisz [32,33]. The summability of multiple Walsh-Fourier series have been investigated in [8,15,16,18,34].Fridli [3] proved that the following theorem is true.…”

Almost everywhere strong summability of cubic partial sums of d-dimensional Walsh-Fourier series

“…The results on strong summation and approximation of trigonometric Fourier series have been extended for several other orthogonal systems. For instance, concerning the Walsh system see Schipp [31,32,33], Fridli and Schipp [2,3], Leindler [20,21,22,23], Totik [36,37,38], Rodin [28], Weisz [41,42], Gabisonia [4], Goginava, Gogoladze [11].…”