On the Fejér means of bounded Ciesielski systems

Abstract

Abstract. We investigate the bounded Ciesielski systems, which can be obtained from the spline systems of order (m, k) in the same way as the Walsh system arises from the Haar system. It is shown that the maximal operator of the Fejér means of the Ciesielski-Fourier series is bounded from the Hardy space H p to L p if 1/2 < p < ∞ and m ≥ 0, |k| ≤ m + 1. Moreover, it is of weak type (1, 1). As a consequence, the Fejér means of the Ciesielski-Fourier series of a function f converges to f a.e. if f ∈ L 1 as n → ∞.

Lebesgue inequalities for Chebyshev thresholding greedy algorithms

“…Examples of such systems in C(K) include the trigonometric system in C[0, 1] (in the real or complex case), as well as certain polygonal versions of the Walsh system [6,16,26], or any reorderings of them (which may cease to be Cesàro bases). (5) As a dual of the previous, if X = L 1 (µ) then every system {e n } ∞ n=1 as in (4) is weakly null, and hence case (1) applies.…”

“…Examples of such systems in C(K) include the trigonometric system in C[0, 1] (in the real or complex case), as well as certain polygonal versions of the Walsh system [6,16,26], or any reorderings of them (which may cease to be Cesàro bases).…”