Results on spline-Fourier and Ciesielski-Fourier series

Abstract

Abstract. Some recent results on spline-Fourier and Ciesielski-Fourier series are summarized. The convergence of spline Fourier series and the basis properties of the spline systems are considered. Some classical topics, that are well known for trigonometric and Walsh-Fourier series, are investigated for Ciesielski-Fourier series, such as inequalities for the Fourier coefficients, convergence a.e. and in norm, Fejér and θ-summability, strong summability and multipliers. The connection between Fourier series and Hardy spaces is studied.1. Introduction. It is known that the spline Fourier series of f ∈ L p converges a.e. and in L p norm to f , whenever 1 ≤ p < ∞. The maximal operator of the partial sums with respect to the spline (or unbounded Ciesielski) systems of order (m, k) is bounded from the Hardy space H p to L p (1/(m − k + 2) < p < ∞) and is of weak type (1, 1). If 1 < p < ∞ then H p is equivalent to L p . Moreover, the spline systems are unconditional and equivalent bases to the Haar system in H p (1/(m − k + 2) < p < ∞).We investigate also the bounded Ciesielski systems, which can be obtained from the spline systems in the same way as the Walsh system from the Haar. Some results, that are well known for trigonometric and Walsh-Fourier series, are extended to CiesielskiFourier series. Paley and Hardy-Littlewood type inequalities are shown for CiesielskiFourier coefficients. If f ∈ L p (1 < p < ∞) then the Ciesielski-Fourier series of f converges