Cesaro Summability of Walsh-Fourier Series

Nj, Fine

doi:10.1073/pnas.41.8.588

Cited by 146 publications

(72 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…convergence of Walsh-Fejér means σ n f was proved by Fine [2]. In 1975 Schipp [11] showed that the maximal operator σ * is of weak type (1, 1) and of type (p, p) for 1 < p ∞.…”

Section: Introductionmentioning

confidence: 97%

On the maximal operator of Walsh-Kaczmarz-Fejér means

Гогинава

Nagy

2011

Czech Math J

View full text Add to dashboard Cite

“…convergence of Walsh-Fejér means σ n f was proved by Fine [2]. In 1975 Schipp [11] showed that the maximal operator σ * is of weak type (1, 1) and of type (p, p) for 1 < p ∞.…”

Section: Introductionmentioning

confidence: 97%

On the maximal operator of Walsh-Kaczmarz-Fejér means

Гогинава

Nagy

2011

Czech Math J

View full text Add to dashboard Cite

“…x ∈ [−π, π], where s k f denotes the kth partial sum of the Fourier series of f . Fejér summability for Walsh-Fourier series is due to Fine [19] (see also Schipp [42]). The Fejér means and the maximal Fejér operator of the Ciesielski-Fourier series are given by…”

Section: 3mentioning

confidence: 99%

Results on spline-Fourier and Ciesielski-Fourier series

Weisz

2006

Approximation and Probability

View full text Add to dashboard Cite

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

show abstract

“…The usual density argument gives This convergence result for the Walsh system is due to Fine [7] (see also Schipp [9] and Weisz [16]). …”

Section: Integrating By Parts We Can See Thatmentioning

confidence: 99%

On the Fejér means of bounded Ciesielski systems

Weisz

2001

Studia Math.

View full text Add to dashboard Cite

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 → ∞.

show abstract

Cesaro Summability of Walsh-Fourier Series

Cited by 146 publications

References 0 publications

On the maximal operator of Walsh-Kaczmarz-Fejér means

On the maximal operator of Walsh-Kaczmarz-Fejér means

Results on spline-Fourier and Ciesielski-Fourier series

On the Fejér means of bounded Ciesielski systems

Contact Info

Product

Resources

About