Almost Everywhere Convergence of Vilenkin-Fourier Series of H 1 Functions

Young, Wo-Sang

doi:10.2307/2048292

Cited by 8 publications

(12 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…p. 235] and for Vilenkin-Fourier series by Young [73]. Now we extend this result to Ciesielski-Fourier series (see Weisz [62]).…”

Section: This Means That Smentioning

confidence: 52%

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

“…p. 235] and for Vilenkin-Fourier series by Young [73]. Now we extend this result to Ciesielski-Fourier series (see Weisz [62]).…”

Section: This Means That Smentioning

confidence: 52%

Results on spline-Fourier and Ciesielski-Fourier series

Weisz

2006

Approximation and Probability

View full text Add to dashboard Cite

show abstract

“…The analogous results for trigonometric Fourier series can be found in Zygmund [17, II. p. 235] and for Vilenkin-Fourier series in Young [15].In this paper we extend these results to Ciesielski-Fourier series, which are generalizations of the Walsh-Fourier series. The Ciesielski systems can be obtained from the spline systems of order (m, k) in the same way as the Walsh system arises from the Haar system (see Ciesielski [2, 1]).…”

mentioning

confidence: 58%

“…p. 235] for trigonometric Fourier series, to Ladhawala and Pankratz [4] for Walsh-Fourier series, and to Young [15] for Vilenkin-Fourier series.…”

Section: Theorem 1 Suppose That the Sublinear Operatormentioning

confidence: 99%

See 1 more Smart Citation

Almost everywhere convergence of Ciesielski-Fourier series of H1functions

Weisz

2004

Arch. Math.

View full text Add to dashboard Cite

It is proved that a lacunary sequence of the Ciesielski-Fourier series of f ∈ H 1 converges almost everywhere to f . Introduction.Let s n f denote the nth partial sum of the Walsh-Fourier series of f ∈ L 1 . Ladhawala and Pankratz [4] proved that if f is in the dyadic Hardy space H 1 and (n i , i ∈ N) is a lacunary sequence of positive integers, then s n i f converges a.e. to f . Moreover, Schipp and Simon [6] verified that if (u) = o(log log u) (u → ∞) then there exists a function in H 1 (H 1 ) whose full sequence of partial sums diverges everywhere. Especially, if (u) = 1(u 1) then we get H 1 (H 1 ) = H 1 , i.e., it follows the existence of f ∈ H 1 such that s n f diverges everywhere (see Ladhawala and Pankratz [4]). The analogous results for trigonometric Fourier series can be found in Zygmund [17, II. p. 235] and for Vilenkin-Fourier series in Young [15].In this paper we extend these results to Ciesielski-Fourier series, which are generalizations of the Walsh-Fourier series. The Ciesielski systems can be obtained from the spline systems of order (m, k) in the same way as the Walsh system arises from the Haar system (see Ciesielski [2, 1]).

show abstract

“…, Karagulian [72,73], Kheladze [76,77], Lebesgue [81] Stechkin [136] Young [187,188,189] . Some estimates of Fourier coefficients and an absolute, point-wise and a.e.…”

Section: January 31 2022mentioning

confidence: 99%

Bounded operators on Martingale Hardy spaces

Tutberidze¹

2022

Preprint

View full text Add to dashboard Cite

The classical Fourier Analysis has been developed in an almost unbelieved way from the first fundamental discoveries by name Fourier. Especially a number of wonderful results have been proved and new directions of such research has been developed e.g. concerning Wavelets Theory, Gabor theory, Time-Frequency Analysis, Fast Fourier Transform, Abstract Harmonic Analysis, etc. One important reason for this is that this development is not only important for improving the "State of the art", but also for its importance in other areas of mathematics and also for several applications (e.g. theory of signal transmission, multiplexing, filtering, image enhancement, coding theory, digital signal processing and pattern recognition.)The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results of Vilenkin series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group.

show abstract

Almost Everywhere Convergence of Vilenkin-Fourier Series of H 1 Functions

Cited by 8 publications

References 0 publications

Results on spline-Fourier and Ciesielski-Fourier series

Results on spline-Fourier and Ciesielski-Fourier series

Almost everywhere convergence of Ciesielski-Fourier series of H1functions

Bounded operators on Martingale Hardy spaces

Contact Info

Product

Resources

About