On the uniform convergence and L¹-convergence of double Walsh-Fourier series

Ferenc Móricz

Displaying similar documents to “On the uniform convergence and L¹-convergence of double Walsh-Fourier series”

On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system

G. Gát (1998)

Studia Mathematica

Similarity:

Let G be the Walsh group. For $f \in L^{1} (G)$ we prove the a. e. convergence σf → f(n → ∞), where $σ_{n}$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator $σ * f ≔ s u p_{n} | σ_{n} f | .$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, $∥ σ * f ∥_{1} \leq c ∥ | f | ∥_{H}$ , where H is the Hardy space on the Walsh group.

Two-parameter Hardy-Littlewood inequality and its variants

Chang-Pao Chen, Dah-Chin Luor (2000)

Studia Mathematica

Similarity:

Let s* denote the maximal function associated with the rectangular partial sums $s_{m n} (x, y)$ of a given double function series with coefficients $c_{j k}$ . The following generalized Hardy-Littlewood inequality is investigated: ${| | s * | |}_{p, μ} \leq C_{p, α, β} {Σ_{j = 0}^{\infty} Σ_{k = 0}^{\infty} {(j ̅)}^{p - α - 2} {(k ̅)}^{p - β - 2} {| c_{j k} |}^{p}}^{1 / p}$ , where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_{j k}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property...

Absolute convergence of Fourier series of a function of Wiener’s class $V_{p}$

Siddiqi, Rafat N. (1979)

Portugaliae mathematica

Similarity:

On the characterization of Hardy-Besov spaces on the dyadic group and its applications

Jun Tateoka (1994)

Studia Mathematica

Similarity:

C. Watari [12] obtained a simple characterization of Lipschitz classes $L i p^{(p)} α (W) (1 \geq p \geq \infty, α > 0)$ on the dyadic group using the $L^{p}$ -modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces $B_{p, q}^{α}$ on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces $B_{p, q}^{α}$ by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality...

Note on the absolute convergence of Fourier series of a function of Wiener’s class $V_{r}$

Okuyama, Yasuo (1983-1984)

Portugaliae mathematica

Similarity:

On the integrability and L¹-convergence of double trigonometric series

Ferenc Móricz (1991)

Studia Mathematica

Similarity:

Integrability theorems for trigonometric series

Bruce Aubertin, John Fournier (1993)

Studia Mathematica

Similarity:

We show that, if the coefficients (an) in a series $a_{0} / 2 + \sum_{n = 1}^{\infty} a_{n} c o s (n t)$ tend to 0 as n → ∞ and satisfy the regularity condition that $\sum_{m = 0}^{\infty} {\sum_{j = 1}^{\infty} [\sum_{n = j 2^{m}}^{(j + 1) 2^{m} - 1} | a_{n} - a_{n + 1} |] ²}^{1 / 2} < \infty$ , then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series $\sum_{n = 1}^{\infty} b_{n} s i n (n t)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if $\sum_{n = 1}^{\infty} | b_{n} | / n < \infty$ . These conclusions were previously known to hold under stronger restrictions on the sizes...

$(H_{p}, L_{p})$ -type inequalities for the two-dimensional dyadic derivative

Ferenc Weisz (1996)

Studia Mathematica

Similarity:

It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space $H_{p, q}$ to $L_{p, q}$ (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type $(L_{1}, L_{1})$ . As a consequence we show that the dyadic integral of a ∞ function $f \in L_{1}$ is dyadically differentiable and its derivative is f a.e.

On convergence of Fourier series of functions of generalized bounded variation

Daniel Waterman (1972)

Studia Mathematica

Similarity:

A note on Fourier coefficients

B. N. Varma (1969)

Rendiconti del Seminario Matematico della Università di Padova

Similarity:

On an infinite linear combination of partial sums of Fourier series

Rajendra Sinha (1976)

Studia Mathematica

Similarity: