Displaying similar documents to “Almost everywhere convergence of Walsh Fourier series of H 1 -functions”

Integrability theorems for trigonometric series

Bruce Aubertin, John Fournier (1993)

Studia Mathematica

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

Oscillations of anharmonic Fourier series and the wave equation.

Alain Haraux, Vilmos Komornik (1985)

Revista Matemática Iberoamericana

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In this paper we have collected some partial results on the sign of u(t,x) where u is a (sufficiently regular) solution of ⎧     utt + (-1)m Δmu = 0     (t,x) ∈ R x Ω ⎨ ⎩     u|Γ = ... = Δm-1 u|Γ = 0     t ∈ R. These results rely on the study of a sign of almost periodic functions of a special...

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

G. Gát (1998)

Studia Mathematica

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Let G be the Walsh group. For f ∈ 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 ≤ c ∥ | f | ∥ H , where H is the Hardy space on the Walsh group.

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

Ferenc Móricz (1992)

Studia Mathematica

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In 1970 C. W. Onneweer formulated a sufficient condition for a periodic W-continuous function to have a Walsh-Fourier series which converges uniformly to the function. In this paper we extend his results from single to double Walsh-Fourier series in a more general setting. We study the convergence of rectangular partial sums in L p -norm for some 1 ≤ p ≤ ∞ over the unit square [0,1) × [0,1). In case p = ∞, by L p we mean C W , the collection of uniformly W-continuous functions f(x, y), endowed...

( H p , L p ) -type inequalities for the two-dimensional dyadic derivative

Ferenc Weisz (1996)

Studia Mathematica

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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 ∈ L 1 is dyadically differentiable and its derivative is f a.e.