Displaying similar documents to “On the embedding H ω ⊂ V p ”

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...

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...