Abstract. We study weighted Lp-integrability (1 ≤ p < ∞) of trigonometric series. It is shown how the integrability of a function with weight x −α depends on some regularity conditions on Fourier coefficients. Criteria for the uniform convergence of trigonometric series in terms of their coefficients are also studied.

Abstract. The double Fourier series of functions of the generalized bounded variation class {n/ ln(n + 1)} * BV are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class Λ * BV, defined here, have quadrant limits at every point and, for f ∈ Λ * BV, there exist at most countable sets P and Q such that, for x / ∈ P and y / ∈ Q, f is continuous at (x, y). It is shown that the previously studied class ΛBV contains essentially discontinuous functions unless the sequence Λ satisfies a strong condition.

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Firstly, we study the uniform convergence of cosine and sine Fourier transforms. Secondly, we obtain Pitt-Boas type results on L p -integrability of Fourier transforms with the power weights. The solutions of both problems are written as criteria in terms of general monotone functions.

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