Abstract

where a > 0, 2o (t) f (X -t) + f (Xt). If [p(t)], is of bounded variation in (0, 7r), then x is said to be a point of de la Vallee-Poussin for f. At such a point, the Fourier series of f is sum-mable I C, a I for every a > 1.1 The point x is said. to he a Dini point of f, if t-lo (t) is integrable in the Lebesgue sense on (0, 7r). Hardy 2 has proved that a Dini point is a point of de la Vallee-Poussin. Evidently, the integrability of t-lp (t) depends only on the behavior of the function f in the neighborhood of the point x. Bosanquet and Kestelman3 have shown that the suinmability I C, 1 1 for a Fourier series at a given point is not a local property of the function under consideration. Therefore, at a Dini point of f, the Fourier series is not necessarily summable I C, 1 | Theorem 1 and Theorem 4 of this paper are generalizations of Hardy's theorem. From these propositions, we derive Theorem 2 and Theorem 3. Direct proofs of the latter theorems are also given.

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