On some Classes of Fourier Series

Abstract

Equation (36) on p. 320 of this paper is false. A consequence of this is that the conditions obtained in §§11, 12, and which are stated to be necessary and sufficient, have only been shown to be sufficient. To obtain conditions which are both necessary and sufficient, we observe that, if {gn(t)} denote a sequence of integrable functions, then, in order that fov every bounded f(t) we have lim n → ∞ ∫ 0 2 π f ( t ) g n ( t ) d t = 0, it is necessary and sufficient that (i) there is a constant K such that ∫ 0 2 π | g n ( t ) | d t < K, and (ii), given a measurable set e in (0, 2π), lim n → ∞ ∫ e g n ( t ) d t = 0 . This is an easy congruence of Lemma 2, p. 293. To obtain condition (2, 6), we must characterize the functions g(t) for which equation (33), p. 320, is true. We amend the definition of class (b) as follows. Amended definition of class (b). Let g(t) be an integrable periodic function. To each n, let there correspond a number vn of non-overlapping intervals in (0, 2π), {( α i ( n ) , β i ( n ) )} (i = 1, 2, …, vn), such that lim n → ∞ ∑ i = 1 v n ( β i ( n ) - α i ( n ) ) = 0 . Let gn(t) = ∑ i = 1 v n { g ( t - β i ( n ) ) - g ( t - α i ( n ) ) } . If there is a constant K such that (2) holds, and if, given a measurable set e in (0, 2π), (3) holds, then g(t) is absolutely continuous in mean. The sequence {gr(t)} of functions is uniformly absolutely continuous in mean, if, with the notation g n ( r ) ( t ) = ∑ i = 1 v n { g r ( t - β i ( n ) ) - g r ( t - α i ( n ) ) } we have ∫ 0 2 π | g n ( r ) ( t ) | d t < K ; lim n → ∞ ∫ e g n ( r ) ( t ) d t = 0 , where K is an absolute constant, and, for a given e, the limit is uniform in r. The propositions enunciated in §11 become true with this amended definition. For §12, we must introduce an amended definition of R-integrability in mean. Amended definition of R-integrability in mean. Let g(t) be an integrable periodic function. To each n let there correspond a number vn of non-overlapping intervals (4) in (0, 2π) such that lim n → ∞ Max 1 ⩽ i ⩽ v n ( β i ( n ) - α i ( n ) = 0 Let γ i ( n ) , δ i ( n ) be arbitrarily chosen to satisfy α i ( n ) ⩽ γ i ( n ) < δ i ( n ) ⩽ β i ( n ) . Let gn(t) = ∑ i = 1 v n { g ( t - δ i ( n ) ) - g ( t - γ i ( n ) ) } ( β i ( n ) - α i ( n ) ) . If there is a constant K such that (2) holds, and if, given a measurable set e in (0, 2π), (3) holds, then g(t) is R-integrable in mean. The sequence {gr(t) is uniformly R-integrable in mean, if, with the notation g n ( r ) ( t ) = ∑ i = 1 v n { g ( r ) ( t - δ i ( n ) ) - g ( r ) ( t - γ i ( n ) ) } ( β i ( n ) - α i ( n ) ) the relations (5) hold, where K is an absolute constant, and, for a given e, the limit is uniform in r. With this amended definition Lemma 8 is true. This lemma gives us a form of the condition (2, 3). I am, however, unable to give an amended definition of a function of R-integrable variation in mean, and thus to give the condition (2, 3) by characterizing the function of which Σ,λn sin nx/n is the Fourier series.