UNIFORM AND L-CONVERGENCE OF LOGARITHMIC MEANS OF WALSH-FOURIER SERIES G. GÁT AND U. GOGINAVA Abstract. The (Nörlund) logarithmic means of the Fourier series of the integrable function f is: 1 l n n-1  k=1 S k (f ) n − k , where l n := n-1  k=1 1 k . In this paper we discuss some convergence and divergence properties of this logarithmic means of the Walsh-Fourier series of functions in the uniform, and in the L 1 Lebesgue norm. Among others, as an application of our divergence results we give a negative answer for a question of Móricz concerning the convergence of logarithmic means in norm. In the literature it is known the notion of the Riesz’s logarithmic means of a Fourier series. The nth mean of the Fourier series of the integrable function f is defined by 1 l n n−1  k=1 S k (f ) k . This Riesz’s logarithmic means with respect to the trigonometric system has been studied by a lot of authors. We mention for instance the papers of Szász, and Yabuta [11, 13]. This mean with respect to the Walsh, Vilenkin system is discussed by Simon, and Gát [10, 3]. For results of this kind with respect to the Walsh-Kaczmarz system, and some generalization of the Walsh system see [7, 1] Let {q k : k ≥ 0} be a sequence of nonnegative numbers. The Nörlund means for the Fourier series of f are defined by 1 Q n n−1  k=1 q n−k S k (f ), where Q n := ∑ n−1 k=1 q k . If q k = 1 k , then we get the (Nörlund) logarithmic means: 1 l n n−1  k=1 S k (f ) n − k . In this paper we call it - it will not cause any misunderstood - as logarithmic means. Al- though, it is a kind of “reverse” Riesz’s logarithmic means. Móricz [6] investigates the ap- proximation properties of some special Nörlund means of Walsh-Fourier series of L p functions in norm. The case, when q k = 1 k is excluded, since the methods of Móricz are not applicable for logarithmic means. The aim of this paper is to prove some convergence and divergence properties of the logarithmic means of functions in the class of continuous functions, and in The first author is supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. M 36511/2001., and by the Széchenyi fellowship of the Hungarian Ministry of Education Szö 184/2003. 1