Riesz logarithmic means of subsequences of partial sums of trigonometric Fourier series

Let ( a ( j )) be an unbounded convex sequence of natural numbers. In 1983, Carleson [7] proved that a necessary and sufficient condition for the ( C , 1) means of (Sₐ₍ⱼ₎f) ( S a ( j ) f ) (partial Fourier sums) to converge uniformly to f in the supremum norm is ⱼ j-1/2log a(j) <+∞ sup j j - 1 / 2 log a ( j ) < + ∞ . In this paper, we prove that if we consider Riesz logarithmic means instead of ( C , 1) means, then almost everywhere convergence holds for any integrable function and for any convex ( a ( j )). That is (see Corollary 1.3): Let ( a ( j )) be any unbounded convex sequence of natural numbers and f∈ L¹ f ∈ L 1 . Then the following convergence holds almost everywhere: 1/log n∑ ⱼ₌₁ⁿSₐ₍ⱼ₎fj → f. 1 log n ∑ j = 1 n S a ( j ) f j → f . In fact, we verify a more general statement (see Theorem 1.1).