A double sequence x = {xjk: j, k = 0, 1, …} of real numbers is called almost convergent to a limit s ifthat is, the average value of {xjk} taken over any rectangle {(j, k): m ≤ j ≤ m + p − 1, n ≤ k ≤ n + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n. The notion of almost convergence for single sequences was introduced by Lorentz [1].

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Abstract. We prove that the Hausdorff operator generated by a function ϕ ∈ L 1 (R) is bounded on the real Hardy space H 1 (R). The proof is based on the closed graph theorem and on the fact that if a function f in L 1 (R) is such that its Fourier transform f (t) equals 0 for t < 0 (or for t > 0), then f ∈ H 1 (R).