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is a continuous function, by uniform convergence. We recall (see [4, pp. 43–44]) that a periodic function φ belongs to the Lipschitz class Lipα for some α > 0 if there exists a constant C = C(φ) such that for all x and h we have |φ(x+ h)− φ(x)| ≤ C|h|. The following theorems by Boas [1] give necessary and sufficient conditions for a sine series to belong to Lipα, where 0 < α ≤ 1. Theorem A. Let {ai : i = 1, 2, . . .} be a sequence of nonnegative numbers such that condition (1.1) is satisfied and let fs be defined by (1.2). If

Double sine series with nonnegative coefficients and Lipschitz classes