On the strong lacunary convergence and strong Cesàro summability of sequences of real-valued functions.
In the present paper we consider a new class of sequences called GM(β,r), which is the generalization of a class defined by Tikhonov in [15]. We obtain sufficient and necessary conditions for uniform convergence of weighted trigonometric series with (β,r)-general monotone coefficients.
A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced.