Mixed Moduli of Smoothness in L_p, 1 < p < ∞: A Survey

In this paper we survey recent developments over the last 25 years on the mixed fractional moduli of smoothness of periodic functions from L_p, 1 < p < ∞. In particular, the paper includes monotonicity properties, equivalence and realization results, sharp Jackson, Marchaud, and Ul'yanov inequalities, interrelations between the moduli of smoothness, the Fourier coefficients, and "angular" approximation. The sharpness of the results presented is discussed