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SAT 8 (2013), 1

Surveys in Approximation Theory, 8 (2013), 1-57.

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

M. K. Potapov, B. V. Simonov, and S. Yu. Tikhonov

Abstract. In this paper we survey recent developments over the last 25 years on the mixed fractional moduli of smoothness of periodic functions from Lp, 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.

E-print: arXiv:1304.3329

Published: 9 April 2013.

M. Potapov
Department of Mechanics and Mathematics
Moscow State University
Moscow 119991 Russia

B. Simonov
Department of Applied Mathematics
Volgograd State Technical University
Volgograd 400005 Russia

S. Tikhonov
ICREA and Centre de Recerca Matemtàtica
Apartat 50, Bellaterra, Barcelona 08193 Spain