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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
E-mail: mkpotapov@mail.ru
B. Simonov
Department of Applied Mathematics
Volgograd State Technical University
Volgograd 400005 Russia
E-mail: simonov-b2002@yandex.ru
S. Tikhonov
ICREA and Centre de Recerca Matemtàtica
Apartat 50, Bellaterra, Barcelona 08193 Spain
E-mail:stikhonov@crm.cat
Homepage: http://www.icrea.cat/Web/ScientificStaff/Sergey-Tikhonov-479