Ukrainian Mathematical Bulletin

Volume 2, Issue 2, 2005 pp. 239-284.

Fourier Multipliers and $K$-Functionals in Spaces of Smooth Functions

Authors: Roal'd M. Trigub
Author institution: Donetsk National University, 24 Universitetskaya Str., 83055, Donetsk, Ukraine

Summary: We present a survey of the results concerning Fourier multipliers in the spaces $C$ and $L$. The method of multipliers is applied to the investigation of approximating properties of various procedures of summation of simple and multiple Fourier series and integrals and evaluation, to within equivalence, of $K$-functionals in the spaces of smooth functions (on the torus ${\Bbb T}^m$ and in the space ${\Bbb R}^m$) specified by various differential operators. This case is characterized by the appearance of new modules of smoothness. We present both exact and asymptotically exact results. For detailed proofs, see the bibliography (87 cited items) and the monograph by R. Trigub and E. Belinsky, {\it Fourier Analysis and Approximations of Functions,} Kluwer, Dordrecht (2004). We also discuss some unsolved problems.

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