Methods of Summability of Fourier Series. Moduli of Smoothness and K-Functionals
In this chapter general results on multipliers (Chapter 7) and sufficient conditions for representing a function as the Fourier transform (Chapter 6) are applied (Section 8.1) to the study of regularity (in one or another sense) of summability methods of simple and multiple Fourier series (classical and nonclassical), to defining exact rate of approximation of an individual function (Sections 8.2 – 8.4) via moduli of smoothness (defined in a special way for the multiple case and for p < 1). By this, K-functionals of a couple of spaces of smooth functions on the torus and poly-disk are computed as well (Section 8.3); this play a significant role in the real method of interpolation (see A.8.5).
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