By V.P. Khavin, N.K. Nikol'skii, J. Peetre, Sh.A. Alimov, R.R. Ashurov, E.M. Dyn'kin, S.V. Kislyakov, A.K. Pulatov

With the basis laid within the first quantity (EMS 15) of the Commutative Harmonic research subseries of the Encyclopaedia, the current quantity takes up 4 complex issues within the topic: Littlewood-Paley concept for singular integrals, extraordinary units, a number of Fourier sequence and a number of Fourier integrals. The authors suppose that the reader understands the basics of harmonic research and with easy useful research. The exposition begins with the fundamentals for every subject, additionally taking account of the ancient improvement, and ends by way of bringing the topic to the extent of present learn. desk of Contents I. a number of Fourier sequence and Fourier Integrals. Sh.A.Alimov, R.R.Ashurov, A.K.Pulatov II. tools of the idea of Singular Integrals. II: Littlewood Paley idea and its purposes E.M.Dyn'kin III.Exceptional units in Harmonic research S.V.Kislyakov

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**Example text**

Ifhowever s < N;l the localization principle breaks down. 10) < lxi< 26 and f(x) = 0 otherwise. 10), so that the localization principle cannot be true. 4. Equiconvergence of the Fourier Series and the Fourier Integral. 11) 1. Multiple Fourier Series and Fourier Integrals let us fix our attention to the fact that the operator f integral operator (T~(X, f) = f D~(x jTN with kernel D~(x 33 I--t (T~(x, f) is an y)f(y)dy, - y) (the Dirichlet kernei) given by D~(x-y)=(27r)-N L Inl<#' (1_ln~2)8ein(x_y).

Otherwise put, the system {einx}~=_oo constitutes a basis in Lp(T1 ) for 1 < P < 00. In the extremal cases p = 1 and p = 00 this is not true. For rectangular summation the basis property in Lp(TN ), 1 < P < 00, remains in force for any N > 1, as was proved by Sokol-Sokolowski (1947). In fact, the following more general statement is true: if the operators Pn : Lp(T1 ) - t Lp(T1 ) and Qn : Lp(T1 ) - t Lp(T1 ) are uniformly bounded, then the operators Rnmf(x, y) = PnQmf(x, y), where Pn acts on the variable x E Tl and Qm on the variable Y E Tl, are likewise uniformly bounded.

They eonsist of those functions which admit an approximation in the Lp('JI'N)-metric by trigonometrie polynomials of degree n 6 with remainder O(n- l ). Finally, a very general approach to this problem is provided by the Besov classes B~8 with an auxiliary parameter (): if p = () these are the Sobolev-Slobodeckiz classes, if () = 00 the Nikol'skil classes and if p = () = 2 the Liouville classes L~ (Nikol'skil (1977)). In order not to eomplieate the presentation we will in the sequel limit ourselves to the Liouville classes L~, justifying this ehoice by the fact that all the above classes approximately eoincide with L~ (Nikol'skil (1977)): where A~ denotes any of the classes of Sobolev-Slobodeckil, Nikol'skil or Besov (the latter with arbitrary ()).