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 foundation laid within the first quantity (EMS 15) of the Commutative Harmonic research subseries of the Encyclopaedia, the current quantity takes up 4 complex subject matters within the topic: Littlewood-Paley thought for singular integrals, unheard of units, a number of Fourier sequence and a number of Fourier integrals. The authors think that the reader knows the basics of harmonic research and with easy practical research. The exposition starts off with the fundamentals for every subject, additionally taking account of the old improvement, and ends through 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 speculation of Singular Integrals. II: Littlewood Paley thought and its functions E.M.Dyn'kin III.Exceptional units in Harmonic research S.V.Kislyakov

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V. Zhizhiashvili (1971), (1973). 30) then the Fourier series converges uniformly to f on']['N, provided we use reet angular summation. 28) but now with x E ']['N, Y E ']['N. I. 30) is sharp, and also in the case of quadratic summability it cannot be improved: there exists a function f such that w(6, f) 1)-N ' = 0(1) (log t 6 -+ 0, but whose Fourier series diverges at the point x = 0 under quadratic summation. 30) is based on the fact that the Dirichlet kernel Dm(x) for reet angular partial sums is a product of onedimensional Dirichlet kernels (cf.

22) The convergence of EI'I(O) for all 1 E L~(TN) of the above type would imply, in view of the Banach~Steinhaus theorem, the boundedness of the 7 7mnslator's Note. That is, a function with compact support. This terminology is, regretfully, rarely used in the Western literat ure. 36 Sh. A. Alimov, R. R. Ashurov, A. K. L(x) in the Lq(E) metric, where ~ = 1- ~. 19). 20), where gE Lp(E), with a divergent Fourier series at the origin for spherical summation. With the aid of this function f it is also not hard to construct a function satisfying all the conditions of Il'in's theorem.

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). L) at the point { = n. L). 13) constitutes one of the main tools with the aid of which one studies spherical partial sums of multiple Fourier series. L). 7), we obtain If s ]RN: > N ;1 the series to the right is uniformly convergent for x E {x E IXjl :::; 27r - ö}.

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