Some Topics in Fourier Analysis and Approximation Theory

This manuscript presents shortly the results obtained by participants of the scientific seminar which is held more than twenty years under leadership of the author at Donetsk University. In the list of references main publications are given. These results are published in serious scientific journals and reported at various conferences, including international ones at Moscow,ICM66; Kaluga,1975; Kiev,1983; Haifa,1994; Z\"urich,ICM94; Moscow,1995. The area of investigation is the Fourier analysis and the theory of approximation of functions. Used are methods of classical analysis including special functions, Banach spaces, etc., of harmonic analysis in finitedimensional Euclidean space, of Diophantine analysis, of random choice, etc. The results due to the author and active participants of the seminar, namely E. S. Belinskii, O. I. Kuznetsova, E. R. Liflyand, Yu. L. Nosenko, V. A. Glukhov, V. P. Zastavny, Val. V. Volchkov, V. O. Leontyev, and others, are given. Besides the participants of the seminar and other mathematicians from Donetsk, many mathematicians from other places were speakers at the seminar, in particular, A.A. Privalov, Z.A. Chanturia, Yu.A. Brudnyi, N.Ya. Krugljak, V.N. Temlyakov, B.D. Kotlyar, A.N. Podkorytov, M.A. Skopina, A.A. Ligun, A.S. Romanyuk, V.A. Martirosyan. Besides the papers of the participants of the seminar, only monographs and survey papers are given in the list of references. The most part of this preprint will appear soon in extended form in the author's book "Fourier Analysis and Approximation Theory", World Federation Publishers, Inc., P.O.Box 48654, Atlanta, GA 30362-0654, USA.

[1]  N. Wiener.  The Fourier Integral: and certain of its Applications , 1933, Nature.

[2]  G. Tolstov.  Fourier Series , 1962 .

[3]  N. Akhiezer,et al.  The Classical Moment Problem. , 1968 .

[4]  W. Rudin.  Function theory in polydiscs , 1969 .

[5]  J. Kahane.  Séries de Fourier absolument convergentes , 1970 .

[6]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[7]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[8]  W. Trebels.  Multipliers for (C,α)-bounded Fourier expansions in Banach spaces and approximation theory , 1973 .

[9]  Le Baron O. Ferguson,et al.  Approximation by Polynomials with Integral Coe cients , 1980 .

[10]  B. Golubov,et al.  Multiple fourier series and integrals , 1984 .

[11]  A. Olevskiǐ.  Modifications of functions and Fourier series , 1985 .

[12]  G. Andrews,et al.  Classical orthogonal polynomials , 1985 .

[13]  R. Varga.  Functional Analysis and Approximation Theory in Numerical Analysis , 1987 .

[14]  V. N. Temli︠a︡kov.  Approximation of functions with bounded mixed derivative , 1989 .

[15]  B. Golubov,et al.  Walsh Series and Transforms , 1991 .

[16]  V. Ivanov,et al.  Exact Constants in Approximation Theory , 1991 .

[17]  M. I. D’yachenko.  Some problems in the theory of multiple trigonometric series , 1992 .

[18]  V. N. Temli︠a︡kov.  Approximation of periodic functions , 1993 .

[19]  A. Timan.  Theory of Approximation of Functions of a Real Variable , 1994 .

[20]  I. Netuka,et al.  Mean Value Property and Harmonic Functions , 1994 .