2014-№2(43) Article 14

A.N. Konyonkov

Conjugation operator in multidimensional spaces of periodic functions. P. 159-167.

UDC517.5

The article focuses on conjugators for spaces of periodic functions with several variables. The author presents a scale-space which depends on four parameters. This scale-space is a generalization of a scale of Besov spaces of periodic functions and when parameter 4 is zero the scale-space equals the scale of Besov spaces. The paper maintains that conjugator operates with a certain shift of parameter 4. When space dimension is more than 1, the smoothness of function is reduced. The paper proves that for multidimensional Gelder spaces the module of continuity of a complementary function can differ from the module of contitnuity of a Gelder space for the logarithm power, which depends on space dimension.

Gilbert transformation, complementary function, Besov space, Gelder space, Sigmund space.

References:

1. Akhiyezer, N.I. Lektsii po teorii approksimatsii [Lectures on the Theory of Approximation] [Text]. – Moscow: Science, 1965.

2. Bari, N.K. Nailuchshiye priblizheniya i differentsial’nyye svoystva dvukh sopryazhennykh funktsiy [Best approximations and differential properties of two conjugate functions] [Text] / N.K. Bari, Stechkin S.B. // 1956. – Vol. 5. – P. 483–522.

3. Zhak, I.Ye. Po povodu odnoy teoremy L. Chezari o sopryazhennykh funktsiyakh dvukh peremennykh [On a theorem of Cesari on conjugate functions of two variables] [Text] // Reports of AN. – 1952. – Vol. 87. – N 6. – P. 887–880.

4. Konenkov, A.N. Ogranichennost’ operatora sopryazheniya v prostranstvakh Besova [Boundedness of the operator interface in Besov’s spaces] [Text] / A.N. Konenkov, A.I. Syusyukalov // Doklady RAN. – Reports of the Academy of Sciences. – 2004. – Vol. 397. – N 4. – P. 449–452.

5. Tribel, KH. Teoriya funktsional’nykh prostranstv [Theory of function spaces] [Text]. – Moscow: Peace, 1986.

6. Tribel’, K.H. Teoriya interpolyatsii, funktsional’nyye prostranstva, differentsial’nyye operatory [Interpolation theory, function spaces, differential operators] [Text]. – Moscow: Peace, 1980.

7. Chesari, L. Sulle serie di Fourier delle funzioni lipschitziane di piu variabili [Text] // Annali della Scuola Normale Superiore di Pisa. – 1938. – N 2. – P. 279–295.

8. Privaloff, I. Sur les fonctions conjuguees [Text] // Bull. Soc. Math. Fr. – 1916. – Vol. 44. – P. 100–103.