Speciality:
01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
01.01.1947
E-mail: ,
Website: https://www.rodin-va.narod.ru Keywords: strong summability of the simple and multiple Fourier series in the trigonometric system and Price system; BMO-property of the partial sums; Hardy, Bellman and Cesaro transforms in Fourier analysis; Rademacher and Fourier series in symmetric spaces; multiplication operator on the Rademacher series; the fractal condition in town-planning.
Subject:
BMO-property of the sequence of partial sums of the Fourier series of an summable function is proved. This property has allowed to prove a conjecture of V. Totik about $L_M$ ($M(u)=\exp|u|-1$) — strong summability of the Fourier series almost everywhere. The description of points in which summation is received occurs. The uniform scheme of the proof for trigonometric series and for series in system of characters of various zero-dimensional groups is received. Researches finish a cycle of works of many mathematicians (Hardy, Littlewood, Marcinkiewicz, Totik, Schipp, Gabisoniya, Oskolkov, Gogoladse etc.). As has shown Karaguliyan these investigation in the certain sense is final. For multiple Fourier series it is proved p-strong summability of Fourier series of the function from appropriate Orliczclass. It is established the tensor BMO-property (TBMO) and it is shown, that actually BMO-property in multiple case is absent. The general assertion, relating the phenomenon of rectangular oscillation of the sequence of rectangular partial sums of a multiple Fourier series and the strong summability of that series, is established. A number of papers (with E. M. Semenov) were devoted to studying of the Rademacher series in symmetric spaces. Together with G. Kurbera (Spain) the behaviour of the operator of multiplication to the Rademacher series in symmetric spaces is investigated. In a class of symmetric spaces the exact bounds of shift of spaces BMO, Marcinkiewicz and Orlicz spaces located "close" to the space $L_\infty$ under action on a trigonometrical series of the Hardy operator, Bellman and Chezaro operator are received. With E. V. Rodina the new phenomenon in changes of the megacities connected to town-planning transformations (on an example of areas of Tokyo) is established. The phenomenon connected with the "fractal dimension of Tokyo's streets" is revealed.
Main publications:
Rodin V. A., Semenov E. M. Rademacher series in symmetric spaces // Analysis Math. V. 1, no. 3. 1975. P. 207–222.
Rodin V. A. The BMO-property of the partial sums of a Fourier series // American Math. Soc. Soviet Math. Dokl. V. 44, no. 1. 1992. P. 294–296.
Rodin V. A. Shift of Spaces by Means of the Hardy and Bellman Transforms // American Math. Soc. Functional Analysis and Its Applications, V. 34, no. 2. 2000. P. 151–152.
Rodin V. A., Rodina E. V. The fractal dimension of Tokyo"s streets // FRACTALS, V. 8, no. 4. 2000. P. 413–418.
Curbera G. P., Rodin V. A. Multiplication operator on the Rademacher series in the Orlicz spaces which are "close to $L_\infty$" // Mathematical Proceedings of the Cambridge Philosophical Society. 2002 (to appear).
