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.
Biography
Graduated from Faculty of Mathematics and Mechanics of Voronezh State University (VSU) in 1969 (department of theory of function and geometry). Ph.D. thesis was defended in 1973. D.Sci. thesis was in 1993. A list of my works contains more than 133 titles. Since 2001 I and I. Y. Novikov have led the research seminar at VSU on Fourier analysis and Wavelets theory.
I worked in 1997 at university of city of Budapest under the invitation of the professor F. Schipp, in 2000 at university of city of Seville under the invitation of the professor G. Curbera.
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).
S. Krivobokova, V. A. Rodin, “Algorithm and program for graphical selection of the Pareto set in a point array”, Applied Mathematics & Physics, 53:2 (2021), 125–131
V. A. Rodin, S. V. Sinegubov, “Generalized Kelly strategy”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 14:2 (2021), 100–107
2020
3.
A. V. Kalach, V. A. Rodin, S. V. Sinegubov, “Optimizing fire-fighting water supply systems using spatial metrics”, J. Comp. Eng. Math., 7:4 (2020), 3–16
V. A. Rodin, S. V. Sinegubov, “Stochastic modeling of surfaces with modified Gauss functions”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 172 (2019), 96–103
5.
V. A. Rodin, S. V. Sinegubov, “On reliability of large-scale nets constructed from identical elements”, Izv. Vyssh. Uchebn. Zaved. Mat., 2019, no. 5, 56–62; Russian Math. (Iz. VUZ), 63:5 (2019), 51–56
6.
V. A. Rodin, S. V. Sinegubov, “Mathematical terrain modelling with the help of modified Gaussian functions”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 12:3 (2019), 63–73
2005
7.
V. N. Dumachev, V. A. Rodin, “Evolution of antagonistic-cooperating populations on base of two-parametrical Ferhjust–Pirls model”, Matem. Mod., 17:7 (2005), 11–22
G. P. Curbera, V. A. Rodin, “Multipliers on the Set of Rademacher Series in Symmetric Spaces”, Funktsional. Anal. i Prilozhen., 36:3 (2002), 87–90; Funct. Anal. Appl., 36:3 (2002), 244–246
V. A. Rodin, “Shift of Spaces by Means of the Hardy and Bellman Transforms”, Funktsional. Anal. i Prilozhen., 34:2 (2000), 89–91; Funct. Anal. Appl., 34:2 (2000), 154–155
10.
S. K. Gorlov, I. Ya. Novikov, V. A. Rodin, “Correction of Haar polynomials used in the compression of graphical information”, Izv. Vyssh. Uchebn. Zaved. Mat., 2000, no. 7, 6–10; Russian Math. (Iz. VUZ), 44:7 (2000), 4–8
1999
11.
V. A. Rodin, “Hardy and Bellman transformations in spaces close to $L_\infty$ and to $L_1$”, Zap. Nauchn. Sem. POMI, 262 (1999), 204–213; J. Math. Sci. (New York), 110:5 (2002), 3016–3021
V. A. Rodin, “Strong means and oscillation of multiple Fourier series in multiplicative systems”, Mat. Zametki, 63:4 (1998), 607–616; Math. Notes, 63:4 (1998), 533–541
V. A. Rodin, “Strong means and the oscillation of multiple Fourier–Walsh series”, Mat. Zametki, 56:3 (1994), 102–117; Math. Notes, 56:3 (1994), 948–959
V. A. Rodin, “Extensions of a Certain Weak Type Operator”, Funktsional. Anal. i Prilozhen., 27:1 (1993), 83–86; Funct. Anal. Appl., 27:1 (1993), 70–73
15.
V. A. Rodin, “The tensor BMO-property of the sequence of partial sums of a multiple Fourier series”, Mat. Sb., 184:10 (1993), 91–106; Russian Acad. Sci. Sb. Math., 80:1 (1995), 211–224
V. A. Rodin, “Rectangular oscillation of the sequence of partial sums of a multiple Fourier series and absence of the BMO property”, Mat. Zametki, 52:2 (1992), 152–154; Math. Notes, 52:2 (1992), 863–865
I. Ya. Novikov, V. A. Rodin, “Characterization of points of $p$-strong summability of trigonometric series, $p\geq 2$”, Izv. Vyssh. Uchebn. Zaved. Mat., 1988, no. 9, 58–62; Soviet Math. (Iz. VUZ), 32:9 (1988), 86–91
1982
23.
V. I. Ovchinnikov, V. D. Raspopova, V. A. Rodin, “Sharp estimates of the Fourier coefficients of summable functions and $K$-functionals”, Mat. Zametki, 32:3 (1982), 295–302; Math. Notes, 32:3 (1982), 627–631
V. A. Rodin, E. M. Semenov, “Complementability of the subspace generated by the Rademacher system in a symmetric space”, Funktsional. Anal. i Prilozhen., 13:2 (1979), 91–92; Funct. Anal. Appl., 13:2 (1979), 150–151
V. A. Rodin, “Membership of the sum of a cosine series with monotone coefficients in a symmetric space”, Izv. Vyssh. Uchebn. Zaved. Mat., 1979, no. 8, 60–64; Soviet Math. (Iz. VUZ), 3:8 (1979), 61–65
A. B. Gulisashvili, V. A. Rodin, E. M. Semenov, “Fourier coefficients of summable functions”, Mat. Sb. (N.S.), 102(144):3 (1977), 362–371; Math. USSR-Sb., 31:3 (1977), 319–328
V. A. Rodin, “The Hardy-Littlewood theorem for the cosine series in a symmetric space”, Mat. Zametki, 20:2 (1976), 241–246; Math. Notes, 20:2 (1976), 693–696
A. O. Vatulyan, V. A. Kabel'kov, T. N. Kabel'kova, S. B. Klimentov, A. I. Kondratenko, A. G. Kusraev, A. N. Nikiforov, A. È. Pasenchuk, V. A. Rodin, V. G. Safronenko, S. M. Sitnik, A. N. Tkachev, V. G. Fetisov, “Aleksandr Nikolaevich Kabel'kov (1947–2011)”, Vladikavkaz. Mat. Zh., 14:2 (2012), 74–77
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