01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
23.07.1939
E-mail:
Keywords:
Fourier series,
Fourier transforms,
Walsh series,
Hardy operator,
Bellman operator,
Hardy–Littlewood operator,
dyadic integral,
dyadic derivative,
approximation by the convolutions,
bases of shifts of a function,
functions of bounded generalized variation.
Subject:
The Gibbs phenomenon for Riesz spherical means of multiple Fourier series was discovered and the Gibbs constants for these means from below were estimated. The necessary and sufficient conditions for convergence in Pringsheim sense of multiple Fourier series of functions of bounded $\Phi$-variation of Hardy type were obtained. The boundedness of the Hardy operator in real Hardy spaces $H(R)$ and $H(T)$ was proved. The similar result for dyadic Hardy operator was also obtained. The analogue of tauberian theorem of Wiener in dyadic harmonic analysis was proved. As a corollary the following two criteria were obtained: 1) the linear hull of the set $\{f(\cdot\oplus y):y\ge0\}$ of dyadic shifts of a given function $f\in L(\mathbb{R}_+)$ is dens in the space $L(\mathbb{R}_+)$ iff the Walsh–Fourier transform $\tilde f(x)$ is not equal to zero on positive half-line $\mathbb{R}_+$ (dyadic analogue of the criterion of Wiener); 2) in order the linear hull of the set $\{f(\cdot\oplus y):0\le y<1\}$ of all dyadic shifts of the given function $f\in L[0,1)$ be dens in the space $L[0,1)$, it is necessary and sufficient that all Walsh–Fourier coefficients of the function $f\in L[0,1)$ are not equal to zero.
Biography
Graduated from Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University (MSU) in 1961 (department of theory of functions and functional analysis). Ph.D. thesis was defended in 1964. D.Sci. thesis was defended in 1975. A list of my works contains more than 80 titles. I am the member of Organizing Committees of Saratov Winter Schools on Function Theory (since 1988), Voronezh Winter Schools on Function Theory (since 1993) and Kazan Summer Schools on Function Theory (since 1995).
Main publications:
B. I. Golubov, A. V. Efimov, V. A. Skvortsov. Ryady i preobrazovaniya Uolsha. Teoriya i primeneniya. M.: Nauka, 1987. (B. Golubov, A. Efimov, V. Skvortsov. Walsh series and transforms. Theory and applications. Kluver Academic Publishers, Dordrecht, Boston, London, 1991).
B. I. Golubov. Elementy dvoichnogo analiza. M.: MGUP, 2005.
B. I. Golubov. Ogranichennost operatorov Khardi i Khardi–Littlvuda v prostranstvakh Re H i BMO // Matem. sb., t. 188, # 7 (1997), 93–106.
B. I. Golubov. Ob analoge neravenstva Khardi dlya preobrazovaniya Fure–Uolsha // Izv. RAN. Ser. matem., t. 65, # 3 (2001), 3–14.
B. I. Golubov. Dvoichnyi analog tauberovoi teoremy Vinera i smezhnye voprosy // Izv. RAN. Ser. matem., t. 67, # 1, (2003), 33–58.
B. I. Golubov. O modifitsirovannom silnom dvoichnom integrale i proizvodnoi // Matem. sb., t. 193, # 4 (2002), 37–60.
S. S. Volosivets, B. I. Golubov, “Modified Hardy and Hardy–Littlewood fractional operators in Morrey–Herz spaces and their commutators in weighted spaces”, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 171 (2019), 70–77
2.
S. S. Volosivets, B. I. Golubov, “Fractional modified Hardy and Hardy–Littlewood operators and their commutators”, Izv. Vyssh. Uchebn. Zaved. Mat., 2019, 9, 16–26
2018
3.
S. S. Volosivets, B. I. Golubov, “Generalized absolute convergence of series from Fourier coeficients by systems of Haar type”, Izv. Vyssh. Uchebn. Zaved. Mat., 2018, 1, 10–20; Russian Math. (Iz. VUZ), 62:1 (2018), 7–16
2015
4.
S. S. Volosivets, B. I. Golubov, “Uniform Convergence and Integrability of Multiplicative Fourier Transforms”, Mat. Zametki, 98:1 (2015), 44–60; Math. Notes, 98:1 (2015), 53–67
2013
5.
S. S. Volosivets, B. I. Golubov, “Fourier transforms in generalized Lipschitz classes”, Tr. Mat. Inst. Steklova, 280 (2013), 126–137; Proc. Steklov Inst. Math., 280 (2013), 120–131
2012
6.
B. I. Golubov, “Absolute convergence of double series of Fourier–Haar coefficients for functions of bounded $p$-variation”, Izv. Vyssh. Uchebn. Zaved. Mat., 2012, 6, 3–13; Russian Math. (Iz. VUZ), 56:6 (2012), 1–10
7.
B. I. Golubov, “Spherical Jump of a Function and the Bochner–Riesz Means of Conjugate Multiple Fourier Series and Fourier Integrals”, Mat. Zametki, 91:4 (2012), 506–514; Math. Notes, 91:4 (2012), 479–486
2010
8.
S. S. Volosivets, B. I. Golubov, “Weighted integrability of multiplicative Fourier transforms”, Tr. Mat. Inst. Steklova, 269 (2010), 71–81; Proc. Steklov Inst. Math., 269 (2010), 65–75
2008
9.
S. S. Volosivets, B. I. Golubov, “Hardy and Bellman operators in spaces connected with $H(\mathbb T)$ and $BMO(\mathbb T)$”, Izv. Vyssh. Uchebn. Zaved. Mat., 2008, 5, 4–13; Russian Math. (Iz. VUZ), 52:5 (2008), 1–8
2007
10.
B. I. Golubov, “Dyadic distributions”, Mat. Sb., 198:2 (2007), 67–90; Sb. Math., 198:2 (2007), 207–230
2006
11.
B. I. Golubov, “Modified Dyadic Integral and Fractional Derivative on $\mathbb R_+$”, Mat. Zametki, 79:2 (2006), 213–233; Math. Notes, 79:2 (2006), 196–214
2005
12.
B. I. Golubov, “Fractional Modified Dyadic Integral and Derivative on $\mathbb{R}_+$”, Funktsional. Anal. i Prilozhen., 39:2 (2005), 64–70; Funct. Anal. Appl., 39:2 (2005), 64–70
2003
13.
B. I. Golubov, “A dyadic analogue of Wiener's Tauberian theorem and some related questions”, Izv. RAN. Ser. Mat., 67:1 (2003), 33–58; Izv. Math., 67:1 (2003), 29–53
2002
14.
B. I. Golubov, “A modified strong dyadic integral and derivative”, Mat. Sb., 193:4 (2002), 37–60; Sb. Math., 193:4 (2002), 507–529
2001
15.
B. I. Golubov, “On an analogue of Hardy's inequality for the Walsh–Fourier”, Izv. RAN. Ser. Mat., 65:3 (2001), 3–14; Izv. Math., 65:3 (2001), 425–435
1999
16.
B. I. Golubov, “On dyadic analogues of Hardy and Hardy–Littlewood operators”, Sibirsk. Mat. Zh., 40:6 (1999), 1244–1252; Siberian Math. J., 40:6 (1999), 1051–1058
1998
17.
B. I. Golubov, “The Hardy and Bellman transforms of the spaces $H^1$ and BMO”, Mat. Zametki, 63:3 (1998), 475–478; Math. Notes, 63:3 (1998), 418–421
18.
B. I. Golubov, “An analogue of a theorem of Titchmarsh for Walsh-Fourier transformations”, Mat. Sb., 189:5 (1998), 69–86; Sb. Math., 189:5 (1998), 707–725
1997
19.
B. I. Golubov, “Boundedness of the Hardy and the Hardy–Littlewood operators in the spaces $\operatorname {Re}H^1$ and $\mathrm {BMO}$”, Mat. Sb., 188:7 (1997), 93–106; Sb. Math., 188:7 (1997), 1041–1054
1994
20.
B. I. Golubov, “On a theorem of Bellman on Fourier coefficients”, Mat. Sb., 185:11 (1994), 31–40; Russian Acad. Sci. Sb. Math., 83:2 (1995), 321–330
1985
21.
B. I. Golubov, “Absolute convergence of multiple Fourier series”, Mat. Zametki, 37:1 (1985), 13–24; Math. Notes, 37:1 (1985), 8–15
1982
22.
B. I. Golubov, “Multiple series and Fourier integrals”, Itogi Nauki i Tekhn. Ser. Mat. Anal., 19 (1982), 3–54; J. Soviet Math., 24:6 (1984), 639–673
1981
23.
B. I. Golubov, “Asymptotic behavior of singular multiple integrals for differentiable functions”, Mat. Zametki, 30:5 (1981), 749–762; Math. Notes, 30:5 (1981), 873–880
24.
B. I. Golubov, “A generalized symmetric derivative and the summability of multiple trigonometric series by the Lebesgue method”, Sibirsk. Mat. Zh., 22:6 (1981), 15–21; Siberian Math. J., 22:6 (1981), 815–820
1980
25.
B. I. Golubov, “On the rate of convergence of integrals of Gauss–Weierstrass type for functions of several variables”, Izv. Akad. Nauk SSSR Ser. Mat., 44:6 (1980), 1255–1278; Math. USSR-Izv., 17:3 (1981), 455–475
26.
B. I. Golubov, “The Abel-Poisson summation method for multiple Fourier series”, Mat. Zametki, 27:1 (1980), 49–59; Math. Notes, 27:1 (1980), 28–33
1979
27.
B. I. Golubov, “On convergence of singular integrals of Gauss–Weierstrass type for functions of several variables”, Dokl. Akad. Nauk SSSR, 248:5 (1979), 1044–1048
28.
B. I. Golubov, “On the summability method of Abel–Poisson type for multiple Fourier integrals”, Mat. Sb. (N.S.), 108(150):2 (1979), 229–246; Math. USSR-Sb., 36:2 (1980), 213–229
1977
29.
B. I. Golubov, “On the summability of Fourier integrals by Riesz spherical means”, Mat. Sb. (N.S.), 104(146):4(12) (1977), 577–596; Math. USSR-Sb., 33:4 (1977), 501–518
1976
30.
B. I. Golubov, “The summability of conjugate multiple Fourier integrals by Riesz means”, Uspekhi Mat. Nauk, 31:5(191) (1976), 237–238
1975
31.
B. I. Golubov, “Approximation of functions of several variables by spherical Riesz means”, Mat. Zametki, 17:2 (1975), 181–191; Math. Notes, 17:2 (1975), 108–113
32.
B. I. Golubov, “On convergence of Riesz spherical means of multiple Fourier series”, Mat. Sb. (N.S.), 96(138):2 (1975), 189–211; Math. USSR-Sb., 25:2 (1975), 177–197
1974
33.
B. I. Golubov, “Convergence of Riesz spherical means of multiple Fourier series”, Dokl. Akad. Nauk SSSR, 215:1 (1974), 31–34
34.
B. I. Golubov, “The approximation of a Hölder class of two variables by Riesz spherical means”, Mat. Zametki, 15:1 (1974), 33–43; Math. Notes, 15:1 (1974), 20–25
35.
B. I. Golubov, “The convergence of the double Fourier series of functions of bounded generalized variation. II”, Sibirsk. Mat. Zh., 15:4 (1974), 767–783; Siberian Math. J., 15:4 (1974), 546–557
36.
B. I. Golubov, “The convergence of the double Fourier series of functions of bounded generalized variation. I”, Sibirsk. Mat. Zh., 15:2 (1974), 262–291; Siberian Math. J., 15:2 (1974), 183–204
1973
37.
B. I. Golubov, “The asymptotic $L_p$-norm of differentiated Fourier sums of functions of bounded variation”, Izv. Akad. Nauk SSSR Ser. Mat., 37:2 (1973), 399–421; Math. USSR-Izv., 7:2 (1973), 401–423
1972
38.
B. I. Golubov, “Functions of generalized bounded variation, convergence of their Fourier series and conjugate trigonometric series”, Dokl. Akad. Nauk SSSR, 205:6 (1972), 1277–1280
39.
B. I. Golubov, “Double Fourier series, and functions of bounded variation”, Izv. Vyssh. Uchebn. Zaved. Mat., 1972, 12, 55–68
40.
B. I. Golubov, “Determination of the jump of a function of bounded $p$-variation by its Fourier series”, Mat. Zametki, 12:1 (1972), 19–28; Math. Notes, 12:1 (1972), 444–449
41.
B. I. Golubov, “Asymptotic behavior of the $L_p$-norms of differentiated Fourier sums of functions of bounded variation”, Uspekhi Mat. Nauk, 27:6(168) (1972), 235–236
42.
B. I. Golubov, “On the convergence of Riesz spherical means of multiple Fourier series and integrals of functions of bounded generalized variation”, Mat. Sb. (N.S.), 89(131):4(12) (1972), 630–653; Math. USSR-Sb., 18:4 (1972), 635–658
43.
B. I. Golubov, “Best approximations of functions in the $L_p$ metric by Haar and Walsh polynomials”, Mat. Sb. (N.S.), 87(129):2 (1972), 254–274; Math. USSR-Sb., 16:2 (1972), 265–285
44.
B. I. Golubov, “Tests of the continuity of functions of bounded $p$-variation”, Sibirsk. Mat. Zh., 13:5 (1972), 1002–1015; Siberian Math. J., 13:5 (1972), 693–702
1971
45.
B. I. Golubov, “Series in the Haar system”, Itogi Nauki. Ser. Matematika. Mat. Anal. 1970, 1971, 109–146; J. Soviet Math., 1:6 (1973), 704–726
46.
B. I. Golubov, “The $p$-variation of functions of two variables”, Izv. Vyssh. Uchebn. Zaved. Mat., 1971, 9, 40–49
1969
47.
B. I. Golubov, “The $p$-variation of functions”, Mat. Zametki, 5:2 (1969), 195–204; Math. Notes, 5:2 (1969), 119–124
1968
48.
B. I. Golubov, “On functions of bounded $p$-variation”, Izv. Akad. Nauk SSSR Ser. Mat., 32:4 (1968), 837–858; Math. USSR-Izv., 2:4 (1968), 799–819
49.
B. I. Golubov, “The Fourier integral and the continuity of functions of bounded $p$-variation”, Izv. Vyssh. Uchebn. Zaved. Mat., 1968, 11, 83–92
50.
B. I. Golubov, “Functions of bounded $p$-variation”, Uspekhi Mat. Nauk, 23:1(139) (1968), 219–220
51.
B. I. Golubov, “A certain class of complete orthogonal systems”, Sibirsk. Mat. Zh., 9:2 (1968), 297–314; Siberian Math. J., 9:2 (1968), 225–239
1967
52.
B. I. Golubov, “Continuous functions of bounded $p$-variation”, Mat. Zametki, 1:3 (1967), 305–312; Math. Notes, 1:3 (1967), 203–207
1966
53.
B. I. Golubov, A. I. Rubinshtein, “A class of convergence systems”, Mat. Sb. (N.S.), 71(113):1 (1966), 96–115
1965
54.
B. I. Golubov, “On absolute convergence of series in Haar's system”, Uspekhi Mat. Nauk, 20:5(125) (1965), 198–202
1964
55.
B. I. Golubov, “Fourier series of continuous functions relative to a Haar system”, Dokl. Akad. Nauk SSSR, 156:2 (1964), 247–250
56.
B. I. Golubov, “On Fourier series of continuous functions with respect to a Haar system”, Izv. Akad. Nauk SSSR Ser. Mat., 28:6 (1964), 1271–1296
57.
B. I. Golubov, “On the summability of sequences”, Izv. Vyssh. Uchebn. Zaved. Mat., 1964, 4, 47–55
2018
58.
B. I. Golubov, B. S. Kashin, L. Yu. Kossovich, S. P. Sidorov, A. P. Khromov, A. N. Chumachenko, “19th International Saratov Winter School “Contemporary problems of function theory and their applications"”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 18:3 (2018), 354–365
2016
59.
B. I. Golubov, B. S. Kashin, L. Yu. Kossovich, S. P. Sidorov, A. P. Khromov, A. N. Chumachenko, “18th International Saratov Winter School “Contemporary Problems of Function Theory and Their Applications””, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 16:4 (2016), 485–487
60.
M. V. Balashov, O. V. Besov, B. I. Golubov, V. V. Goryainov, V. N. Diesperov, S. I. Dudov, G. E. Ivanov, S. P. Konovalov, R. V. Konstantinov, A. B. Kurzhanskii, S. R. Nasyrov, A. G. Sergeev, V. V. Starkov, V. M. Tikhomirov, M. I. Shabunin, “Evgenii Sergeevich Polovinkin (on his 70th birthday)”, Uspekhi Mat. Nauk, 71:5(431) (2016), 187–190; Russian Math. Surveys, 71:5 (2016), 983–987
61.
B. I. Golubov, B. S. Kashin, T. P. Lukashenko, M. G. Plotnikov, M. A. Skopina, A. P. Solodov, A. M. Stepin, N. N. Kholshchevnikova, “Valentin Anatol'evich Skvortsov (on his 80th birthday)”, Uspekhi Mat. Nauk, 71:1(427) (2016), 184–186; Russian Math. Surveys, 71:1 (2016), 175–177
2015
62.
B. I. Golubov, B. S. Kashin, L. Yu. Kossovich, S. P. Sidorov, A. P. Khromov, “XVII International Saratov Winter School «Contemporary Problems of the Function Theory and its Applications». Dedicated to the 150th Anniversary of V. A. Steklov”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 15:3 (2015), 357–359
2012
63.
B. I. Golubov, B. S. Kashin, L. Yu. Kossovich, S. P. Sidorov, A. P. Khromov, “16 Saratov winter school “Contemporary problems of function theory and its applications””, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 12:2 (2012), 114–115
2011
64.
O. V. Besov, S. V. Bochkarev, B. I. Golubov, A. A. Gonchar, M. I. D'yachenko, V. V. Kozlov, S. V. Konyagin, Yu. V. Malykhin, S. M. Nikol'skii, M. K. Potapov, V. A. Sadovnichii, S. A. Telyakovskii, “Boris Sergeevich Kashin (on his 60th birthday)”, Uspekhi Mat. Nauk, 66:4(400) (2011), 189–191; Russian Math. Surveys, 66:4 (2011), 825–828
2008
65.
B. I. Golubov, B. S. Kashin, “Introduction”, Izv. Vyssh. Uchebn. Zaved. Mat., 2008, 5, 3
66.
B. I. Golubov, A. A. Gonchar, B. S. Kashin, S. M. Nikol'skii, A. M. Olevskii, M. K. Potapov, “On the 80th birthday of Petr Lavrent'evich Ul'yanov”, Uspekhi Mat. Nauk, 63:5(383) (2008), 203–207; Russian Math. Surveys, 63:5 (2008), 989–994
2006
67.
B. I. Golubov, S. M. Nikol'skii, S. A. Telyakovskii, P. L. Ul'yanov, “Károly Tandori (obituary)”, Uspekhi Mat. Nauk, 61:1(367) (2006), 165–168; Russian Math. Surveys, 61:1 (2006), 161–164
1982
68.
B. Golubov, “Cohn D. L. Measure theory. Boston etc.: Birkhäuser, 1980, IX+373 p. (Book review)”, Zh. Vychisl. Mat. Mat. Fiz., 22:4 (1982), 1016–1017; U.S.S.R. Comput. Math. Math. Phys., 22:4 (1982), 258–259
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