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Molchanov, Vladimir Fedorovich

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Total publications: 32
Scientific articles: 26

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Professor
Doctor of physico-mathematical sciences (1988)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 27.02.1939
E-mail: ,
Keywords: theory of group representations; symmetric spaces; harmonic analysis on homogeneous spaces; quantization; canonical representations; boundary representations.

Subject:

In a series of my works (the 60 80 ies) the construction of harmonic analysis on semisimple symmetric spaces $G/H$ (non-Riemannian) of rank one was begun and completed. A description of the corresponding principal non-unitary series of representations was given. Principal notions of the theory were introduced ($H$-invariants, Fourier transform, Poisson transform, spherical functions) and corresponding methods were worked out. Plancherel formula was obtained explicitly ( in different variants, one of them is expansion of the delta function in terms of spherical functions). The Berezin quantization was transferred from Hermitian symmetric spaces to symplectic semisimple symmetric spaces. In particular, an important case of quantizations was described — the so–called polynomial quantization. A new form of the deformation decomposition (the decomposition of the Berezin transform) was offered using "generalized powers" (generalized Pochhammer symbols) instead of usual powers of a parameter. This form makes the decomposition natural and apparent and allows to compute it explicitly. Canonical representations on these symplectic spaces were studied - in connection with the construction of quantizations (decompositions into irreducible constitutients — right up to explicit formulae for one rank spaces). The canonical representations (sometimes called the Berezin representations) on Hermitian symmetric spaces were introduced by Berezin and Vershik–Gelfand–Graev. They are unitary representations. We consider the canonical representations in a much wider sense: we give up the condition of unitarity, they act on sufficiently extensive function spaces, in paricular, on spaces of distributions. Also boundary representations generated by canonical representations were studied. In particular, appearance of Jordan blocks in the decomposition of these representations was discovered. It is found that the decomposition of boundary representations is intimately connected with the meromorphic structure of Poisson and Fourier transforms associated with the canonical representations. These results (quantizations, canonical and boundary representations) can be transferred to a certain extent to some semisimple symmetric spaces which are not symplectic, for example, to hyperboloids of arbitrary signature. This work (quantizations, canonical and boundary representations etc.) is a part of what I call a non-unitary version of harmonic analysis, a new and promising field of research. For hyperboloids of Hermitian type, the holomorphic discrete series was investigated, Cauchy-Szego kernels were computed, projection operators on analytic and antianalytic series of irreducible unitary reresentations were explicitly found, an analogue of the Hilbert transform was introduced and computed. One of results — separation of series — was carried over to hyperboloids of arbitrary signature. For finite reflection groups, Poincare polynomials and series were explicitly computed.

Biography

Graduated with a first–class honours degree from Faculty of Mathematics and Mechanics of Lomonosov Moscow State University in 1962 (chair of theory of functions and functional analysis). Candidate dissertation (Ph.D. thesis) was defended in 1967. Doctor dissertation was defended in 1987. A list of my works contains about 100 titles. I have led the research seminar at Derzhavin Tambov State University on functional analysis.

Member of Moscow Mathematical Society Corresponding member of RANS (Russian Academy of Natural Science).

   
Main publications:
  • Molchanov V. F. Quantization on para-Hermitian symmetric spaces // Amer. Math. Soc. Transl. Ser. 2, vol. 175 (Adv. Math. Sci., 31), 1996, 81–96.
  • Dijk. G. van, Molchanov V. F. The Berezin form for rank one para-Hermitian symmetric spaces // J. Math. Pures Appl., 1998, 77, no. 8, 747–799.
  • Dijk. G. van, Molchanov V. F. Tensor products of maximal degenerate series representations of the group $SL(n,\Bbb R)$ // J. Math. Pures Appl., 1999, 78, no. 1, 99–119.

http://www.mathnet.ru/eng/person17888
List of publications on Google Scholar
List of publications on ZentralBlatt
https://mathscinet.ams.org/mathscinet/MRAuthorID/195585

Publications in Math-Net.Ru
2019
1. V. F. Molchanov, E. S. Yuryeva, “Integer triangles, Pell's equation and Chebyshev polynomials”, Russian Universities Reports. Mathematics, 24:126 (2019),  179–186  mathnet
2018
2. V. F. Molchanov, E. E. Kryukova, “Placements without neighbours”, Tambov University Reports. Series: Natural and Technical Sciences, 23:124 (2018),  655–665  mathnet
3. V. F. Molchanov, “Polynomial quantiztion and overalgebra for hyperboloid of one sheet”, Tambov University Reports. Series: Natural and Technical Sciences, 23:123 (2018),  353–360  mathnet
2017
4. V. F. Molchanov, “Berezin quantization as a part of the representation theory”, Tambov University Reports. Series: Natural and Technical Sciences, 22:6 (2017),  1235–1246  mathnet
2015
5. V. F. Molchanov, “Poisson and Fourier Transforms for Tensor Products”, Funktsional. Anal. i Prilozhen., 49:4 (2015),  50–60  mathnet  elib; Funct. Anal. Appl., 49:4 (2015), 279–288  isi  scopus
2012
6. V. F. Molchanov, “Radon transform on a space over a residue class ring”, Mat. Sb., 203:5 (2012),  119–134  mathnet  mathscinet  zmath  elib; Sb. Math., 203:5 (2012), 727–742  isi  scopus
2006
7. V. F. Molchanov, “Canonical representations on two-sheeted hyperboloids”, Zap. Nauchn. Sem. POMI, 331 (2006),  91–124  mathnet  mathscinet  zmath  elib; J. Math. Sci. (N. Y.), 141:4 (2007), 1432–1451  elib  scopus
2005
8. V. F. Molchanov, “Canonical Representations and Overgroups for Hyperboloids”, Funktsional. Anal. i Prilozhen., 39:4 (2005),  48–61  mathnet  mathscinet  zmath  elib; Funct. Anal. Appl., 39:4 (2005), 284–295  isi  scopus
1999
9. V. F. Molchanov, “Representations of pseudo-unitary groups associated with a cone”, Lobachevskii J. Math., 3 (1999),  221–241  mathnet  mathscinet  zmath
1997
10. V. F. Molchanov, “Separation of Series for Hyperboloids”, Funktsional. Anal. i Prilozhen., 31:3 (1997),  35–43  mathnet  mathscinet  zmath  elib; Funct. Anal. Appl., 31:3 (1997), 176–182  isi
1992
11. V. F. Molchanov, “On the Poincaré series of representations of finite reflection groups”, Funktsional. Anal. i Prilozhen., 26:2 (1992),  82–85  mathnet  mathscinet  zmath; Funct. Anal. Appl., 26:2 (1992), 143–145  isi
1990
12. V. F. Molchanov, “Harmonic analysis on homogeneous spaces”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 59 (1990),  5–144  mathnet  mathscinet  zmath
1986
13. V. F. Molchanov, “The Plancherel formula for pseudo-Riemannian symmetric spaces of rank $1$”, Dokl. Akad. Nauk SSSR, 290:3 (1986),  545–549  mathnet  mathscinet  zmath
14. V. F. Molchanov, “Spherical functions on pseudo-Riemannian symmetric spaces of rank $1$”, Dokl. Akad. Nauk SSSR, 287:5 (1986),  1054–1058  mathnet  mathscinet  zmath
1983
15. V. F. Molchanov, “Orbits of a stationary subgroup on a pseudo-Riemannian symmetric space of rank one”, Uspekhi Mat. Nauk, 38:5(233) (1983),  203–204  mathnet  mathscinet  zmath; Russian Math. Surveys, 38:5 (1983), 158–159  isi
1982
16. V. F. Molchanov, “Poincaré polynomials of representations of finite groups generated by reflections”, Mat. Zametki, 31:6 (1982),  837–845  mathnet  mathscinet  zmath; Math. Notes, 31:6 (1982), 423–427  isi
17. V. F. Molchanov, “Harmonic analysis on pseudo-Riemannian symmetric spaces of the group $SL(2,\mathbf R)$”, Mat. Sb. (N.S.), 118(160):4(8) (1982),  490–503  mathnet  mathscinet  zmath; Math. USSR-Sb., 46:4 (1983), 493–506
1981
18. V. F. Molchanov, “The Plancherel formula for the tangent bundle of a projective space”, Dokl. Akad. Nauk SSSR, 260:5 (1981),  1067–1070  mathnet  mathscinet  zmath
1980
19. V. F. Molchanov, “Quantization on the imaginary Lobachevskii plane”, Funktsional. Anal. i Prilozhen., 14:2 (1980),  73–74  mathnet  mathscinet  zmath; Funct. Anal. Appl., 14:2 (1980), 142–144
20. V. F. Molchanov, “Plancherel's formula for hyperboloids”, Trudy Mat. Inst. Steklov., 147 (1980),  65–85  mathnet  mathscinet  zmath; Proc. Steklov Inst. Math., 147 (1981), 63–83
1979
21. V. F. Molchanov, “Tensor products of unitary representations of the three-dimensional Lorentz group”, Izv. Akad. Nauk SSSR Ser. Mat., 43:4 (1979),  860–891  mathnet  mathscinet  zmath; Math. USSR-Izv., 15:1 (1980), 113–143  isi
1978
22. V. F. Molchanov, “Elementary representations of the Laguerre group”, Mat. Zametki, 23:1 (1978),  31–40  mathnet  mathscinet  zmath; Math. Notes, 23:1 (1978), 19–23
23. V. F. Molchanov, “Reduction of representations of the complementary series of the $2+3$ de Sitter group with respect to the Lorentz group”, TMF, 37:2 (1978),  274–280  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 37:2 (1978), 1017–1022
1977
24. V. F. Molchanov, “The restriction of a representation of the complementary series of a pseudo-orthogonal group to a pseudo-orthogonal group of lower dimension”, Dokl. Akad. Nauk SSSR, 237:4 (1977),  782–785  mathnet  mathscinet  zmath
1976
25. V. F. Molchanov, “Spherical functions on hyperboloids”, Mat. Sb. (N.S.), 99(141):2 (1976),  139–161  mathnet  mathscinet  zmath; Math. USSR-Sb., 28:2 (1976), 119–139  isi
1975
26. V. F. Molchanov, “Decomposition of the tensor square representation of the complementary series of a group”, Funktsional. Anal. i Prilozhen., 9:4 (1975),  79–80  mathnet  mathscinet  zmath; Funct. Anal. Appl., 9:4 (1975), 344–345
1971
27. V. F. Molchanov, “On the caluculation of weight multiplicity”, TMF, 8:2 (1971),  251–254  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 8:2 (1971), 810–812
1970
28. V. F. Molchanov, “Representations of pseudo-orthogonal groups associated with a cone”, Mat. Sb. (N.S.), 81(123):3 (1970),  358–375  mathnet  mathscinet  zmath; Math. USSR-Sb., 10:3 (1970), 333–347
1968
29. V. F. Molchanov, “An analog of Plancherel's formula for hyperboloids”, Dokl. Akad. Nauk SSSR, 183:2 (1968),  288–291  mathnet  mathscinet  zmath
1966
30. V. F. Molchanov, “Harmonic analysis on a hyperboloid of one sheet”, Dokl. Akad. Nauk SSSR, 171:4 (1966),  794–797  mathnet  mathscinet  zmath

2019
31. A. M. Borodin, Aleksandr I. Bufetov, Aleksei I. Bufetov, A. M. Vershik, V. E. Gorin, A. I. Molev, V. F. Molchanov, R. S. Ismagilov, A. A. Kirillov, M. L. Nazarov, Yu. A. Neretin, N. I. Nessonov, A. Yu. Okounkov, L. A. Petrov, S. M. Khoroshkin, “Grigori Iosifovich Olshanski (on his 70th birthday)”, Uspekhi Mat. Nauk, 74:3(447) (2019),  193–213  mathnet  elib; Russian Math. Surveys, 74:3 (2019), 555–577  isi
2013
32. A. M. Vershik, A. A. Kirillov, V. F. Molchanov, Yu. A. Neretin, G. I. Olshanski, V. V. Ryzhikov, V. M. Tikhomirov, A. A. Shkalikov, “Rais Sal'manovich Ismagilov (on his 75th birthday)”, Uspekhi Mat. Nauk, 68:4(412) (2013),  185–190  mathnet  mathscinet  elib; Russian Math. Surveys, 68:4 (2013), 783–788  isi  scopus
2008
33. A. M. Vershik, I. M. Gel'fand, S. G. Gindikin, A. A. Kirillov, G. L. Litvinov, V. F. Molchanov, Yu. A. Neretin, V. S. Retakh, “Mark Iosifovich Graev (to his 85th brithday)”, Uspekhi Mat. Nauk, 63:1(379) (2008),  169–182  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 63:1 (2008), 173–188  isi
1997
34. V. F. Molchanov, “Tambov School-Seminar on Harmonic Analysis”, Uspekhi Mat. Nauk, 52:6(318) (1997),  216  mathnet
1989
35. A. A. Kirillov, V. I. Man'ko, V. F. Molchanov, I. I. Shitikov, “School-Seminar “Group Presentations in Physics””, Uspekhi Mat. Nauk, 44:6(270) (1989),  171–172  mathnet
1988
36. S. G. Gindikin, V. F. Molchanov, Yu. G. Reshetnyak, I. I. Shitikov, “XII School on Operator Theory in Functional Spaces”, Uspekhi Mat. Nauk, 43:1(259) (1988),  223–224  mathnet

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