Two-dimensional half-strong real moment problem and the corresponding block matrices. Part I

  • M. E. Dudkin Nat. tech. University of Ukraine "KPI them. I. Sikorsky ”, Kyiv https://orcid.org/0000-0002-5554-182X
  • O. Yu. Dyuzhenkova Nat. tech. University of Ukraine "KPI them. I. Sikorsky ”, Kyiv
Keywords: moment problem, block Jaconi matrices

Abstract

УДК 517.9

The relationship between the classical moment problem and the spectral theory of Jacobi matrices is generalized. We present the solution of the two-dimensional half-strong moment problem and suggest an analog of Jacobi-type matrices associated with the two-dimensional half-strong moment problem and the corresponding system of polynomials orthogonal with respect to a measure with compact support in the real plane.

References

N. I. Akhiezer, Классическая проблема моментов (Russian) [[Klassicheskaya problema momentov]], Fizmatgiz, Moskva (1961).

Yu. M. Berezansky, Разложения по собственным функциям самосопряженных операторов (Russian) [[Razlozheniya po sobstvenny`m funkcziyam samosopryazhenny`kh operatorov]], Nauk. dumka, Kiev (1965).

Yu. M. Berezansky, Самосопряженные операторы в пространствах функций бесконечного числа переменных (Russian) [[Samosopryazhenny`e operatory` v prostranstvakh funkczij beskonechnogo chisla peremenny`kh]], Nauk. dumka, Kiev (1978).

Yu. M. Berezansky, Yu. G. Kondrat`ev, Cпектральные методы в бесконечномерном анализе (Russian) [[Spektral`ny`e metody` v beskonechnomernom analize]], Nauk. dumka, Kiev (1988).

Yu. M. Berezansky, G. F. Us, Z. G. Sheftel`, Функциональный анализ: Курс лекций (Russian) [[Funkczional`ny`j analiz: Kurs lekczij]], Vishha shk., Kiyiv (1990).

Yu. M. Berezansky, Some generalizations of the classical moment problem, Integr. Еquat. and Oper. Theory, 44, 255 – 289 (2002), https://doi.org/10.1007/BF01212034 . DOI: https://doi.org/10.1007/BF01212034

Yu. M. Berezansky, M. E. Dudkin, The complex moment problem in the exponential form, Methods Funct. Anal. and Topology, 10, № 4, 1 – 10 (2004).

Yu. M. Berezansky, M. E. Dudkin, The direct and inverce spectral problems for the block Jacobi type unitary matrices, Methods Funct. Anal. and Topology, 11, № 4, 327 – 345 (2005).

Yu. M. Berezansky, M. E. Dudkin, On the complex moment problem, Math. Nachr., 280, № 1-2, 60 – 73 (2007), https://doi.org/10.1002/mana.200410464 DOI: https://doi.org/10.1002/mana.200410464

Yu. M. Berezansky, M. E. Dudkin, Якобiєвi матрицi i проблема моментiв (Ukrainian) [[Yakobiyevi matriczi i problema momentiv]], Praczi In-tu matematiki NAN Ukrayini, 105 (2019).

C. Berg, J. P. R. Christensen, C. U. Jessel, A remark on the multidimension moment problem, Math. Ann., 243, 163 – 169 (1979), https://doi.org/10.1007/BF01420423 DOI: https://doi.org/10.1007/BF01420423

T. M. Bisgaard, On note on factoring of positive definite functions on semigroups, Math. Nachr., 236, 31 – 46 (2002), https://onlinelibrary.wiley.com/doi/10.1002/1522-2616(200203)236:1%3C31::AID-MANA31%3E3.0.CO;2-D DOI: https://doi.org/10.1002/1522-2616(200203)236:1<31::AID-MANA31>3.0.CO;2-D

M. J. Cantero, L. Moral, L. Velázquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra and Appl., 362, 29 – 56 (2003), https://doi.org/10.1016/S0024-3795(02)00457-3 DOI: https://doi.org/10.1016/S0024-3795(02)00457-3

T. Carleman, Les fonctions quasi analytiques, Paris (1926)

A. Devinatz, Integral representations of positive definite functions. II, Trans. Amer. Math. Soc., 77, 455 – 480 (1954), https://doi.org/10.2307/1990835 DOI: https://doi.org/10.2307/1990835

A. Devinatz, Two parameter moment problems, Duke Math. J., 24, 481 – 498 (1957), http://projecteuclid.org/euclid.dmj/1077467615

M. E. Dudkin, The exact inner structure of the block Jacobi type unitary matrices connected with the corresponding direct and inverse spectral problems matrices, Methods Funct. Anal. and Topology, 14, № 2, 168 – 176 (2008).

M. E. Dudkin, The complex moment problem in the exponential form with direct and inverse spectral problems for the block Jacobi type correspondence matrices, Methods Funct. Anal. and Topology, 18, № 2, 111 – 139 (2012).

M. E. Dudkin, The inner structure of the Jacobi-Laurent matrix related to the strong Hamburger moment problem, Methods Funct. Anal. and Topology, 19, № 2, 97 – 107 (2013).

M. E. Dudkin, V. I. Kozak, Direct and inverse spectral problems for the block Jacobi type bounded symmetric matrices related to the two dimensional moment problem, Methods Funct. Anal. and Topology, 20, № 3, 219 – 251 (2014).

M. E. Dudkin, V. I. Kozak, , Пряма спектральна задача з блочними матрицями типу Якобi, що вiдповiдають сильнiй двовимiрнiй проблемi моментiв (Ukrainian) [[Pryama spektral`na zadacha z blochnimi matriczyami tipu Yakobi, shho vidpovidayut` sil`nij dvovimirnij problemi momentiv]], Nauk. zap. NaUKMA, Fiz.-mat. nauki, 178, 16 – 22 (2016).

W. B. Jones, W. J. Thron, O. Njåstad, Orthogonal Laurent polynomials and strong Hamburger moment problem, J. Math. Anal. and Appl., 98, № 2, 528 – 554 (1984), https://doi.org/10.1016/0022-247X(84)90267-1 DOI: https://doi.org/10.1016/0022-247X(84)90267-1

W. B. Jones, O. Njåstad, Orthogonal Laurent polynomials and strong moment theorey: a survey, Continued fractions and geometric function theory (CONFUN) (Trondheim, 1997), J. Comput. and Appl. Math., 105, № 1-2, 51 – 91 (1999), https://doi.org/10.1016/S0377-0427(99)00027-8 DOI: https://doi.org/10.1016/S0377-0427(99)00027-8

R. B. Zarkhina, О двумерной проблеме моментов (Russian) [[O dvumernoj probleme momentov]], Dokl. AN SSSR, 124, № 4, 743 – 746 (1959)

I. S. Kacz, A. A. Nudel`man, Сильная проблема моментов Cтилтьеса (Russian) [[Sil`naya problema momentov Ctilt`esa]], Algebra i analiz, 8, № 6, 26 – 56 (1996)

V. I. Kozak, Побудова блочних матриць типу Якобi, вiдповiдних сильнiй двовимiрнiй дiйснiй проблемi моментiв (Russian) [[Pobudova blochnikh matricz` tipu Yakobi, vidpovidnikh sil`nij dvovimirnij dijsnij problemi momentiv]], Nauk. zap. NaUKMA, Fiz.-mat. nauki, 165, 19 – 26 (2015).

A. G. Kostyuchenko, B. S. Mityagin, Многомерная проблема моментов (Russian) [[Mnogomernaya problema momentov]], Dokl. AN SSSR, 131, № 6, 1249 – 1252 (1960).

A. G. Kostyuchenko, B. S. Mityagin, Положительно-определенные функционалы на ядерных пространствах (Russian) [[Polozhitel`no-opredelenny`e funkczionaly` na yaderny`kh prostranstvakh]], Tr. Mosk. mat. o-va, 9, 283 – 316 (1960).

M. G. Krejn, Об одном общем методе разложения положительно определенных ядер на элементарные произведения (Russian) [[Ob odnom obshhem metode razlozheniya polozhitel`no opredelenny`kh yader na e`lementarny`e proizvedeniya]], Dokl. AN SSSR, 53, № 1, 3 – 6 (1946).

M. G. Krejn, Про ермiтовi оператори з напрямними функцiоналами (Ukrainian) [[Pro ermitovi operatori z napryamnimi funkczionalami]], Zb. nauk. pr. In-t matematiki AN URSR, № 10, 83 – 106 (1948).

S. Mandelbrojt, Séries adhérentes, régularisation des suites, applications(French), Gauthier-Villars, Paris xiv+277 pp. (1952)

E. Nelson, Analytic vectors, Ann. Math., (2) 70, 572 – 614 (1959), https://doi.org/10.2307/1970331 DOI: https://doi.org/10.2307/1970331

A. E. Nussbaum, Quasi-analytic vectors, Ark. Math., 6, № 10, 179 – 191 (1965), https://doi.org/10.1007/BF02591357 DOI: https://doi.org/10.1007/BF02591357

A. E. Nussbaum, A note on quasi-analytic vectors, Stud. Math., 33, 305 – 309 (1969), https://doi.org/10.4064/sm-33-3-305-309 DOI: https://doi.org/10.4064/sm-33-3-305-309

L. C. Petersen, On the relation between the multidimensional moment problem and the one-dimensional moment problem, Math. Scand., 51, 361 – 366 (1982), https://doi.org/10.7146/math.scand.a-11986 DOI: https://doi.org/10.7146/math.scand.a-11986

K. K. Simonov, Strong matrix moment problem of Hamburger, Methods Funct. Anal. and Topology, № 2, 183 – 196 (2006).

K. K. Simonov, Ортогональные матричные полиномы Лорана (Russian) [[Ortogonal`ny`e matrichny`e polinomy` Lorana]], Mat. zametki, 79, № 2, 316 – 320 (2006).

B. Fuglede, The multidimensional moment problem, Expo. Math., 1, № 1, 47 – 65 (1983)

E. K. Haviland, On the moment problem for distribution functions in more than one dimension, Amer. J. Math., 57, 562 – 572 (1995), https://doi.org/10.2307/2371187 DOI: https://doi.org/10.2307/2371187

E. K. Haviland, On the moment problem for distribution functions in more than one dimension. II, Amer. J. Math., 58, 164 – 168 (1996), https://doi.org/10.2307/2371063 DOI: https://doi.org/10.2307/2371063

Y. Xu, On ortogonal polinomials in several variables, Amer. Math. Soc., 14, 247 – 270 (1997).

Y. Xu, Block Jacobi matrices and zeros of multivariate ortogonal polynomials, Amer. Math. Soc., 342, № 2, 855 – 866 (1994), https://doi.org/10.2307/2154656 DOI: https://doi.org/10.2307/2154656

G. I. Eskin, Достаточное условие разрешимости многомерной проблемы моментов (Russian) [[Dostatochnoe uslovie razreshimosti mnogomernoj problemy` momentov]]>, Dokl. AN SSSR, 133, № 3, 540 – 543 (1960).

Published
18.08.2020
How to Cite
Dudkin , M. E., and O. Y. Dyuzhenkova. “Two-Dimensional Half-Strong Real Moment Problem and the Corresponding Block Matrices. Part I”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 8, Aug. 2020, pp. 1047-63, doi:10.37863/umzh.v72i8.6062.
Section
Research articles