Terms Uniqueness Extent Appropriate Points in the Two-Dimensional Problem

Микола Євгенович Дудкін, Валентина Іванівна Козак

Abstract


Background. We continue to study the properties of block Jacobi matrices corresponding to the two-dimensional real problem. Repeating the reasoning applied to probabilistic measure with compact support, we get similar matrices associated with Borel measure without limitation. The difficulty in our research is that the probability measure on compact corresponds uniquely to the block Jacobi type matrices. If the measure is arbitrary, then the same set of matrices can fit infinite number of measures.

Objective. The objective of research is to find conditions under which some Borel measure without limitation corres­ponds to only one pair of block matrices.

Methods. Using previous publications established the form of block Jacobi matrix type, by coefficients of these matrices one can inferred about the above mentioned bijection.

Results. The result of research is a conditions on the coefficients in the form of divergent series in which the one-to-one correspondence holds true.

Conclusions. Using the solution of direct and inverse spectral problems for two-dimensional real moment problem of the previous work we found its condition to be determined (unique) by the coefficients of the block Jacobi type matrix. The result is a two-dimensional analogue of the well-known case in classical Hamburger moment problem.


Keywords


Two-dimensional moment problem; Block Jacobi type matrix; Determinism of a two-dimensional moment problem

References


M.G. Krein, “On the general method of decomposition of positive defined kernels on elementary products”, Dokl. Acad. Nauk SSSR, vol. 53, no. 1, pp. 3–6, 1946.

M.G. Krein, “On Hermitian operators with directing functionals”, Zbirnyk prac' Inst. Mat. AN USSR, no. 10, pp. 83–106, 1948.

N.I. Akhiezer, The Classical Moment Problem. Мoscow, USSR: Gos. fiz.-mat. lit, 1961 (in Russian).

Yu.M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators. Kyiv, USSR: 1965 (in Russian) (Eng. transl.: Providence, R.I.: Amer. Math., Soc., 1968).

Yu.M. Berezansky and Yu.G. Kondratiev, Spectral Methods in Infnite-Dimensional Analysis. Kyiv, USSR: Naukova Dumka, 1988 (in Russian).

Yu.M. Berezansky and M.E. Dudkin, “The direct and inverce spectral problems for the block Jacobi type unitary matrices”, Methods Funct. Anal. Topology, vol. 11, no. 4, pp. 327–345, 2005.

Yu.M. Berezansky and M.E. Dudkin, “The complex moment problem and direct and inverse spectral problems for the block Jacobi type bounded normal matrices”, Methods Funct. Anal. Topology, vol. 12, no. 1, pp. 1–32, 2005.

V.I. Kozak, “Inverse spectral problem for a block matrix of Jacobi type corresponds to the real two dimensional moment problem”, Naukovi Visti NTUU KPI, no. 4, pp. 10–15, 2013 (in Ukrainian).

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

M.E. Dudkin and V.I. Kozak, “Second order polynomials in the two dimensional moment problem”, Naukovi Visti NTUU KPI, no. 4, pp. 41–46, 2015 (in Ukrainian).

P.K. Suetin, Orthogonal Polynomials on Two Variables. Moscow, USSR: Nauka, 1988 (in Russian).

E. Nelson, “Analytic vectors”, Ann. Math., vol. 70, pp. 572–614, 1959.


GOST Style Citations


  1. Krein M.G. On the general method of decomposition of positive defined kernels on elementary products // Докл. АН СССР. – 1946. – 53. – № 1. – P. 3–6.

  2. Krein M.G. On Hermitian operators with directing functionals // Сборник трудов Ин-та мат. АН СРСР. – 1948. – № 10. – P. 83–106.

  3. Ахиезер Н.И. Классическая проблема моментов. – М.: Гос. физ.-мат. лит, 1961. – 312 с.

  4. Березанский Ю.М. Разложение по собственным функциям самосопряженных операторов. – К.: Наук. думка, 1965. – 450 с. – [Eng. transl.: Providence, R.I.: Amer. Math., Soc., 1968. – 450 p.].

  5. Березанский Ю.М., Кондратьев Ю.Г. Спектральные методы в бесконечномерном анализе. – К.: Наук. думка, 1988. – 800 с.

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

  7. Berezansky Yu.M., Dudkin M.E. The complex moment problem and direct and inverse spectral problems for the block Jacobi type bounded normal matrices // Methods Funct. Anal. Topology. – 2005. – 12, № 1. – P. 1–32.

  8. Козак В.I. Обернена спектральна задача для блочних матриць типу Якобi, вiдповiдних дiйснiй двовимiрнiй проблемi моментiв // Науковi вiстi НТУУ “КПI”. – 2013. – № 4. – С. 10–15.

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

  10. Дудкiн М.Є., Козак В.I. Полiноми другого роду у двовимiрнiй проблемi моментiв // Науковi вiстi НТУУ “КПI”. – 2015. – № 4. – P. 41–46.

  11. Суетин П.К. Ортогональные многочлены по двум переменным. – М.: Наука, 1988. – 384 с.

  12. E. Nelson. Analytic vectors // Ann. Math. – 1959. – 70. – P. 572–614.




DOI: https://doi.org/10.20535/1810-0546.2016.4.59942

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