Direct Spectral Problems for the Block Jacobi Type Bounded Symmetric Matrices Related to the Two Dimensional Real Moment Problem

Микола Євгенович Дудкін, Валентина Іванівна Козак

Abstract


The generalization of the classical moment problem and the spectral theory of self-adjoint Jacobi block matrix are well-known in one-dimensional case and it generalized on the two-dimensional case. Finite and infinite moment problem is solved using Yu.M. Berezansky generalized eigenfunction expansion method for respectively finite and infinite family of commuting self-adjoint operators. In the classical case one orthogonalize a family of polynomials \[x^{n},n\in \mathbb{N}_{0}\] with respect to a measure on the real axis and shift operator on takes the form of ordinary Jacobi  matrix. Jacobi matrix determines the difference equation. Solving the difference equation, we obtain the corresponding polynomial that called the direct problem. The construction of the matrix is called inverse problem. In this publication we orthogonalize two-indexes family of polynomials \[x^{n},y^{m},n,m\in \mathbb{N}_{0}\] with respect to a measure on the real plane. For orthogonalization order should be chosen. In this case we have two shift operators on  and on . According to the chosen order, these operators take the form of block Jacobi matrices of special form. The main result is the solution of the direct problem, which consists in the following: to solve the system of two difference equations generated by block Jacobi type matrices, i.e., to obtain the corresponding polynomials but in two variables. The correctness of the solution is guaranteed again by Yu.M. Berezanskyi generalized eigenfunction expansion method for a pair of commuting self-adjoint operators. Constructions are connected with an application in spring pendulum in the plane

Keywords


Block Jacobi matrixі; Moment problem; Generalized eigenfunction expansion; Difference equations

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. ANUSSR, no. 10, pp. 83–106, 1948.

N.I. Akhiezer, The Classical Moment Problem and Some Belated Questions in Analysis. New York: Hafner, 1965 (Rus. ed.: Moscow: Fizmatgiz, 1961).

Yu.M. Berezansky, “The expansions in eigenfunctions of partial difference equations of order two”, Trudy Moskov. Mat. Obshch., vol. 5, pp. 203–268, 1956.

Yu.M. Berezansky, “Expansions in Eigenfunctions of Self adjoint Operators”, Amer. Math. Soc., Providence, RI, 1968 (Rus. ed.: Kiev: Naukova Dumka, 1965).

Yu.M. Berezansky and Yu.G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, Vols. 1. 2. Dordrecht, Boston, London: Kluwer Academic Publishers, 1995 (Rus. ed.: Kiev: Naukova Dumka, 1988).

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”, Ibid, vol. 12, no. 1, pp. 1–32, 2005.

A. Devinatz, “Integral representations of positive definite functions, II”, Trans. Amer. Math. Soc., vol. 77, pp. 455–480, 1954.

A. Devinatz, “Two parameter moment problems”, Duke Math. J., vol. 24, pp. 481– 498, 1957.

Козак В.І. Обернена спектральна задача для блочних матриць типу Якобі, відповідних дійсній двовимірній проблемі моментів // Наукові вісті НТУУ “КПІ”. – 2013. – № 4. – С. 73–76.

Yu.М. Berezansky et al., Functional Analysis, Vols. 1. 2. Basel, Boston, Berlin: Birkhauser Verlag, 1996 (Rus. ed.: Kiev: Vyshcha shkola, 1990).

P.K. Suetin, Orthogonal polynomials in Two Variables. Moscow: Nauka, 1988.


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DOI: https://doi.org/10.20535/1810-0546.2014.4.28218

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