Georgian Mathematical Journal 13 (2006), No. 2, 363--382
Copyright Heldermann Verlag 2006
Selfadjoint Operators and Generalized Central Algorithms in Frechet Spaces
Soso Tsotniashvili
Tskhinvali State University, 2 Besiki Street, Gori 1400, Georgia
eredvi@yandex.ru
David Zarnadze
N. Muskhelishvili Inst. Comput. Math., 8 Akuri Street, Tbilisi 0193, Georgia
z-dato@rambler.ru
[Abstract-pdf]
The paper gives an extension of the fundamental principles of selfadjoint operators in Fr\'{e}chet--Hilbert spaces, countable-Hilbert and nuclear Fr\'{e}chet spaces. Generalizations of the well known theorems of von Neumann, Hellinger-Toeplitz, Friedrichs and Ritz are obtained. Definitions of generalized central and generalized spline algorithms are given. The restriction $A^{\infty}$ of a selfadjoint operator $A$ defined on a dense set $D(A)$ of the Hilbert space $H$ to the Frechet space $D(A^{\infty})$ is substantiated. The extended Ritz method is used for obtaining an approximate solution of the equation $A^{\infty} u=f$ in the Frechet space $D(A^{\infty})$. It is proved that approximate solutions of this equation constructed by the extended Ritz method do not depend on the number of norms that generate the topology of the space $D(A^{\infty})$. Hence this approximate method is both a generalized central and generalized spline algorithm. Examples of selfadjoint and positive definite elliptic differential operators satisfying the above conditions are given. The validity of theoretical results in the case of a harmonic oscillator operator is confirmed by numerical calculations.
Keywords: Selfadjoint operator, best approximation, generalized central algorithm, extended Ritz method, energetic Frechet space.
MSC: 7B25, 65D15, 41A65, 65J10, 65L60, 68Q25
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