The method of moments

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1968

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Abstract

The problem of moments is defined and is used to solve the eigenvalue problem and problems of the first and second kind in a finite dimensional space. It is also shown that the method of moments can be used to determine a sequence of operators which converge strongly to an arbitrary preassigned bounded linear operator. It is shown that if A is a completely continuous operator then the method of moments can be used to determine a sequence of operators which converge uniformly to A. This result is used to prove fSeveral theorems concerning the convergence and speed of convergence for various problems involving completely continuous operators. The method of moments is used in combination with the Liouville-Neumann method to speed the convergence of the latter when solving a problem of the second kind where the operator is completely continuous. The method of moments is used to solve various problems involving self-adjoint operators. Also the spectral theory of a bounded self-adjoint operator is derived and the method of moments is used to find a sequence of spectral functions which converge strongly to the spectral function of a given bounded self-adjoint operator. Various applied examples are treated. It is proven that in certain cases the method of moments can successfully be applied to problems involving unbounded operators.

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