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IMI/Publicaţii/BASM/Ediţii/BASM n.2 (48), 2005/

An approximate solution of the Fredholm type equation of the second kind for any λ ≠ 0

Authors: Alexander Kouleshoff

Abstract

Consider the following equation

Assume that the complex-valued kernel K(s,t) is defined on
for some ε > 0 and
,
Consider the following mapping

If the function f is integrable according to definition of the Riemann integral (as the function with values in the space , then the kernel of the square of the integral operator
can be approximated by a finite dimensional kernel. The formula (I - P)+ = (I - P2)+(I + P) and the persistency of the operator (I - P2)+ with respect to perturbations of special type are proved. For any λ≠0 we find approximations of the function φ which minimizes functional and has the least norm in L2[a, b] among all functions minimizing the above mentioned functional. Simultaneously we find approximations of the kernel and orthocomplement to the image of the operator I - λK if λ≠0 is a characteristic number. The corresponding approximation errors are obtained.

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