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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