**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 - P

^{2})

^{+}(I + P) and the persistency of the operator (I - P

^{2})

^{+} 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 L

_{2}[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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