Authors: Alexander Kouleshoff
Abstract
Consider the following equation
![](/files/basm/y2005-n2/y2005-n2-f1.png)
Assume that the complex-valued kernel K(s,t) is defined on
![](/files/basm/y2005-n2/y2005-n2-f2.png)
for some ε > 0 and
![](/files/basm/y2005-n2/y2005-n2-f3.png)
,
![](/files/basm/y2005-n2/y2005-n2-f4.png)
Consider the following mapping
![](/files/basm/y2005-n2/y2005-n2-f5.png)
If the function f is integrable according to definition of the Riemann integral (as the function
with values in the space
![](/files/basm/y2005-n2/y2005-n2-f6.png)
, then the kernel of the square of the integral operator
![](/files/basm/y2005-n2/y2005-n2-f7.png)
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
![](/files/basm/y2005-n2/y2005-n2-f8.png)
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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