An approximate solution of the Fredholm type equation of the second kind for any $\lambda\ne 0$
Alexander Kouleshoff Mechanics and Mathematics Faculty, Belarussian State University,
Minsk, Belarus
Аннотация:
Consider the following equation
\begin{equation*}
\bigl((I-\lambda K)\varphi\bigr)(s)=\varphi(s)-\lambda\int\limits_a^bK(s,t)\varphi(t)\,dt=f(s).
\end{equation*}
Assume that the complex-valued kernel
$K(s,t)$ is defined on $(a-\varepsilon,b+\varepsilon)\times(a-\varepsilon,b+\varepsilon)$
for some
$\varepsilon>0$ and
\begin{gather*}
\|K\|_2^2=\int\limits_a^b|K(s,t)|^2\,ds\,dt,
\\
p(s,t)=\lambda K(s,t)+\overline{\lambda}\,\overline{K(t,s)}-|\lambda|^2\int\limits_a^b\overline{K(\xi,s)}K(\xi,t)\,d\xi.
\end{gather*}
Consider the following mapping
\begin{equation*}
f\colon[a,b]\ni\xi\to p(s,\xi)p(\xi,t)\in L_2([a,b]\times[a,b]).
\end{equation*}
If the function
$f$ is integrable according to definition of the Riemann integral (as the function with values in the space
$L_2([a,b]\times[a,b])$), then the kernel of the square of the integral operator
\begin{equation*}
(P\varphi)(s)=\int\limits_a^bp(s,t)\varphi(t)\,dt
\end{equation*}
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 a special type are proved. For any
$\lambda\neq 0$ we find approximations of the function
$\varphi$ which minimizes
functional
$\|(I-\lambda K)\varphi-f\|_2$ 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
$I-\lambda K$ if
$\lambda\neq 0$ is a characteristic number. The corresponding
approximation errors are obtained.
Ключевые слова и фразы:
Fredholm type integral equation of the second kind, characteristic number, pseudoinversed operator, finite dimentional kernels, spread of two subspaces.
MSC: 30C80 Поступила в редакцию: 16.02.2004
Язык публикации: английский