Around finite sections of integral operators

Abstract

The finite section method is primarily intended to make integral equations over unbounded domains (the axis) practically solvable by reducing them to equations over bounded domains (intervals). For the singular integral equation

$$(a{P_R} + b{Q_R})u = f$$

(1)

with ${P_R} = (I + {S_R})/2, {Q_R} = I - {P_R}, u, f \in L_R^p(\alpha )$, and piecewise continuous coefficients a and b, this requires to solve instead of (1) equations of the form

$${P_t}(a{P_R} + b{Q_R}){P_t}{u_t} = {P_t}f,$$

(2)

where the projections P _t are given by $({P_t}f)(s) = \left\{ {\begin{array}{*{20}{c}} {f(s)} \\ 0 \end{array}} \right.\begin{array}{*{20}{c}} {if}&{\left| s \right| \leqslant t} \\ {if}&{\left| s \right| > t,} \end{array}$ and where t is positive (and large). The finite section method is said to apply to equation (1) if the equations (2) are uniquely solvable for all right sides f and for all t large enough (say, $t \geqslant {t_0}$), and if their solutions u _t converge to a solution u of (1) as $t \to \infty$ in the norm of $L_R^p(\alpha )$.