AN APPROXIMATE HILBERT TRANSFORM AND ITS INVERSION

Abstract

If η > 0 and f is in a certain space of generalized functions, define
{F_η }(x) = ‹ {f(t), {{t - x} \over {{{(t - x)}^2} + {η ^2}}}} › .
It is shown that
\mathop {\lim }\limits_{η → 0 +} \left( { - {1 \over {{π ^2}}}} \ ight)P∫ _{- ∞} ^∞ {{{{F_η }(x)dx} \over {x - y}}} = f(y)
in the weak distributional sense.