  In this paper we prove pointwise and distributional Fourier transform
inversion theorems for functions on the real line that are locally of bounded
variation, while in a neighbourhood of infinity are Lebesgue integrable or have
polynomial growth. We also allow the Fourier transform to exist in the
principal value sense. A function is called regulated if it has a left limit
and a right limit at each point. The main inversion theorem is obtained by
solving the differential equation $df(t)-i\omega f(t)=g(t)$ for a regulated
function $f$, where $\omega$ is a complex number with positive imaginary part.
This is done using the Henstock--Stieltjes integral. This is an integral
defined with Riemann sums and a gauge. Some variants of the integration by
parts formula are also proved for this integral. When the function is of
polynomial growth its Fourier transform exists in a distributional sense,
although the inversion formula only involves integration of functions and
returns pointwise values.
