In this paper three methods are derived for approximating f, given its Laplace transform g on
(
0
,
∞
)
(0,\infty )
, i.e.,
∫
0
∞
f
(
t
)
exp
(
−
s
t
)
d
t
=
g
(
s
)
\smallint _0^\infty {f(t)\exp ( - st)\,dt = g(s)}
. Assuming that
g
∈
L
2
(
0
,
∞
)
g \in {L^2}(0,\infty )
, the first method is based on a Sinc-like rational approximation of g, the second on a Sinc solution of the integral equation
∫
0
∞
f
(
t
)
exp
(
−
s
t
)
d
t
=
g
(
s
)
\smallint _0^\infty {f(t)\exp ( - st)\,dt = g(s)}
via standard regularization, and the third method is based on first converting
∫
0
∞
f
(
t
)
exp
(
−
s
t
)
d
t
=
g
(
s
)
\smallint _0^\infty {f(t)\exp ( - st){\mkern 1mu} dt = g(s)}
to a convolution integral over
R
\mathbb {R}
, and then finding a Sinc approximation to f via the application of a special regularization procedure to solve the Fourier transform problem. We also obtain bounds on the error of approximation, which depend on both the method of approximation and the regularization parameter.