Error estimates of the real inversion formulas of the Laplace transform

Let f be the Laplace transform of F = F(t) such that (0)integral (infinity) or \F(t)](2)t(1-2q) dt < <infinity> for some q > 0. We set F-N(t) = (0)integral (infinity) f(x) e (-xt) P-N,P-q(xt) dx (N = 0, 1, 2, ...) for the polynomials [GRAPHICS] Let max (1/2, 2q - 1) < <alpha> < 1 and let <beta> > 0 satisfy alpha less than or equal to beta < q + <alpha>/2. If the function f(x) satisfies f(z)z(beta) is an element of Hq-beta+alpha /2(R+), where Hq-beta+alpha /2(R+) is the Bergman-Selberg space on the right half complex plane {Re z > 0}, then the error estimate as N --> infinity \F(t) - F-N(t)\ = t(q-1+alpha /2) o(N(1-2 alpha)/4) is given. Here o(N(1-2 alpha)/4) is independent of t. Moreover we characterize functions F guaranteeing the above error estimate.