Mellin inversion theorem

Method

If φ ( s ) is analytic in the strip a < ℜ ( s ) < b , and if it tends to zero uniformly as ℑ ( s ) → ± ∞ for any real value c between a and b, with its integral along such a line converging absolutely, then if

we have that

Conversely, suppose f(x) is piecewise continuous on the positive real numbers, taking a value halfway between the limit values at any jump discontinuities, and suppose the integral

is absolutely convergent when a < ℜ ( s ) < b . Then f is recoverable via the inverse Mellin transform from its Mellin transform φ .

Boundedness condition

We may strengthen the boundedness condition on φ ( s ) if f(x) is continuous. If φ ( s ) is analytic in the strip a < ℜ ( s ) < b , and if | φ ( s ) | < K | s | − 2 , where K is a positive constant, then f(x) as defined by the inversion integral exists and is continuous; moreover the Mellin transform of f is φ for at least a < ℜ ( s ) < b .

On the other hand, if we are willing to accept an original f which is a generalized function, we may relax the boundedness condition on φ to simply make it of polynomial growth in any closed strip contained in the open strip a < ℜ ( s ) < b .

We may also define a Banach space version of this theorem. If we call by L ν , p ( R + ) the weighted Lp space of complex valued functions f on the positive reals such that

where ν and p are fixed real numbers with p>1, then if f(x) is in L ν , p ( R + ) with 1 < p ≤ 2 , then φ ( s ) belongs to L ν , q ( R + ) with q = p / ( p − 1 ) and

Here functions, identical everywhere except on a set of measure zero, are identified.

Since the two-sided Laplace transform can be defined as

these theorems can be immediately applied to it also.