Mellin inversion theorem

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In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions under which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function.

Method[edit]

If \varphi(s) is analytic in the strip a < \Re(s) < b, and if it tends to zero uniformly as   \Im(s) \to \pm \infty  for any real value c between a and b, with its integral along such a line converging absolutely, then if

f(x)= \{ \mathcal{M}^{-1} \varphi \} = \frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} \varphi(s)\, ds

we have that

\varphi(s)= \{ \mathcal{M} f \} = \int_0^{\infty} x^s f(x)\,\frac{dx}{x}.

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

\varphi(s)=\int_0^{\infty} x^s f(x)\,\frac{dx}{x}

is absolutely convergent when a < \Re(s) < b. Then f is recoverable via the inverse Mellin transform from its Mellin transform \varphi[citation needed].

Boundedness condition[edit]

We may strengthen the boundedness condition on \varphi(s) if f(x) is continuous. If \varphi(s) is analytic in the strip a < \Re(s) < b, and if |\varphi(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 \varphi for at least a < \Re(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 \varphi to simply make it of polynomial growth in any closed strip contained in the open strip a < \Re(s) < b.

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

\|f\| = \left(\int_0^\infty |x^\nu f(x)|^p\, \frac{dx}{x}\right)^{1/p} < \infty

where ν and p are fixed real numbers with p>1, then if f(x) is in L_{\nu, p}(R^{+}) with 1 < p \le 2, then \varphi(s) belongs to L_{\nu, q}(R^{+}) with q = p/(p-1) and

f(x)=\frac{1}{2 \pi i} \int_{\nu-i \infty}^{\nu+i \infty} x^{-s} \varphi(s)\,ds.

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

Since the two-sided Laplace transform can be defined as

 \left\{\mathcal{B} f\right\}(s) = \left\{\mathcal{M} f(- \ln x) \right\}(s)

these theorems can be immediately applied to it also.

See also[edit]

References[edit]

  • P. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoretical Computer Science, 144(1-2):3-58, June 1995
  • McLachlan, N. W., Complex Variable Theory and Transform Calculus, Cambridge University Press, 1953.
  • Polyanin, A. D. and Manzhirov, A. V., Handbook of Integral Equations, CRC Press, Boca Raton, 1998.
  • Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals, Oxford University Press, second edition, 1948.
  • Yakubovich, S. B., Index Transforms, World Scientific, 1996.
  • Zemanian, A. H., Generalized Integral Transforms, John Wiley & Sons, 1968.

External links[edit]