Mellin inversion theorem

Summary

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   is analytic in the strip  , and if it tends to zero uniformly as   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   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  . Then   is recoverable via the inverse Mellin transform from its Mellin transform  . These results can be obtained by relating the Mellin transform to the Fourier transform by a change of variables and then applying an appropriate version of the Fourier inversion theorem.[1]

Boundedness condition edit

The boundedness condition on   can be strengthened if   is continuous. If   is analytic in the strip  , and if  , where K is a positive constant, then   as defined by the inversion integral exists and is continuous; moreover the Mellin transform of   is   for at least  .

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

We may also define a Banach space version of this theorem. If we call by   the weighted Lp space of complex valued functions   on the positive reals such that

 

where ν and p are fixed real numbers with  , then if   is in   with  , then   belongs to   with   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.

See also edit

References edit

  1. ^ Debnath, Lokenath (2015). Integral transforms and their applications. CRC Press. ISBN 978-1-4822-2357-6. OCLC 919711727.
  • Flajolet, P.; Gourdon, X.; Dumas, P. (1995). "Mellin transforms and asymptotics: Harmonic sums" (PDF). Theoretical Computer Science. 144 (1–2): 3–58. doi:10.1016/0304-3975(95)00002-E.
  • McLachlan, N. W. (1953). Complex Variable Theory and Transform Calculus. Cambridge University Press.
  • Polyanin, A. D.; Manzhirov, A. V. (1998). Handbook of Integral Equations. Boca Raton: CRC Press. ISBN 0-8493-2876-4.
  • Titchmarsh, E. C. (1948). Introduction to the Theory of Fourier Integrals (Second ed.). Oxford University Press.
  • Yakubovich, S. B. (1996). Index Transforms. World Scientific. ISBN 981-02-2216-5.
  • Zemanian, A. H. (1968). Generalized Integral Transforms. John Wiley & Sons.

External links edit

  • Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.