Mellin inversion theorem

Mellin inversion theorem

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.

If φ(s) is analytic in the strip a < \Re(s) < b, and if it tends to zero uniformly with increasing \Im(s) 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 φ.

We may strengthen the boundedness condition on φ(s) if f(x) is continuous. If φ(s) is analytic in the strip a < \Re(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 < \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 φ 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 Lν,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 Lν,p(R + ) with 1 < p \le 2, then φ(s) belongs to Lν,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

References

  • 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


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Mellin transform — In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used …   Wikipedia

  • Transformation de Mellin — En mathématiques, la transformation de Mellin est une transformation intégrale qui peut être considérée comme la version multiplicative (en) de la transformation de Laplace bilatérale. Cette transformation intégrale est fortement reliée à la …   Wikipédia en Français

  • Convolution theorem — In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) equals point wise… …   Wikipedia

  • Prime number theorem — PNT redirects here. For other uses, see PNT (disambiguation). In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are… …   Wikipedia

  • Liste de théorèmes — par ordre alphabétique. Pour l établissement de l ordre alphabétique, il a été convenu ce qui suit : Si le nom du théorème comprend des noms de mathématiciens ou de physiciens, on se base sur le premier nom propre cité. Si le nom du théorème …   Wikipédia en Français

  • List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

  • Меллин, Ялмар — Ялмар Меллин Hjalmar Mellin Роберт Я …   Википедия

  • Mertens conjecture — In mathematics, the Mertens conjecture is the incorrect statement that the Mertens function M(n) is bounded by √n, which implies the Riemann hypothesis. It was conjectured by Stieltjes in a 1885 letter to Hermite (reprinted in Stieltjes 1905) and …   Wikipedia

  • Riemann zeta function — ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): dark colors denote values close to zero and hue encodes the value s argument. The white spot at s = 1 is the pole of the zeta function; the black spots on the… …   Wikipedia

  • Inverse Laplace transform — Contents 1 Mellin s inverse formula 2 Post s inversion formula 3 See also 4 References 5 Ext …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”