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# Tauberian Theorem

A Tauberian theorem is a theorem that deduces the convergence of an series on the basis of the properties of the function it defines and any kind of auxiliary hypothesis which prevents the general term of the series from converging to zero too slowly. Hardy (1999, p. 46) states that "a 'Tauberian' theorem may be defined as a corrected form of the false converse of an 'Abelian theorem.' "

Wiener's Tauberian theorem states that if , then the translates of span a dense subspace iff the Fourier transform is nonzero everywhere. This theorem is analogous with the theorem that if (for a Banach algebra with a unit), then spans the whole space if and only if the Gelfand transform is nonzero everywhere.

Abelian Theorem, Hardy-Littlewood Tauberian Theorem

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## References

Bromwich, T. J. I'A. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 2nd ed. New York: Chelsea, p. 256, 1991.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 31 and 46, 1999.Katznelson, Y. An Introduction to Harmonic Analysis. New York: Dover, 1976.Loomis, L. H. An Introduction to Abstract Harmonic Analysis. Princeton, NJ: Van Nostrand, 1953.Wiener, N. The Fourier Integral and Certain of Its Applications. New York: Dover, 1951.

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Tauberian Theorem

## Cite this as:

Weisstein, Eric W. "Tauberian Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TauberianTheorem.html