Abelian and Tauberian theorems relating the local behavior of an integrable function to the tail behavior of its Fourier transform Academic Article uri icon


  • We describe the precise relationship between the local smoothness behavior of an integrable function and the asymptotic tail behavior of its Fourier transform. This has special relevance for probability and spectral density functions with a discontinuous mth derivative. Simply stated, if the function's Fourier transform behaves asymptotically as the product of the Fourier series for a discrete complex measure and a regularly varying function, then an mth derivative of the function behaves at its discontinuities like the density of a regularly varying function. With side conditions, the converse also holds. 1991.

published proceedings

  • Journal of Mathematical Analysis and Applications

author list (cited authors)

  • Cline, D.

citation count

  • 6

complete list of authors

  • Cline, Daren BH

publication date

  • January 1991