Fourier inversion for multidimensional characteristic functions

Mark A. Pinsky*

*Corresponding author for this work

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

A density function f(x), x∈Rn is said to be piecewise smooth if for each x∈Rn, the mean value function {Mathematical expression} is piecewise C with compact support. (dω is normalized surface measure on the unit sphere). The Fourier transform is {Mathematical expression} with spherical partial sum {Mathematical expression}. Theorem. For such f, limr↑∞fR(x)=M0+f(x) if and only if r→Mrf(x) has k=[(n-3)/2] continuous derivatives. ([]=integer part). Otherwise we have lim {Mathematical expression} where ν≥0 is uniquely determined.

Original languageEnglish (US)
Pages (from-to)187-193
Number of pages7
JournalJournal of Theoretical Probability
Volume6
Issue number1
DOIs
StatePublished - Jan 1 1993

Keywords

  • characteristic function
  • Fourier transform
  • Gibbs' phenomenon
  • spherical partial sum

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)

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