### Abstract

A density function f(x), x∈R^{n} is said to be piecewise smooth if for each x∈R^{n}, 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, lim_{r↑∞}f_{R}(x)=M_{0}+f(x) if and only if r→M_{r}f(x) has k=[(n-3)/2] continuous derivatives. ([]=integer part). Otherwise we have lim {Mathematical expression} where ν≥0 is uniquely determined.

Original language | English (US) |
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Pages (from-to) | 187-193 |

Number of pages | 7 |

Journal | Journal of Theoretical Probability |

Volume | 6 |

Issue number | 1 |

DOIs | |

State | Published - 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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## Cite this

Pinsky, M. A. (1993). Fourier inversion for multidimensional characteristic functions.

*Journal of Theoretical Probability*,*6*(1), 187-193. https://doi.org/10.1007/BF01046775