# table of generalized Fourier and measured groupoid transforms

## 0.1 Generalized Fourier transforms

Fourier-Stieltjes transforms and measured groupoid transforms are useful generalizations of the (much simpler) Fourier transform^{}, as concisely shown in the following table-
with the same format as C. Woo’s Feature on Fourier transforms (http://planetmath.org/TableOfFourierTransforms)
- for the purpose of direct comparison with the latter transform. Unlike the more general Fourier-Stieltjes
transform, the Fourier transform exists if and only if the function to be transformed is Lebesgue integrable^{} over the whole real axis for $t\in \mathbb{R}$, or over the entire $\u2102$ domain when $\stackrel{\u02c7}{m}(t)$ is a complex function.

###### Definition 0.1.

Fourier-Stieltjes transform.

Given a *positive definite ^{}, measurable function^{}* $f(x)$ on the interval
$(-\mathrm{\infty},\mathrm{\infty})$ there exists a monotone increasing, real-valued bounded
function $\alpha (t)$ such that:

$$f(x)={\int}_{\mathbb{R}}{e}^{itx}d(\alpha (t),$$ | (0.1) |

for all $x\in \mathbb{R}$ except a small set. When $f(x)$ is defined as above and if $\alpha (t)$ is nondecreasing and bounded then the measurable function defined by the above integral is called *the Fourier-Stieltjes transform of* $\alpha (t)$, and it is continuous^{} in addition to being positive definite.

### FT Generalizations

Title | table of generalized Fourier and measured groupoid transforms |
---|---|

Canonical name | TableOfGeneralizedFourierAndMeasuredGroupoidTransforms |

Date of creation | 2013-03-22 18:10:27 |

Last modified on | 2013-03-22 18:10:27 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 46 |

Author | bci1 (20947) |

Entry type | Topic |

Classification | msc 55U99 |

Synonym | Fourier-Stieltjes transforms |

Related topic | FourierTransform |

Related topic | TwoDimensionalFourierTransforms |

Defines | Fourier-Stieltjes and measured groupoid transforms |

\@unrecurse |