### Abstract

Original language | English |
---|---|

Pages (from-to) | 471-490 |

Number of pages | 20 |

Journal | Integral Transforms and Special Functions |

Volume | 20 |

Issue number | 6 |

DOIs | |

Publication status | Published - 13 May 2009 |

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### Keywords

- fractional integral transforms
- semigroups of operators
- generalised functions
- mathematics
- statistics

### Cite this

*Integral Transforms and Special Functions*,

*20*(6), 471-490. https://doi.org/10.1080/10652460802646063

}

*Integral Transforms and Special Functions*, vol. 20, no. 6, pp. 471-490. https://doi.org/10.1080/10652460802646063

**Fractional transformations of generalised functions.** / Khan, Khaula Naeem; Lamb, Wilson; McBride, Adam C.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Fractional transformations of generalised functions

AU - Khan, Khaula Naeem

AU - Lamb, Wilson

AU - McBride, Adam C.

PY - 2009/5/13

Y1 - 2009/5/13

N2 - A distributional theory of fractional transformations is developed. A constructive approach, based on the eigenfunction expansion method pioneered by A. H. Zemanian, is used to produce an appropriate space of test functions and corresponding space of generalised functions. The fractional transformations that are defined are shown to form an equicontinuous group of operators on the space of test functions and a weak continuous group on the space of generalised functions. Integral representations for the fractional transformations are also obtained under certain conditions. The fractional Fourier transformation is considered as a particular case of our general theory.

AB - A distributional theory of fractional transformations is developed. A constructive approach, based on the eigenfunction expansion method pioneered by A. H. Zemanian, is used to produce an appropriate space of test functions and corresponding space of generalised functions. The fractional transformations that are defined are shown to form an equicontinuous group of operators on the space of test functions and a weak continuous group on the space of generalised functions. Integral representations for the fractional transformations are also obtained under certain conditions. The fractional Fourier transformation is considered as a particular case of our general theory.

KW - fractional integral transforms

KW - semigroups of operators

KW - generalised functions

KW - mathematics

KW - statistics

U2 - 10.1080/10652460802646063

DO - 10.1080/10652460802646063

M3 - Article

VL - 20

SP - 471

EP - 490

JO - Integral Transforms and Special Functions

JF - Integral Transforms and Special Functions

SN - 1065-2469

IS - 6

ER -