### Citations

295 |
The theory of Fourier integrals
- Titchmarsh
- 1948
(Show Context)
Citation Context ... modified Yν-transform and derive its inverse. In Section 3 we state and prove the uncertainty principle for the modified Yν-transform. Our approach depends heavily on the use of the Mellin transform =-=[16]-=-. 2. THE MODIFIED Yν-TRANSFORM In this section we show that the modified Yν-transform is an invertible transform from L2(R+, dwν) onto L2(R+, dw−1ν ) if −1 < ν < 0. Further, we derive its inverse and ... |

71 |
A theorem concerning Fourier transforms.
- Hardy
- 1933
(Show Context)
Citation Context ...n. That is, it is impossible for a nonzero function and its Fourier transform to be simultaneously small. ”Smallness” had taken different interpretation in different contexts. Hardy and Cowling ([3], =-=[5]-=-) for example showed such impossibility when ”smallness” is interprted as sharp decay. Results in [5] for the Fourier transform had been extended to Euclidean spaces and to some non-commutitative grou... |

24 |
Uncertainty principles on certain Lie groups,
- Sitaram, Sundari, et al.
- 1995
(Show Context)
Citation Context ...ple showed such impossibility when ”smallness” is interprted as sharp decay. Results in [5] for the Fourier transform had been extended to Euclidean spaces and to some non-commutitative groups ([12], =-=[13]-=-, [14]), and similar results to that in [3] had been obtained on certain Lie groups [2]. The classical uncertainty principle has been extended to other transforms such as the Hankel transform [10], an... |

21 | An uncertainty principle for Hankel transforms
- RÖSLER, VOIT
- 1999
(Show Context)
Citation Context ...2], [13], [14]), and similar results to that in [3] had been obtained on certain Lie groups [2]. The classical uncertainty principle has been extended to other transforms such as the Hankel transform =-=[10]-=-, and the Dunkl transform [9]. The principle, in fact, has been extended to integral operators with a bounded kernel for which there is a Received April 17, 2002 1056-2176 $15.00 c©Dynamic Publishers,... |

19 | An analogue of Hardy’s theorem for very rapidly decreasing functions on semi-simple Lie groups
- Sitaram, Sundari
- 1997
(Show Context)
Citation Context ...r example showed such impossibility when ”smallness” is interprted as sharp decay. Results in [5] for the Fourier transform had been extended to Euclidean spaces and to some non-commutitative groups (=-=[12]-=-, [13], [14]), and similar results to that in [3] had been obtained on certain Lie groups [2]. The classical uncertainty principle has been extended to other transforms such as the Hankel transform [1... |

12 |
Uncertainty principles like Hardy’s theorem on some Lie groups
- Bagchi, Ray
- 1998
(Show Context)
Citation Context ...[5] for the Fourier transform had been extended to Euclidean spaces and to some non-commutitative groups ([12], [13], [14]), and similar results to that in [3] had been obtained on certain Lie groups =-=[2]-=-. The classical uncertainty principle has been extended to other transforms such as the Hankel transform [10], and the Dunkl transform [9]. The principle, in fact, has been extended to integral operat... |

11 | An uncertainty principle for the Dunkl transform
- Rösler
- 1999
(Show Context)
Citation Context ...esults to that in [3] had been obtained on certain Lie groups [2]. The classical uncertainty principle has been extended to other transforms such as the Hankel transform [10], and the Dunkl transform =-=[9]-=-. The principle, in fact, has been extended to integral operators with a bounded kernel for which there is a Received April 17, 2002 1056-2176 $15.00 c©Dynamic Publishers, Inc. 452 FADHEL AL-MUSALLAM ... |

8 |
Generalizations of Heisenberg’s inequality
- Cowling, Price
- 1983
(Show Context)
Citation Context ...cision. That is, it is impossible for a nonzero function and its Fourier transform to be simultaneously small. ”Smallness” had taken different interpretation in different contexts. Hardy and Cowling (=-=[3]-=-, [5]) for example showed such impossibility when ”smallness” is interprted as sharp decay. Results in [5] for the Fourier transform had been extended to Euclidean spaces and to some non-commutitative... |

5 |
An uncertainty principle for integral operators
- Jeu
(Show Context)
Citation Context ...extended to integral operators with a bounded kernel for which there is a Received April 17, 2002 1056-2176 $15.00 c©Dynamic Publishers, Inc. 452 FADHEL AL-MUSALLAM AND VU KIM TUAN Plancherel theorem =-=[4]-=-. The purpose of this paper is to obtain an uncertainty principle for the modified Yν-transform, which we define below. Let ν ∈ R. Let the measures wν and w−1ν on [0,∞) be given by dwν(x) = x 1−2νdx, ... |

3 | Hardy’s theorem for the n-dimensional Euclidean motion group
- Sundari
- 1998
(Show Context)
Citation Context ...owed such impossibility when ”smallness” is interprted as sharp decay. Results in [5] for the Fourier transform had been extended to Euclidean spaces and to some non-commutitative groups ([12], [13], =-=[14]-=-), and similar results to that in [3] had been obtained on certain Lie groups [2]. The classical uncertainty principle has been extended to other transforms such as the Hankel transform [10], and the ... |

1 |
Titchmarsh, A pair of inversion formulae
- C
- 1923
(Show Context)
Citation Context ...nverse and estimate its norm as well as that of its inverse. The modified Yν-transform is related to the Yν-transform of order ν, and the latter is defined and bounded on L2(R+), when −1 < ν < 1, as (=-=[15]-=-, [16]) (2) (Yνf)(x) = ∞∫ 0 √ xy Yν(xy)f(y)dy, (x ∈ R+). A simple algebraic manipulation shows that for f ∈ L2(R+, dwν) one has (3) f̃ν(x) = x 1 2 −ν(Yν(y 12−νf(y))(x), (x ∈ R+). We can derive the inv... |