### Abstract

We obtain some uncertainty inequalities for the Jacobi transform fα,β (λ), where we suppose α, β ∈ R and ρ = α +β +1 ≥ 0. As in the Euclidean case, analogues of the local and global uncertainty principles hold for fα,β. In this paper, we shall obtain a new type of an uncertainty inequality and its equality condition: When β ≤ 0 or β ≤ α, the L2-norm of fα,β (λ)λ is estimated below by the L2-norm of ρf (x)(cosh x)−1. Otherwise, a similar inequality holds. Especially, whenβ > α+1, the discrete part of f appears in the Parseval formula and it influences the inequality. We also apply these uncertainty principles to the spherical Fourier transform on SU(1, 1). Then the corresponding uncertainty principle depends, not uniformly on the K-types of f.

Original language | English |
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Pages (from-to) | 127-146 |

Number of pages | 20 |

Journal | Tokyo Journal of Mathematics |

Volume | 31 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2008 Jan 1 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Uncertainty principles for the jacobi transform.** / Kawazoe, Takeshi.

Research output: Contribution to journal › Article

*Tokyo Journal of Mathematics*, vol. 31, no. 1, pp. 127-146. https://doi.org/10.3836/tjm/1219844827

}

TY - JOUR

T1 - Uncertainty principles for the jacobi transform

AU - Kawazoe, Takeshi

PY - 2008/1/1

Y1 - 2008/1/1

N2 - We obtain some uncertainty inequalities for the Jacobi transform fα,β (λ), where we suppose α, β ∈ R and ρ = α +β +1 ≥ 0. As in the Euclidean case, analogues of the local and global uncertainty principles hold for fα,β. In this paper, we shall obtain a new type of an uncertainty inequality and its equality condition: When β ≤ 0 or β ≤ α, the L2-norm of fα,β (λ)λ is estimated below by the L2-norm of ρf (x)(cosh x)−1. Otherwise, a similar inequality holds. Especially, whenβ > α+1, the discrete part of f appears in the Parseval formula and it influences the inequality. We also apply these uncertainty principles to the spherical Fourier transform on SU(1, 1). Then the corresponding uncertainty principle depends, not uniformly on the K-types of f.

AB - We obtain some uncertainty inequalities for the Jacobi transform fα,β (λ), where we suppose α, β ∈ R and ρ = α +β +1 ≥ 0. As in the Euclidean case, analogues of the local and global uncertainty principles hold for fα,β. In this paper, we shall obtain a new type of an uncertainty inequality and its equality condition: When β ≤ 0 or β ≤ α, the L2-norm of fα,β (λ)λ is estimated below by the L2-norm of ρf (x)(cosh x)−1. Otherwise, a similar inequality holds. Especially, whenβ > α+1, the discrete part of f appears in the Parseval formula and it influences the inequality. We also apply these uncertainty principles to the spherical Fourier transform on SU(1, 1). Then the corresponding uncertainty principle depends, not uniformly on the K-types of f.

UR - http://www.scopus.com/inward/record.url?scp=84952330294&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84952330294&partnerID=8YFLogxK

U2 - 10.3836/tjm/1219844827

DO - 10.3836/tjm/1219844827

M3 - Article

VL - 31

SP - 127

EP - 146

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 1

ER -