ABSTRACT

The aim of this paper is to develop a new harmonic analysis related to a Bessel type operator ${\mathrm{\Delta }}_{\nu }^{\mathbf{m}}$ on the real line: We define the canonical Fourier Bessel transform ${\mathcal{F}}_{\nu }^{\mathbf{m}}$ and study some of its important properties. We prove a Riemann–Lebesgue lemma, inversion formula and operational formulas for this transformation. We derive Plancherel theorem and Babenko inequality for ${\mathcal{F}}_{\nu }^{\mathbf{m}}.$ In the present paper, several uncertainty inequalities and theorems for the canonical Fourier Bessel transform are given, including the Heisenberg inequality, Hardy theorem, Nash-type inequality, Carlson-type inequality, global uncertainty principle, local uncertainty principle, logarithmic uncertainty principle in terms of entropy and Miyachi uncertainty principle.

摘要

本文的目的是开发一种与贝塞尔类型算子有关的新谐波分析 ${\mathrm{\Delta }}_{\nu }^{\mathbf{米}}$ 实线：我们定义了经典的傅立叶贝塞尔变换 ${\mathcal{F}}_{\nu }^{\mathbf{米}}$并研究其一些重要特性。我们证明了这种变换的黎曼-莱贝格引理，反演公式和运算公式。我们推导了Plancherel定理和Babenko不等式${\mathcal{F}}_{\nu }^{\mathbf{米}}。$ 本文给出了经典傅里叶贝塞尔变换的几个不确定性不等式和定理，包括海森堡不等式，哈代定理，纳什型不等式，卡尔森型不等式，全局不确定性原理，局部不确定性原理，对数不确定性原理的熵和宫内不确定性原理。