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  Volume 7, Issue 3, Article 80
On The Sharpened Heisenberg-Weyl Inequality

    Authors: John Michael Rassias,  
    Keywords: Sharpened, Heisenberg-Weyl inequality, Gram determinant.  
    Date Received: 21/06/05  
    Date Accepted: 21/07/06  
    Subject Codes:

26, 33, 42, 60, 52.

    Editors: Saburou Saitoh,  

The well-known second order moment Heisenberg-Weyl inequality (or uncertainty relation) in Fourier Analysis states: Assume that $ f:mathbb{R}to mathbb{C}$ is a complex valued function of a random real variable $ x$ such that $ fin L^2(mathbb{R})$. Then the product of the second moment of the random real $ x$ for $ leftvert f rightvert^2$ and the second moment of the random real $ xi $ for $ leftvert {hat{f}} rightvert^2$ is at least $ {E_{leftvert f rightvert^2} } mathord{left/ {vphantom {{E_{leftvert f rightvert^2} } {4pi }}} right. kern-nulldelimiterspace} {4pi }$, where $ hat{f}$ is the Fourier transform of $ f$, such that $ hat{f} left( xi right)=int_mathbb{R} {e^{-2ipi xi x}} fleft( x right)dx$, $ fleft( x right)=int_mathbb{R} {e^{2ipi xi x}} hat {f}left( xi right)dxi ,$ and $ E_{leftvert f rightvert^2} =int_mathbb{R} {leftvert {fleft( x right)} rightvert^2dx} $.

This uncertainty relation is well-known in classical quantum mechanics. In 2004, the author generalized the afore-mentioned result to higher order moments and in 2005, he investigated a Heisenberg-Weyl type inequality without Fourier transforms. In this paper, a sharpened form of this generalized Heisenberg-Weyl inequality is established in Fourier analysis. Afterwards, an open problem is proposed on some pertinent extremum principle.These results are useful in investigation of quantum mechanics.

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