The N-dimensional Uncertainty Principle For The Free Metaplectic Transformation On Function Spaces

The free metaplectic transformation on a function space is a linear regular transformation in N dimensions.The transformation operator is especially useful for signal processing applications.In order to simultaneously describe the function(or signal)itself and the information after the free metaplectic transformation,we focus on several typical uncertainty principles.Including the classic Heisenberg uncertainty principle,Nazarov uncertainty principle,Donoho-Stark uncertainty principle,Hardy uncertainty principle,Beurling uncertainty principle,logarithmic uncertainty principle and entropy uncertainty principle.These uncertainty principles in Fourier transform domain have been deeply studied by many scholars,and the extension of these uncertainty principles(UP)into the uncertainty principle of the free metaplectic transformation domain is the focus of this paper.In the second chapter,the basic theory of the free metaplectic transformation,other important definitions and the lemma of the relationship between Fourier transform and the free metaplectic transformation will be introduced.In Chapter 3,based on the relationship between the two transformations,the uncertainty principle in Fourier transform domain is extended.Based on the singular value and norm theory of matrix,the classical uncertainty principle is obtained;taking the minimum support as the criterion,the Nazarov uncertainty principle is obtained;considering the energy concentration degree of a certain area,Donoho-stark uncertainty principle are obtained;taking the measurement of the attenuation velocity of the two functions before and after transformation as the standard,Hardy uncertainty principle and Beurling uncertainty principle are obtained;based on the logarithmic study of Pitt inequality,the logarithmic uncertainty principle is obtained;Taking Shannon entropy as the measurement tool,the entropy uncertainty principle is obtained,and more generally Renyi entropy uncertainty principle and its extension are obtained.The fourth chapter summarizes the existing theoretical results and points out the direction of further research.