Uncertainty principle on weighted spheres, balls and simplexes

2013/04/22 by Yuan Xu, Xu, Yuan
Mathematics · Medicine · Earth and Planetary Sciences · #Mathematical Analysis and Transform Methods #Medical Imaging Techniques and Applications #Seismic Imaging and Inversion Techniques

paper · pdf · doi:10.48550/arxiv.1304.6135

Abstract

For a family of weight functions h_κ that are invariant under a reflection group, the uncertainty principle on the unit sphere in the form of min_{1 ≤ i ≤ d} ∫_\mathbbS^d-1 (1- x_i) |f(x)|² h_κ²(x) dσ∫_\mathbbS^d-1 |∇₀ f(x) |² h_κ²(x) dσ≥ c is established for invariant functions f that have unit norm and zero mean, where ∇₀ is the spherical gradient. In the same spirit, uncertainty principles for weighted spaces on the unit ball and on the standard simplex are established, some of them hold for all admissible functions instead of invariant functions.

Related

• Uncertainty Principles on weighted spheres, balls and simplexes
• The Hardy-Rellich inequality and uncertainty principle on the sphere
• An uncertainty principle related to the Euclidean motion group
• Generalized uncertainty principles
• Weighted Approximation of functions on the unit sphere