On the Connection of Uncertainty Principles for Functions on the Circle and on the Real Line

Prestin, Jürgen; Quak, Ewald; Rauhut, Holger und Selig, Kathi (2003): On the Connection of Uncertainty Principles for Functions on the Circle and on the Real Line. In: Journal of Fourier Analysis and Applications, Vol. 9, No. 4: pp. 387-409

Abstract

An uncertainty principle for 2π-periodic functions and the classical Heisenberg uncertainty principle are shown to be linked by a limit process. Dependent on a parameter, a function on the real line generates periodic functions either by periodization or sampling. It is proven that under certain smoothness conditions, the periodic uncertainty products of the generated functions converge to the real-line uncertainty product of the original function if the parameter tends to infinity. These results are used to find asymptotically optimal sequences for the periodic uncertainty principle, based either on Theta functions or trigonometric polynomials obtained by sampling B-splines.

• Item Type: Journal article
• Faculties: Mathematics, Computer Science and Statistics > Mathematics > Chair of Mathematics of Information Processing
• Subjects: 500 Science > 510 Mathematics
• ISSN: 1069-5869
• Language: English