Nonexistence results for relaxation spectra with compact support. (19th February 2016)

• Record Type: Journal Article
• Authors: Douglas, R J; Gruffudd, H R Whittle
• Is Part Of: Inverse problems. Volume 32:Number 3(2016:Mar.)
• Publication Date: 2016-02-19
• Subjects: Fourier transform -- inverse problems -- Paley–Wiener theorem -- relaxation spectrum; 42A38 -- 45Q05 -- 46F12; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/3/035006
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: In this paper we consider the problem of recovering the (transformed) relaxation spectrum h from the (transformed) loss modulus g by inverting the integral equation, where denotes convolution, using Fourier transforms. We are particularly interested in establishing properties of h, having assumed that the Fourier transform of g has entire extension to the complex plane. In the setting of square integrable functions, we demonstrate that the Paley–Wiener theorem cannot be used to show the existence of non-trivial relaxation spectra with compact support. We prove a stronger result for tempered distributions: there are no non-trivial relaxation spectra with compact support. Finally we establish necessary and sufficient conditions for the relaxation spectrum h to be strictly positive definite.