Nonexistence results for relaxation spectra with compact support

作者：Douglas R J^*; Gruffudd H R Whittle

来源：Inverse Problems, 2016, 32(3): 035006.

DOI：10.1088/0266-5611/32/3/035006

• Fourier transform
• inverse problems
• Paley-Wiener theorem
• relaxation spectrum

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 g = sech * h, 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.

• 出版日期2016-3