Wavelet regularization and the continuous relaxation spectrum

The purpose of this paper is to show how continuous wavelet analysis can be used to establish natural models for the continuous relaxation spectrum of a polymeric material. A method of wavelet regularization is proposed for the practical recovery of the continuous spectrum over a limited range of relaxation times. Working with logarithmic variables (log–frequency and log–time), it may be seen that the loss modulus is a scaling function transform of the continuous relaxation spectrum. It is shown how the decomposition formula of Calderón and Mallat may be used to reconstruct the spectrum from measurements of storage and loss moduli. At practical levels of resolution, the spectrum may be represented as a finite sum of hyperbolic scaling functions. There are two principal regularization mechanisms, namely, sparsity (the number of terms in the sum), and scale (which controls both resolution and smoothness). The method of wavelet regularization is illustrated by recovering spectra from both synthetic and real data.