On the rate of growth of condition numbers for convolution matrices

When analyzing linear systems of equations, the most important indicator of potential instability is the condition number of the matrix. For a convolution matrix W formed from a series w (where
W_{ij} = w_{i-j+1}, 1 \leq i-j + 1 \leq k, W_{ij} = 0
otherwise), this condition number defines the stability of the deconvolution process. For the larger convolution matrices commonly encountered in practice, direct computation of the condition number (e.g., by singular value decomposition) would be extremely time consuming. However, for convolution matrices, an upper bound for the condition number is defined by the ratio of the maximum to the minimum values of the amplitude spectrum of w. This bound is infinite for any series w with a zero value in its amplitude spectrum; although for certain such series, the actual condition number for W may in fact be relatively small. In this paper we give a new simple derivation of the upper bound and present a means of defining the rate of growth of the condition number of W for a band-limited series by means of the higher order derivatives of the amplitude spectrum of w at its zeros. The rate of growth is shown to be proportional to m
^p
, where m is the column dimension of W and p is the order of the zero of the amplitude spectrum.