Toeplitz band matrices with exponentially growing condition numbers

Abstract

The paper deals with the spectral condition numbers (Tn(b)) of sequences of Toeplitz matrices Tn(b) = (bj k) n j;k=1 as n goes to in nity. The function b(ei ) = P k bke ik is referred to as the symbol of the sequence fTn(b)g. It is well known that (Tn(b)) may increase exponentially if the symbol b has very strong zeros on the unit circle T = fz 2 C : jzj = 1g, for example, if b vanishes on some subarc of T. If b is a trigonometric polynomial, in which case the matrices Tn(b) are band matrices, then b cannot have strong zeros unless it vanishes identically. It is shown that the condition numbers (Tn(b)) may nevertheless grow exponentially or even faster to in nity. In particular, it is proved that this always happens if b is a trigonometric polynomial which has no zeros on T but nonzero winding number about the origin. The techniques employed in this paper are also applicable to Toeplitz matrices generated by rational symbols b and to the condition numbers associated with lp norms (1 p 1) instead of the l norm.