Toeplitz band matrices with exponentially growing condition numbers

Abstract

The paper deals with the spectral condition numbers κ(T n(b)) of sequences of Toeplitz matrices Tn(b) = (b j-k)j,k=1n as n goes to infinity. The function b(eiθ) = ∑κ bκe iκθ is referred to as the symbol of the sequence {T n(b)}. It is well known that κ(Tn(b)) may increase exponentially if the symbol b has very strong zeros on the unit circle double-struck T = {z ε Cℂ : |z| = 1}, for example, if b vanishes on some subarc of double-struck 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 κ(Tn(b)) may nevertheless grow exponentially or even faster to infinity. In particular, it is proved that this always happens if b is a trigonometric polynomial which has no zeros on double-struck 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 ≤ ∞) instead of the l2 norm.