Trace formulas for Toeplitz matrices with piecewise continuous symbols (Q1086811)

For a function \(\phi\) defined on the unit circle with Fourier coefficients \(\phi_ k\), consider the Toeplitz matrices \(T_ n[\phi]=(\phi_{j-k}\), \(j,k=0,...,n.\) This paper considers the problem of computing asymptotically trace f(T\({}_ n[\phi])\), \(n\to \infty\) for certain functions f and for \(\phi\), a piecewise continuous function. The asymptotic expansion takes the form \[ trace f(T_ n[\phi])=a_ 1n+a_ 2\log n+a_ 3o(1) \] where \(a_ 1\), \(a_ 2\) and \(a_ 3\) are determined. The basic idea is to use already known results when f is the logarithm and complex analysis.