An \(N\)-dimensional analogue of Szegö's limit theorem (Q1916774)

Szegö's limit theorem referred to in the title states: If \(\phi\) is a real, bounded function, \(T_p(\phi)\) the Toeplitz matrix with symbol \(\phi\), and \(F\) a Riemann integrable function, then \[ \text{tr }F(T_p(\phi))= p(F(\phi))^\wedge(0)+ o(p), \] asymptotically as \(p\to\infty\) (where \(\{\widehat h(m)\}\) denotes the Fourier coefficients of a function \(h\)). In this paper, an \(N\)-dimensional analogue of this theorem is introduced. In particular, \(N+1\) terms of the asymptotic expansion of \(\text{tr }F(T_p)(\phi)\) are computed.