Approximating real symmetric Toeplitz matrices using the nearest circulant

The nearest circulant approximation of a real Toeplitz matrix in the Frobenius norm is derived. This matrix is symmetric. It is proven that symmetric circulant matrices are the only real circulant matrices with all real eigenvalues. The Frobenius norm of the difference between this approximation and the Toeplitz matrix for the case of a Toeplitz matrix displaying exponential decay is evaluated using an expression of $\sum_{n = 1}^N n^k p^n$ in terms of the first $k$ geometric moments. Compared to a classic approximation the nearest circulant displays dramatically better behaviour in any finite cases, though both share the same leading term for large $M$.