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Asymptotic expansion of the eigenvalues of a Toeplitz matrix with realsymbol. This work provides two results obtained as a consequence of aninversion formula for Toeplitz matrices with real symbol. First we obtain ansymptotic expression for the minimal eigenvalues of a Toeplitz matrix with asymbol which is periodic, even and derivable on $[0, 2\pi[$. Next we prove thata Toeplitz band matrix with a symbol without zeros on the united circle isinvertible with an inverse which is essentially a band matrix. As a consequenceof this last statement we give an asymptotic estimation for the entries of theinverse of a Toplitz matrix with a regular symbol.

Asymptotic expansion of the eigenvalues of a Toeplitz matrix with a real symbol