Asymptotic behavior of the eigenvalues of Toeplitz matrices with even symbol

In this paper we consider an interval [ θ 1 , θ 2 ] ⊂ ] 0 , π [ and f a periodic and even function in C 4 [ 0 , 2 π ] such that f ( θ ) ∈ [ f ( θ 1 ) , f ( θ 2 ) ] ⇔ θ ∈ [ θ 1 , θ 2 ] and f ∈ 𝒜 + [ θ 1 , θ 2 ] (resp. f ∈ 𝒜 - [ θ 1 , θ 2 ] , that means f ′ ( θ ) > 0 (resp. f ′ ( θ ) < 0 ) for all θ ∈ [ θ 1 , θ 2 ] . Then we obtain a higher order asymptotic formula for all the eigenvalues of the Toeplitz matrix T N ( f ) as N → + ∞ which belong to [ f ( θ 1 ) ,