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Asymptotic inverse of the Toeplitz matrix and Green kernel. (Inverse asymptotique de la matrice de Toeplitz et noyau de Green.) (French) Zbl 0965.15002
Summary: Two generalizations of a theorem of F. L. Spitzer and C. J. Stone [Ill. J. Math. 4, 253-277 (1960; Zbl 0124.34403)] are given. This theorem shows a deep link between the asymptotic behaviour of the elements of \((T_N(f))^{-1}\), where \(T_N(f)\) is the Toeplitz matrix associated to the singular symbol \(f\) of order \(2\), and the Green kernel of the second derivative operator. An exact expression of the inverse is obtained for a particular family, the second result is concerned by an asymptotic expansion for a general family.

MSC:
15A09 Theory of matrix inversion and generalized inverses
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
15B57 Hermitian, skew-Hermitian, and related matrices
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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