In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue λ1(n) of symmetric (Hermitian) n × nToeplitz matrices Tn(f) generated by an integrable function f defined in [-π, π]. In [7, 8, 11] it is shown that λ1(n) tends to essinf f = mf in the following way: λ1(n) - mf ∼ 1/n2k. These authors use three assumptions: A1) f - mf has a zero in x = x0 of order 2k. A2) f is continuous and at least C2k in a neighborhood of x0. A3) x = x0 is the unique global minimum of f in [-π, π]. In [10] we have proved that the hypothesis of smoothness A2 is not necessary and that the same result holds under the weaker assumption that f ∈ L1[-π,π]. In this paper we further extend this theory to the case of a function f ∈ L1 [-π, π] having several global minima by suppressing the hypothesis A3 and by showing that the maximal order 2k of the zeros of f - mf is the only parameter which characterizes the rate of convergence of λ1(n) to mf

On the extreme spectral properties of toeplitz matrices generated by L1 functions with several minima/maxima

Serra Capizzano, S.
1996-01-01

Abstract

In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue λ1(n) of symmetric (Hermitian) n × nToeplitz matrices Tn(f) generated by an integrable function f defined in [-π, π]. In [7, 8, 11] it is shown that λ1(n) tends to essinf f = mf in the following way: λ1(n) - mf ∼ 1/n2k. These authors use three assumptions: A1) f - mf has a zero in x = x0 of order 2k. A2) f is continuous and at least C2k in a neighborhood of x0. A3) x = x0 is the unique global minimum of f in [-π, π]. In [10] we have proved that the hypothesis of smoothness A2 is not necessary and that the same result holds under the weaker assumption that f ∈ L1[-π,π]. In this paper we further extend this theory to the case of a function f ∈ L1 [-π, π] having several global minima by suppressing the hypothesis A3 and by showing that the maximal order 2k of the zeros of f - mf is the only parameter which characterizes the rate of convergence of λ1(n) to mf
1996
BIT
Extreme eigenvalues; Toeplitz matrices
Serra Capizzano, S.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2119584
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 40
  • ???jsp.display-item.citation.isi??? 36
social impact