On the extreme spectral properties of Toeplitz matrices generated byL
1 functions with several minima/maxima |
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Authors: | Stefano Serra |
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Institution: | (1) Dipartimento di Informatica, University of Pisa, Corso Italia 40, I-56100 Pisa, Italia |
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Abstract: | In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue
1
(n)
of symmetric (Hermitian)n ×n Toeplitz matricesT
n
(f) generated by an integrable functionf defined in – , ]. In 7, 8, 11] it is shown that
1
(n)
tends to essinff =m
f
in the following way:
1
(n)
–m
f
1/n
2k
. These authors use three assumptions:A1)f –m
f
has a zero inx =x
0 of order 2k.A2)f is continuous and at leastC
2k
in a neighborhood ofx
0.A3)x =x
0 is the unique global minimum off in – , ]. In 10] we have proved that the hypothesis of smoothnessA2 is not necessary and that the same result holds under the weaker assumption thatf L
1– , ]. In this paper we further extend this theory to the case of a functionf L
1– , ] having several global minima by suppressing the hypothesisA3 and by showing that the maximal order 2k of the zeros off –m
f
is the only parameter which characterizes the rate of convergence of
1
(n)
tom
f
. |
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Keywords: | Toeplitz matrices extreme eigenvalues |
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