International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 45   | 1 | 2 |

Section 1.3.2.6.9.3. Asymptotic distribution of eigenvalues of Toeplitz forms

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.6.9.3. Asymptotic distribution of eigenvalues of Toeplitz forms

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The eigenvalues of the Hermitian form [T_{n} [\,f]] are defined as the [n + 1] real roots of the characteristic equation [\det \{T_{n} [\,f - \lambda]\} = 0]. They will be denoted by[\lambda_{1}^{(n)}, \lambda_{2}^{(n)}, \ldots, \lambda_{n + 1}^{(n)}.]

It is easily shown that if [m \leq f(x) \leq M] for all x, then [m \leq \lambda_{\nu}^{(n)} \leq M] for all n and all [\nu = 1, \ldots, n + 1]. As [n \rightarrow \infty] these bounds, and the distribution of the [\lambda^{(n)}] within these bounds, can be made more precise by introducing two new notions.

  • (i) Essential bounds: define ess inf f as the largest m such that [f(x) \geq m] except for values of x forming a set of measure 0; and define ess sup f similarly.

  • (ii) Equal distribution. For each n, consider two sets of [n + 1] real numbers:[a_{1}^{(n)}, a_{2}^{(n)}, \ldots, a_{n + 1}^{(n)}, \quad\hbox{and}\quad b_{1}^{(n)}, b_{2}^{(n)}, \ldots, b_{n + 1}^{(n)}.]Assume that for each [\nu] and each n, [|a_{\nu}^{(n)}| \,\lt\, K] and [|b_{\nu}^{(n)}| \,\lt\, K] with K independent of [\nu] and n. The sets [\{a_{\nu}^{(n)}\}] and [\{b_{\nu}^{(n)}\}] are said to be equally distributed in [[-K, +K]] if, for any function F over [[-K, +K]],[\lim\limits_{n \rightarrow \infty} {1 \over n + 1} \sum\limits_{\nu = 1}^{n + 1} [F (a_{\nu}^{(n)}) - F (b_{\nu}^{(n)})] = 0.]

We may now state an important theorem of Szegö (1915[link], 1920[link]). Let [f \in L^{1}], and put [m = \hbox{ess inf}\, f], [M = \hbox{ess sup}\,f]. If m and M are finite, then for any continuous function [F(\lambda)] defined in the interval [m, M] we have[\lim\limits_{n \rightarrow \infty} {1 \over n + 1} \sum\limits_{\nu = 1}^{n + 1} F (\lambda_{\nu}^{(n)}) = \int\limits_{0}^{1} F[\,f(x)] \,\hbox{d}x.]In other words, the eigenvalues [\lambda_{\nu}^{(n)}] of the [T_{n}] and the values [f[\nu/(n + 2)]] of f on a regular subdivision of ]0, 1[ are equally distributed.

Further investigations into the spectra of Toeplitz matrices may be found in papers by Hartman & Wintner (1950[link], 1954[link]), Kac et al. (1953)[link], Widom (1965)[link], and in the notes by Hirschman & Hughes (1977)[link].

References

Hartman, P. & Wintner, A. (1950). On the spectra of Toeplitz's matrices. Am. J. Math. 72, 359–366.
Hartman, P. & Wintner, A. (1954). The spectra of Toeplitz's matrices. Am. J. Math. 76, 867–882.
Hirschman, I. I. Jr & Hughes, D. E. (1977). Extreme Eigenvalues of Toeplitz Operators. Lecture Notes in Mathematics, Vol. 618. Berlin: Springer-Verlag.
Kac, M., Murdock, W. L. & Szegö, G. (1953). On the eigenvalues of certain Hermitian forms. J. Rat. Mech. Anal. 2, 767–800.
Szegö, G. (1915). Ein Grenzwertsatz uber die Toeplitzschen Determinanten einer reellen positiven Funktion. Math. Ann. 76, 490–503.
Szegö, G. (1920). Beitrage zur Theorie der Toeplitzchen Formen (Erste Mitteilung). Math. Z. 6, 167–202.
Widom, H. (1965). Toeplitz matrices. In Studies in Real and Complex Analysis, edited by I. I. Hirschmann Jr, pp. 179–209. MAA Studies in Mathematics, Vol. 3. Englewood Cliffs: Prentice-Hall.








































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