| 概要 | The theory of orthogonal polynomials on the unit circle (OPUC)
dates back to Szegő's work of 1915-21, and has been given a great
impetus by the recent work of Simon, in particular his two-volume book
[Si4,5], the survey paper (or summary of the book) [Si3], and the book
[Si9], whose title we allude to in ours. Simon's motivation comes from
spectral theory and analysis. Another major area of application of OPUC
comes from probability, statistics, time series and prediction theory;
see for instance the book by Grenander and Szegő [GrSz]. Coming to the
subject from this background, our aim here is to complement [Si3] by
giving some probabilistically motivated results. We also advocate a new
definition of long-range dependence.
References.
[Si3] B. Simon, OPUC on one foot. Bull. Amer. Math. Soc. 42 (2005), 431-460.
[Si4,5] B. Simon, Orthogonal polynomials on the unit circle. Part 1:
Classical theory. Part 2: Spectral theory. American Math. Soc.,
Providence RI, 2005.
[Si9] B. Simon, Szegő's theorem and its descendants: Spectral theory for
L2 perturbations of orthogonal polynomials. Princeton University Press,
2011.
[GrSz] U. Grenander and G. Szegő, Toeplitz forms and their applications.
University of California Press, Berkeley CA, 1958. |