On the Asymptotic Behavior of the Prediction Error

Orthogonal Polynomials and Rational Modifications of Measures

“…where /3 n +\ is given by (6). where e n is given by (9) and j=0 7=0 ^W SKETCH OF PROOF. The proof is a straightforward calculation analogous to that used to prove Proposition 1.…”

“…The measure d\i\ gives a special case of ultraspherical polynomials on the unit circle. Ultraspherical polynomials on the unit circle have been considered earlier by Delsarte and Genin [1,5], but also by Golinskii [9]. We use again Proposition 5.…”

Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences

“…The prediction problem stated above goes back to classical works of Kolmogorov (1941a, 1941b), Szegő (1915, 1967) and Wiener (1949). It was then considered by many authors for different classes of non‐deterministic processes (see, e.g., Baxter, 1962, Devinatz, 1964, Golinskii, 1974, Golinskii and Ibragimov, 1971, Grenander and Rosenblatt, 1954, 1957, Grenander and Szegő, 1958, Helson and Szegő, 1960, Ibragimov, 1964, Ibragimov and Solev, 1968, Inoue, 2002, Pourahmadi, 2001, Rozanov, 1967, and reference therein). More references can be found in the survey articles Bingham (2012) and Ginovian (1999).…”

On the Polynomials Orthogonal on Regular Polygons

“…These polynomials have a trivial weight function w s =1ÂN which can be obtained from (3.4) by the above mentioned limiting transition. Note also that in the limit N Ä we get circle analogs of the ultraspherical polynomials (see, e.g., [12,14]) having the recurrence parameters…”