Proceedings of the American Mathematical Society

Author(s): R. Cheng; A. G. Miamee; M. Pourahmadi
Journal: Proc. Amer. Math. Soc. 126 (1998), 2333-2340.
MSC (1991): Primary 42A10, 60G25
Retrieve article in: PDF
This article is available free of charge

Abstract: Fix a positive integer $n$

and $1<p<\infty$ . We provide expressions for the weighted $L^{p}$ distance

$\begin{displaymath}\inf _{f} \int^{2 \pi}_{0} | 1 - f |^p w\,d\lambda, \end{displaymath}$

where $d\lambda$ is normalized Lebesgue measure on the unit circle, $w$

is a nonnegative integrable function, and $f$

ranges over the trigonometric polynomials with frequencies in

$\begin{displaymath}S_1 = \{ \ldots, -3, -2, -1\}\cup\{ 1, 2, 3,\ldots, n\},\end{displaymath}$

$\begin{displaymath}S_2=\{ \ldots, -3, -2,-1\}\setminus\{-n\},\end{displaymath}$

These distances are related to other extremal problems, and are shown to be positive if and only if $\log w$ is integrable. In some cases they are expressed in terms of the series coefficients of the outer functions associated with $w$

1.: S. Cambanis and A.R. Soltani (1984). Prediction of stable processes: Spectral and moving average representations, Z.Wahrsch. Verw. Geb. 66, 593-612. MR 86g:60054
2.: P.L. Duren (1970). Theory of spaces. Academic Press, New York. MR 42:3552
3.: T.W. Gamelin (1967). Uniform Algebras. Prentice Hall, Englewood Cliffs, N.J. MR 53:14137
4.: H. Helson (1964). Lectures On Invariant Subspaces. Academic Press, New York. MR 30:1409
5.: A.G. Miamee (1993). On basicity of exponentials in $L^p (d \mu)$ and general prediction problems, Period.Math. Hungar. 26, 115-124. MR 94k:60066
6.: A.G. Miamee and M. Pourahmadi (1988). Best approximation in $L^p (d \mu)$ and prediction problems of Szegö, Kolmogorov, Yaglom and Nakazi, J. London Math. Soc. (2) 38, 133-145. MR 90g:60042
7.: T. Nakazi (1984). Two problems in prediction theory, Studia Math. 78, 7-14. MR 86i:60122
8.: T. Nakazi and K. Takahashi (1980). Prediction $n$ units of time ahead, Proc. Amer. Math. Soc. 80, 658-659. MR 82b:60041
9.: M. Pourahmadi (1992). Alternating projections and interpolation of stationary processes. J. Appl. Probab. 29, 921-931. MR 93m:60077
10.: B.S. Rajput and C. Sundberg (1994). On some extremal problems in and the prediction of -harmonizable stochastic processes, Probab. Theory Related Fields 99, 197-210. MR 95g:30048

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42A10, 60G25

R. Cheng
Affiliation: Department of Mathematics, University of Louisville, Louisville, Kentucky 40292
Address at time of publication: ECI Systems and Engineering, 596 Lynnhaven Parkway, Virginia Beach, Virginia 23452

A. G. Miamee
Affiliation: Department of Mathematics, University of Louisville, Louisville, Kentucky 40292

M. Pourahmadi
Affiliation: Department of Mathematics, University of Louisville, Louisville, Kentucky 40292

DOI: 10.1090/S0002-9939-98-04275-0
PII: S 0002-9939(98)04275-0
Keywords: Prediction error, outer function, dual extremal problem, stationary sequences
Received by editor(s): March 27, 1996
Received by editor(s) in revised form: January 13, 1997
Additional Notes: The second author's research was supported by Office of Naval Research Grant No. N00014-89-J-1824 and U.S. Army Grant No. DAAH04-96-1-0027.
The third author's research was supported in part by NSA Grant No. MDA904-97-1-0013.
Communicated by: J. Marshall Ash
Copyright of article: Copyright 1998, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS