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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A prediction problem in $ L^2 (w)$


Authors: Mohsen Pourahmadi, Akihiko Inoue and Yukio Kasahara
Journal: Proc. Amer. Math. Soc. 135 (2007), 1233-1239
MSC (2000): Primary 54C40, 14E20; Secondary 46E25, 20C20
Published electronically: October 18, 2006
MathSciNet review: 2262930
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Abstract | References | Similar Articles | Additional Information

Abstract: For a nonnegative integrable weight function $ w$ on the unit circle $ T$, we provide an expression for $ p=2$, in terms of the series coefficients of the outer function of $ w$, for the weighted $ L^p$ distance $ \inf_f \int_T\vert 1-f\vert^p wd \mu$, where $ \mu$ is the normalized Lebesgue measure and $ f$ ranges over trigonometric polynomials with frequencies in $ [\{\dots,-3,-2,-1\}\setminus\{-n\}]\cup\{m\}$, $ m \geq 0$, $ n \geq 2$. The problem is open for $ p \neq 2$.


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Additional Information

Mohsen Pourahmadi
Affiliation: Division of Statistics, Northern Illinois University, DeKalb, Illinois 60115-2854
Email: pourahm@math.niu.edu

Akihiko Inoue
Affiliation: Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan
Email: inoue@math.sci.hokudai.ac.jp

Yukio Kasahara
Affiliation: Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan
Email: y-kasa@math.sci.hokudai.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08575-3
PII: S 0002-9939(06)08575-3
Keywords: Duality and orthogonalization, extremal problems, stationary processes
Received by editor(s): October 4, 2005
Received by editor(s) in revised form: November 17, 2005
Published electronically: October 18, 2006
Additional Notes: The work of the first author was supported by NSF grants DMS-0307055 and DMS-0505696.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2006 American Mathematical Society