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

 

Some extremal problems in $L^{p}(w)$


Authors: R. Cheng, A. G. Miamee and M. Pourahmadi
Journal: Proc. Amer. Math. Soc. 126 (1998), 2333-2340
MSC (1991): Primary 42A10, 60G25
MathSciNet review: 1443377
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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}

or

\begin{displaymath}S_3 =\{ \ldots, -3, -2, - 1\}\cup\{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 $w$.


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Additional Information

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: http://dx.doi.org/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
Article copyright: © Copyright 1998 American Mathematical Society