Some extremal problems in $L^p(w)$
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- by R. Cheng, A. G. Miamee and M. Pourahmadi PDF
- Proc. Amer. Math. Soc. 126 (1998), 2333-2340 Request permission
Abstract:
Fix a positive integer $n$ and $1<p<\infty$. We provide expressions for the weighted $L^{p}$ distance \[ \inf _{f} \int ^{2 \pi }_{0} | 1 - f |^p w d\lambda , \] 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 \[ S_1 = \{ \ldots , -3, -2, -1\}\cup \{ 1, 2, 3,\ldots , n\},\] \[ S_2=\{ \ldots , -3, -2,-1\}\setminus \{-n\},\] or \[ S_3 =\{ \ldots , -3, -2, - 1\}\cup \{n\}.\] 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 $w$.References
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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
- MR Author ID: 141590
- Received by editor(s): March 27, 1996
- Received by editor(s) in revised form: January 13, 1997
- Additional Notes: The second authorβs 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 authorβs research was supported in part by NSA Grant No. MDA904-97-1-0013. - Communicated by: J. Marshall Ash
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 2333-2340
- MSC (1991): Primary 42A10, 60G25
- DOI: https://doi.org/10.1090/S0002-9939-98-04275-0
- MathSciNet review: 1443377