Polynomial extremal problems in $L^{p}$
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- by E. Beller
- Proc. Amer. Math. Soc. 30 (1971), 249-259
- DOI: https://doi.org/10.1090/S0002-9939-1971-0281884-9
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Abstract:
For $p > 2$, let ${m_{p,n}}$ be the minimum of the ${L^p}$ norm all nth degree polynomials ${\sum ^n} {a_k} {e^{ikt}}$ which satisfy $|{a_k}| = 1, k = 0, 1, \cdots , n$. We exhibit certain polynomials ${P_n}$ whose ${L^p}$ norm $( 2 < p < \infty )$ is asymptotic to $\surd {n}$, thereby proving that ${m_{p,n}}$ is itself asymptotic to $\surd {n}$. We also show that the sup norm of (essentially) the same polynomials is asymptotic to $(1.1716 \ldots ) \times \surd {n}$.References
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Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 249-259
- MSC: Primary 30.10
- DOI: https://doi.org/10.1090/S0002-9939-1971-0281884-9
- MathSciNet review: 0281884