An $l_{1}$ extremal problem for polynomials
HTML articles powered by AMS MathViewer
- by E. Beller and D. J. Newman PDF
- Proc. Amer. Math. Soc. 29 (1971), 474-481 Request permission
Abstract:
Let ${\mathfrak {M}_n}$ be the maximum of the ${l_1}$ norm, $\sum \nolimits ^n |{c_k}|$, of all nth degree polynomials satisfying $|\sum \nolimits ^n {c_k}{z^k}| \leqq 1$ for $|z| = 1$. We prove that ${\mathfrak {M}_n}$ is asymptotic to $\surd n$, by exhibiting polynomials ${P_n}$ (which are partial sums of certain Fourier series), whose ${l_1}$ norm is asymptotic to $\surd n$.References
- E. Beller, Polynomial extremal problems in $L^{p}$, Proc. Amer. Math. Soc. 30 (1971), 249β259. MR 281884, DOI 10.1090/S0002-9939-1971-0281884-9
- Paul ErdΕs, Some unsolved problems, Michigan Math. J. 4 (1957), 291β300. MR 98702
- D. J. Newman, An $L^{1}$ extremal problem for polynomials, Proc. Amer. Math. Soc. 16 (1965), 1287β1290. MR 185119, DOI 10.1090/S0002-9939-1965-0185119-4
- Walter Rudin, Some theorems on Fourier coefficients, Proc. Amer. Math. Soc. 10 (1959), 855β859. MR 116184, DOI 10.1090/S0002-9939-1959-0116184-5 H. S. Shapiro, Thesis for S.M. degree, M.I.T., Cambridge, Mass., 1957.
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 29 (1971), 474-481
- MSC: Primary 30.10; Secondary 42.00
- DOI: https://doi.org/10.1090/S0002-9939-1971-0280688-0
- MathSciNet review: 0280688