An extremal problem for trigonometric polynomials
HTML articles powered by AMS MathViewer
- by J. Marshall Ash and Michael Ganzburg PDF
- Proc. Amer. Math. Soc. 127 (1999), 211-216 Request permission
Abstract:
Let $T_{n}(x)=\sum _{k=0}^{n}(a_{k}\cos kx+b_{k}\sin kx)$ be a trigonometric polynomial of degree $n.$ The problem of finding $C_{np},$ the largest value for $C$ in the inequality $\max \{\left | a_{0}\right | ,\left | a_{1}\right | ,...,\left | a_{n}\right | ,\left | b_{1}\right | ,...,\left | b_{n}\right | \}$ $\leq (1/C)\left \| T_{n}\right \| _{p}$ is studied. We find $C_{np}$ exactly provided $p$ is the conjugate of an even integer $2s$ and $n\geq 2s-1,s=1,2,....$ For general $p,1\leq p\leq \infty ,$we get an interval estimate for $C_{np},$ where the interval length tends to $0$ as $n$tends to $\infty .$References
- J. Marshall Ash and Gang Wang, One- and two-dimensional Cantor-Lebesgue type theorems, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1663–1674. MR 1357390, DOI 10.1090/S0002-9947-97-01641-3
- J. Marshall Ash, Gang Wang, and David Weinberg, A Cantor-Lebesgue theorem with variable “coefficients”, Proc. Amer. Math. Soc. 125 (1997), no. 1, 219–228. MR 1350931, DOI 10.1090/S0002-9939-97-03568-5
- S. N. Bernstein, Extremal Properties of Polynomials, GROL, Leningrad, Moscow, 1937. (In Russian.)
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- S. M. Nikol′skiĭ, Priblizhenie funktsiĭ mnogikh peremennykh i teoremy vlozheniya, “Nauka”, Moscow, 1977 (Russian). Second edition, revised and supplemented. MR 506247
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 1, Gordon & Breach Science Publishers, New York, 1986. Elementary functions; Translated from the Russian and with a preface by N. M. Queen. MR 874986
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- A. F. Timan, Theory of approximation of functions of a real variable, A Pergamon Press Book, The Macmillan Company, New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar. MR 0192238
- L. V. Taykov, A circle of extremal problems for trigonometric polynomials, Ukrainskii Mat. Zhurnal, 20(1965), 205–211. (In Russian.)
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- J. Marshall Ash
- Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
- MR Author ID: 27660
- Email: mash@math.depaul.edu
- Michael Ganzburg
- Affiliation: Department of Mathematics, Hampton University, Hampton, Virginia 23668
- Email: ganzbrgm@fusion.hamptonu.edu
- Received by editor(s): January 9, 1997
- Received by editor(s) in revised form: May 12, 1997
- Communicated by: Christopher D. Sogge
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 211-216
- MSC (1991): Primary 42A05; Secondary 41A44
- DOI: https://doi.org/10.1090/S0002-9939-99-04481-0
- MathSciNet review: 1458861