An extremal problem

for trigonometric polynomials

Authors:
J. Marshall Ash and Michael Ganzburg

Journal:
Proc. Amer. Math. Soc. **127** (1999), 211-216

MSC (1991):
Primary 42A05; Secondary 41A44

MathSciNet review:
1458861

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a trigonometric polynomial of degree The problem of finding the largest value for in the inequality is studied. We find exactly provided is the conjugate of an even integer and For general we get an interval estimate for where the interval length tends to as tends to

**[AW]**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**, 10.1090/S0002-9947-97-01641-3**[AWW]**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**, 10.1090/S0002-9939-97-03568-5**[B]**S. N. Bernstein,*Extremal Properties of Polynomials*, GROL, Leningrad, Moscow, 1937. (In Russian.)**[N1]**S. M. Nikol′skiĭ,*Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables*, Trudy Mat. Inst. Steklov., v. 38, Trudy Mat. Inst. Steklov., v. 38, Izdat. Akad. Nauk SSSR, Moscow, 1951, pp. 244–278 (Russian). MR**0048565****[N2]**S. M. Nikol′skiĭ,*Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya*, “Nauka”, Moscow, 1977 (Russian). Second edition, revised and supplemented. MR**506247****[P]**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****[R]**Walter Rudin,*Uniqueness theory for Laplace series*, Trans. Amer. Math. Soc.**68**(1950), 287–303. MR**0033368**, 10.1090/S0002-9947-1950-0033368-1**[Ti]**A. F. Timan,*Theory of approximation of functions of a real variable*, Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, A Pergamon Press Book. The Macmillan Co., New York, 1963. MR**0192238****[Ta]**L. V. Taykov,*A circle of extremal problems for trigonometric polynomials,*Ukrainskii Mat. Zhurnal,**20**(1965), 205-211. (In Russian.)**[Z]**A. Zygmund,*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776**

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Additional Information

**J. Marshall Ash**

Affiliation:
Department of Mathematics, DePaul University, Chicago, Illinois 60614

Email:
mash@math.depaul.edu

**Michael Ganzburg**

Affiliation:
Department of Mathematics, Hampton University, Hampton, Virginia 23668

Email:
ganzbrgm@fusion.hamptonu.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-04481-0

Keywords:
Trigonometric polynomial,
inequalities between different norms,
best constants

Received by editor(s):
January 9, 1997

Received by editor(s) in revised form:
May 12, 1997

Communicated by:
Christopher D. Sogge

Article copyright:
© Copyright 1999
American Mathematical Society