Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [AW] J. M. Ash and G. Wang, One and two dimensional Cantor-Lebesgue type theorems, Trans. Amer. Math. Soc., 349 (1997), 1663-1674. MR 97h:42006
  • [AWW] J. M. Ash, G. Wang, and D. Weinberg, A Cantor-Lebesgue theorem with variable ``coefficients,'' Proc. Amer. Math. Soc., 125(1997), 219-228. MR 97c:42011
  • [B] S. N. Bernstein, Extremal Properties of Polynomials, GROL, Leningrad, Moscow, 1937. (In Russian.)
  • [N1] S. M. Nikolskii, Inequalities for entire functions of finite degree and their applications in the theory of differentiable functions of many variables, Proceedings of MIAN USSR, 38(1951), 244-278. (In Russian.) MR 14:32e
  • [N2] S. M. Nikolskii, Approximation of Functions in Many Variables and Embedding Theorems, Nauka, Moscow, 1977. (In Russian.) MR 81f:46046
  • [P] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. I, Gordon and Breach, New York, 1986. MR 88f:00013
  • [R] W. Rudin, Uniqueness theory for Laplace series, Trans. Amer. Math. Soc., 68(1950), 287-303. MR 11:430a
  • [Ti] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, New York, 1963. MR 33:465
  • [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, Vol. I, 2nd rev. ed., Cambridge Univ. Press, New York, 1959. MR 21:6498

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42A05, 41A44

Retrieve articles in all journals with MSC (1991): 42A05, 41A44

Additional Information

J. Marshall Ash
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614

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

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

American Mathematical Society