Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the $ L\sp 2$ inequalities involving trigonometric polynomials and their derivatives


Author: Weiyu Chen
Journal: Trans. Amer. Math. Soc. 347 (1995), 1753-1761
MSC: Primary 42A05; Secondary 41A17
DOI: https://doi.org/10.1090/S0002-9947-1995-1254834-7
MathSciNet review: 1254834
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we study the upper bound of the integral

$\displaystyle \int_0^\pi {{{({t^{(k)}}(x))}^2}w(x)} dx$

where $ t(x)$ is a trigonometric polynomial with real coefficients such that $ \left\Vert t \right\Vert\infty \leqslant 1$ and $ w(x)$ is a nonnegative function defined on $ [0,\pi ]$. When $ w(x) = \sin ^jx$, where $ j$ is a positive integer, we obtain the exact upper bound for the above integral.

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

  • [1] G. K. Kristiansen, Some inequalities for algebraic and trigonometric polynomials, J. London Math. Soc. 20 (1979), 300-314. MR 551458 (81e:42001)
  • [2] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, Athena Series, New York, 1966. MR 0213785 (35:4642)
  • [3] L. Shen, Comments on an $ {L^2}$ inequality of A. K. Varma involving the first derivative of polynomials, Proc. Amer. Math. Soc. 111 (1991), 955-959. MR 1039265 (91g:26030)
  • [4] G. Szegö, Orthogonal polynomials, 4th ed., Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975.
  • [5] L. V. Taikov, An extension of S. N. Bernstein's inequality, Trudy Mat. Inst. Steklov 78 (1965), 43-47. MR 0190610 (32:8022)
  • [6] A. K. Varma, Markoff type inequalities for curved majorants in $ {L_2}$ norm, Colloq. Math. Soc. János Bolyai: Approximation Theory, Keekemet (Hungary), 1990.
  • [7] -, On some extremal properties of algebraic polynomials, J. Approx. Theory 69 (1992), 48-54. MR 1154222 (92m:41042)
  • [8] A. K. Varma, T. M. Mills, and S. J. Smith, Markoff type inequalities for curved majorants, preprint. MR 1313881 (95m:26027)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42A05, 41A17

Retrieve articles in all journals with MSC: 42A05, 41A17


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1254834-7
Keywords: Bernstein-type inequalities, trigonometric polynomials
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society