Nikolskii-type inequalities for shift invariant function spaces

Authors:
Peter Borwein and Tamás Erdélyi

Journal:
Proc. Amer. Math. Soc. **134** (2006), 3243-3246

MSC (2000):
Primary 41A17

Published electronically:
June 6, 2006

MathSciNet review:
2231907

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a vectorspace of complex-valued functions defined on of dimension over . We say that is shift invariant (on ) if implies that for every , where on . In this note we prove the following.

**Theorem.** *Let be a shift invariant vectorspace of complex-valued functions defined on of dimension over . Let . Then*

*for every and*

**1.**Peter Borwein and Tamás Erdélyi,*Polynomials and polynomial inequalities*, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR**1367960****2.**Peter Borwein and Tamás Erdélyi,*Pointwise Remez- and Nikolskii-type inequalities for exponential sums*, Math. Ann.**316**(2000), no. 1, 39–60. MR**1735078**, 10.1007/s002080050003**3.**Dimiter Dryanov and Qazi Ibadur Rahman,*On certain mean values of polynomials on the unit interval*, J. Approx. Theory**101**(1999), no. 1, 92–120. MR**1724028**, 10.1006/jath.1999.3364**4.**T. Erdélyi,*Markov-Nikolskii-type inequalities for exponential sums on a finite interval*, Adv. in Math., to appear.**5.**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****6.**G. Szegő and A. Zygmund,*On certain mean values of polynomials*, J. Analyse Math.**3**(1954), 225–244. MR**0064910**

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Additional Information

**Peter Borwein**

Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Email:
pborwein@cecm.sfu.ca

**Tamás Erdélyi**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843

Email:
terdelyi@math.tamu.edu

DOI:
https://doi.org/10.1090/S0002-9939-06-08533-9

Keywords:
Nikolskii-type inequalities,
shift invariant function spaces,
exponential sums

Received by editor(s):
May 17, 2005

Published electronically:
June 6, 2006

Communicated by:
David Preiss

Article copyright:
© Copyright 2006
by the authors