Sharp Marchaud and converse inequalities in Orlicz spaces

Authors:
Z. Ditzian and A. V. Prymak

Journal:
Proc. Amer. Math. Soc. **135** (2007), 1115-1121

MSC (2000):
Primary 26A15, 26B99, 41A27; Secondary 41A63, 46E30

Published electronically:
October 27, 2006

MathSciNet review:
2262913

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For spaces on , and , sharp versions of the classical Marchaud inequality are known. These results are extended here to Orlicz spaces (on , and ) for which is convex for some , , where is the Orlicz function. Sharp converse inequalities for such spaces are deduced.

**[Da-Di, 04]**F. Dai and Z. Ditzian,*Combinations of multivariate averages*, J. Approx. Theory**131**(2004), no. 2, 268–283. MR**2106541**, 10.1016/j.jat.2004.10.003**[Da-Di, 05]**F. Dai and Z. Ditzian,*Littlewood-Paley theory and a sharp Marchaud inequality*, Acta Sci. Math. (Szeged)**71**(2005), no. 1-2, 65–90. MR**2160356****[Di, 80]**Z. Ditzian,*Some remarks on approximation theorems on various Banach spaces*, J. Math. Anal. Appl.**77**(1980), no. 2, 567–576. MR**593235**, 10.1016/0022-247X(80)90248-6**[Di, 88]**Z. Ditzian,*On the Marchaud-type inequality*, Proc. Amer. Math. Soc.**103**(1988), no. 1, 198–202. MR**938668**, 10.1090/S0002-9939-1988-0938668-8**[Di, 89]**Z. Ditzian,*Multivariate Landau-Kolmogorov-type inequality*, Math. Proc. Cambridge Philos. Soc.**105**(1989), no. 2, 335–350. MR**974990**, 10.1017/S0305004100067839**[Di, 99]**Z. Ditzian,*A modulus of smoothness on the unit sphere*, J. Anal. Math.**79**(1999), 189–200. MR**1749311**, 10.1007/BF02788240**[Ka-Ma]**Alexei Yu. Karlovich and Lech Maligranda,*On the interpolation constant for Orlicz spaces*, Proc. Amer. Math. Soc.**129**(2001), no. 9, 2727–2739. MR**1838797**, 10.1090/S0002-9939-01-06162-7**[Ma-Tr]**R. P. Maleev and S. L. Troyanski,*On the moduli of convexity and smoothness in Orlicz spaces*, Studia Math.**54**(1975), no. 2, 131–141. MR**0388067****[Ti]**M. F. Timan,*Converse theorem of the constructive theory of functions in the space*, Sb. Math.,**44**(88) (1958), 125-132 [in Russian].**[To]**V. Totik,*Sharp converse theorem of 𝐿^{𝑝}-polynomial approximation*, Constr. Approx.**4**(1988), no. 4, 419–433. MR**956177**, 10.1007/BF02075471**[Zy]**A. Zygmund,*A remark on the integral modulus of continuity*, Univ. Nac. Tucumán. Revista A.**7**(1950), 259–269. MR**0042479**

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Additional Information

**Z. Ditzian**

Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Email:
zditzian@math.ualberta.ca

**A. V. Prymak**

Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Email:
prymak@gmail.com

DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08682-5

Keywords:
Sharp Marchaud inequality,
Orlicz spaces,
power-type $q$.

Received by editor(s):
November 9, 2005

Published electronically:
October 27, 2006

Additional Notes:
The first author was supported by NSERC grant of Canada A4816.

This research was done while the second author visited University of Alberta in 2005; the visit was supported by the first author’s NSERC grant of Canada A4816

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.