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Nikolskii inequality for lacunary spherical polynomials

Authors: Feng Dai, Dmitry Gorbachev and Sergey Tikhonov
Journal: Proc. Amer. Math. Soc. 148 (2020), 1169-1174
MSC (2010): Primary 33C50, 33C55, 42B15, 42C10
Published electronically: September 20, 2019
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Abstract: We prove that for $ d\ge 2$, the asymptotic order of the usual Nikolskii inequality on $ \SS ^d$ (also known as the reverse Hölder inequality) can be significantly improved in many cases, for lacunary spherical polynomials of the form $ f=\sum _{j=0}^m f_{n_j}$ with $ f_{n_j}$ being a spherical harmonic of degree $ n_j$ and $ n_{j+1}-n_j\ge 3$. As is well known, for $ d=1$, the Nikolskii inequality for trigonometric polynomials on the unit circle does not have such a phenomenon.

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Additional Information

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

Dmitry Gorbachev
Affiliation: Department of Applied Mathematics and Computer Science, Tula State University, 300012 Tula, Russia

Sergey Tikhonov
Affiliation: Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C 08193 Bellaterra (Barcelona), Spain; ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain; Department of Mathematics, Building C Science Faculty, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

Keywords: Spherical harmonics, polynomial inequalities
Received by editor(s): April 25, 2019
Received by editor(s) in revised form: July 14, 2019
Published electronically: September 20, 2019
Additional Notes: The first author was supported by NSERC Canada under the grant RGPIN 04702 Dai
The second author was supported by the Russian Science Foundation under grant 18-11-00199
The third author was partially supported by MTM 2017-87409-P, 2017 SGR 358, and by the CERCA Programme of the Generalitat de Catalunya.
Communicated by: Yuan Xu
Article copyright: © Copyright 2019 American Mathematical Society