Nikolskii inequality for lacunary spherical polynomials
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- by Feng Dai, Dmitry Gorbachev and Sergey Tikhonov PDF
- Proc. Amer. Math. Soc. 148 (2020), 1169-1174 Request permission
Abstract:
We prove that for $d\ge 2$, the asymptotic order of the usual Nikolskii inequality on $\mathbb {S}^d$ (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.References
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Additional Information
- Feng Dai
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 660750
- Email: fdai@ualberta.ca
- Dmitry Gorbachev
- Affiliation: Department of Applied Mathematics and Computer Science, Tula State University, 300012 Tula, Russia
- MR Author ID: 633235
- Email: dvgmail@mail.ru
- 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
- MR Author ID: 706641
- Email: stikhonov@crm.cat
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1169-1174
- MSC (2010): Primary 33C50, 33C55, 42B15, 42C10
- DOI: https://doi.org/10.1090/proc/14775
- MathSciNet review: 4055944