Nikolskii inequality for lacunary spherical polynomials

Dai, Feng; Gorbachev, Dmitry; Tikhonov, Sergey

doi:10.1090/proc/14775

Nikolskii inequality for lacunary spherical polynomials
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by Feng Dai, Dmitry Gorbachev and Sergey Tikhonov

Proc. Amer. Math. Soc. 148 (2020), 1169-1174

DOI: https://doi.org/10.1090/proc/14775

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 $\mathbb {S}^d$ (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.

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