Nikolskii inequality for lacunary spherical polynomials
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's 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.00323
 Bibcode:
 2019arXiv190500323D
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 33C55;
 33C50;
 42B15;
 42C10
 EPrint:
 6 pages