# Nikol'skii inequalities and their applications (17rit679)

## Organizers

Sergey Tikhonov (Catalan Institution for Research and Advanced Studies)

Feng Dai (University of Alberta)

Vladimir Temlyakov (University of South Carolina)

## Objectives

We plan to attack the following three specific problems:

1)

**Asymptotic behavior of the Nikol'skii constant.**We will study the Nikol'skii constant $$ C(n,d,p,q):=\sup\frac{\|f\|_{L^q(\mathbb{S}^{d})}} {\|f\|_{L^p(\mathbb{S}^{d})}} $$ for $0

**2) The Nikol'skii constant for large dimensions.** It has been shown in [DGT] that the normalized Nikol'skii constant $$ L_{d}:=\frac{\Gamma(d+1)}{2}|\mathbb{S}^d|\mathcal{L}(d, 1,\infty) $$ decays exponentially fast as the dimension $d\to \infty$: \begin{equation}\label{2} 2^{-d}\leq L_{d}\le (\sqrt{2/e})^{d\left(1+O(d^{-2/3})\right)}. \end{equation} Recently, we observed an interesting connection between the constant $L_d$ and the hypergeometric functions. We plan to continue working on the improvement of \eqref{2}. In particular, we will study whether the upper estimate in \eqref{2} is sharp.\\

**3) Applications in greedy approximation with regard to the system of spherical harmonics.**

Very recently, we observed a new phenomenon for high-dimensional `sparse' spherical polynomials: for $d\ge 2$, the classical Nikol'skii inequality on the sphere $\mathbb{S}^d$ can be significantly improved for `sparse' 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 $n_{j+1}-n_j\ge 3$. It is worthwhile to point out that trigonometric polynomials in one variable do not have such a phenomenon. We expect this new observation will have important applications in sparse spherical harmonic approximation on the sphere. In particular, we will explore how far the results of [te3} on greedy approximation with respect to trigonometric systems can be extended to the high-dimensional sphere. This is an unexplored area. One of the main difficulties comes from the fact that elements in an orthonormal basis of $L^2(\mathbb{S]^d)$ consisting of spherical harmonics are not uniformly bounded.

#### Bibliography

- [te1] A.V. Andrianov, V.N. Temlyakov, \textit{ On two methods of generalization of properties of univariate function systems to their tensor product}, Proc. Steklov Inst. Math.
**219**(1997), 25--35. - [BRV1] A.~Bondarenko, D.~Radchenko, and M.~Viazovska, \textit{Optimal asymptotic bounds for spherical designs}, Ann. of Math. (2)
**178**(2013), no.~2, 443--452. - [BRV2] A.~Bondarenko, D.~Radchenko, and M.~Viazovska, \textit{Well-separated spherical designs}, Constr. Approx.
**41**(2015), no.~1, 93--112. - [DGT} F. ~ Dai, D. ~Gorbachev, S. Tikhonov, \textit{ Nikol'skii constants for polynomials on the unit sphere}, in progress. \bibitem{LL1] E.~Levin, D.~Lubinsky, \textit{$L_p$ Chritoffel functions, $L_p$ universality, and Paley--Wiener spaces}, J.~D'Analyse Math.
**125**(2015), 243--283. - [LL2] E.~Levin, D.~Lubinsky, \textit{Asymptotic behavior of Nikol'skii constants for polynomials on the unit circle}, Comput. Methods Funct. Theory (2015),
**15**(2015), no.~3, 459--468. - [te2] V. Temlyakov, S. Tikhonov \textit{Remez-type inequalities for the hyperbolic cross polynomials}, arXiv:1606.03773.
- [te3] V. N. Temlyakov, \textit{ Sparse approximation and recovery by greedy algorithms in Banach spaces}, Forum Math. Sigma
**2**(2014).