Nikol'skii inequalities and their applications (17rit679)

Arriving in Banff, Alberta Sunday, December 3 and departing Sunday December 10, 2017

Organizers

(Catalan Institution for Research and Advanced Studies)

(University of Alberta)

Vladimir Temlyakov (University of South Carolina)

Objectives

The main objective of our proposed collaboration at BIRS in 2017 is to continue our collaboration that began at CRM in Barcelona during a special semester in 2016. We made interesting progress in sharp Nikol'skii inequality in CRM and plan to further advance these results during the Banff visit.



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



  1. [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.
  2. [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.
  3. [BRV2] A.~Bondarenko, D.~Radchenko, and M.~Viazovska, \textit{Well-separated spherical designs}, Constr. Approx. 41 (2015), no.~1, 93--112.
  4. [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.
  5. [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.
  6. [te2] V. Temlyakov, S. Tikhonov \textit{Remez-type inequalities for the hyperbolic cross polynomials}, arXiv:1606.03773.
  7. [te3] V. N. Temlyakov, \textit{ Sparse approximation and recovery by greedy algorithms in Banach spaces}, Forum Math. Sigma 2 (2014).