Novel bounds of algebraic Nikol’skii constant

Abstract

Let $M_{n}=\sup_{P\in \mathcal{P}_{n}\setminus \{0\}} \frac{\max_{x\in

[-1,1]}|P(x)|}{\int_{-1}^{1}|P(x)|\,dx}$ be the Nikol'skii constant between the

uniform and integral norms for algebraic polynomials with complex coefficients

of degree at most $n$. D. Amir and Z. Ziegler (1976) proved that

$0.125(n+1)^{2}\le M_{n}\le 0.5(n+1)^{2}$ for $n\ge 0$. The same upper bound

was obtained by T.K. Ho (1976). F. Dai, D. Gorbachev, and S. Tikhonov

(2019--2020) refined this result by establishing that $M_{n}=Mn^{2}+o(n^{2})$

for $n\to \infty$, where $M\in (0.141,0.192)$ is the sharp Nikol'skii constant

for entire functions of exponential spherical type in the space

$L^{1}(\mathbb{R}^{2})$ and functions of exponential type in

$L^{1}(\mathbb{R})$ with weight $|x|$.

We prove that for arbitrary $n\ge 0$ one has $M(n+1)^{2}\le M_{n}\le

M(n+2)^{2}$, where $M\in (0.1410,0.1411)$. This statement also allows us to

refine the exact Jackson--Nikol'skii constant for polynomials on the Euclidean

sphere $\mathbb{S}^{2}$. The proof is based on the relationship between the

algebraic Nikol'skii constants and the Bernstein--Nikol'skii trigonometric

constants and our estimates of these constants (2018--2019). We also apply the

characterization of the extremal algebraic polynomial obtained by D. Amir and

Z. Ziegler (1976), V.V. Arestov and M.V. Deikalova (2015). Using this

characterization, we compose a trigonometric system for determining the zeros

of an extremal polynomial, which we solve approximately with the required

accuracy using Newton's method.