Novel bounds of algebraic Nikol’skii constant
Let M<sub>n</sub> = supp<sub>e</sub>p^\{o} <sup>ma</sup>/1<sup>el</sup>|p<sup>1</sup>(>)| ^e 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)<sup>5</sup> < M<sub>n</sub> < 0.5(n + 1)^ for n > 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<sub>n</sub> = Mn<sup>5</sup> + o(n<sup>5</sup>)for n ^ to, where M G (0.141,0.192) is the sharp Nikol’skii constant for entire functions of exponential spherical type in the space Т<sup>Х</sup>(М<sup>5</sup>) and functions of exponential type in М(М) with weight \x\. We prove that for arbitrary n > 0 one has M(n + 1)<sup>5</sup> < M<sub>n</sub> < M(n + 2)<sup>5</sup>, where M G (0.1410,0.1411). This statement also allows us to refine the exact Jackson-Nikol’skii constant for polynomials on the Euclidean sphere S<sup>5</sup>. 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.