Novel bounds of algebraic Nikol’skii constant
Let Mn = suppep^\{o} ma/1el|p1(>)| ^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)5 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 Mn = Mn5 + o(n5)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 ТХ(М5) and functions of exponential type in М(М) with weight \x\. We prove that for arbitrary n > 0 one has M(n + 1)5 n 5, 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 S5. 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.