On Multivariate Polynomial L¹ — Approximation to Zero and Related Coefficient Inequalities

Abstract

It is well known (cf. TIMAN /12/, pp.66) that among all monic univariate polynomials of degree n the n-th normalized orthogonal polynomial with respect to the weight function w_p, given by w_p(x)=(1-x²)^{1/p -1/2},is the best approximation to zero on I = [-1,1] in the L^P-sense,

$$p \in \{ 1,2,\infty \}$$

. It is also true that among all polynomials of degree ≤ n with second leading coefficient equal to 1 the corresponding monic orthogonal polynomial of degree n - 1 deviates least form zero on I in the L^P-sense.