Bernstein's inequality for multivariate polynomials on the standard simplex
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classical Bernstein pointwise estimate of the (first) derivative of a univariate algebraic polynomial on an interval has natural extensions to the multivariate setting. However, in several variables the domain of boundedness, even if convex, has a considerable geometric variety. In 1990, Y. Sarantopoulos satisfactorily settled the case of a centrally symmetric convex body by a method we may call “the method of inscribed ellipses.” On the other hand, for the general case of nonsymmetric convex bodies we are only within a constant factor of an exact inequality. The best known results suggest relevance of the generalized Minkowski functional, and a natural conjecture for the exact Bernstein factor was formulated with this geometric quantity. This work deals with the most natural and simple nonsymmetric case, that of a standard simplex in Rd , and computes the exact yield of the method of inscribed ellipses. Although the known general estimates of the Bernstein factor are improved for the simplex here, we find that not even the exact yield of the inscribed ellipse method reaches the conjecture. However, we also show that for an arbitrary convex body the subset of ridge polynomials satisfies the conjecture. 1. Introduction If a univariate algebraic polynomial p is given with degree at most n, then by the classical Bernstein-Szeg˝o inequality (see [1, 10, 11]), we have 2 n pC[a,b] − p2 (x) p (x) ≤ (b − x)(x − a)
(a < x < b).
(1.1)
This inequality is sharp for every n and every point x ∈ (a,b), as
n : deg p ≤ n, p(x) < pC[a,b] = sup . (b − x)(x − a) p2C[a,b] − p2 (x) p (x)
(1.2)
We may say that the upper estimate (1.1) is exact, and the right-hand side is just the “true Bernstein factor” of the problem. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:2 (2005) 145–163 DOI: 10.1155/JIA.2005.145
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Bernstein’s inequality on the simplex
In the multivariate setting a number of extensions were proved for this classical result. However, due to the geometric variety of possible convex sets replacing intervals of R, our present knowledge is still not final. The exact Bernstein inequality is known only for symmetric convex bodies, and we are within a bound of some constant factor in the general, nonsymmetric case. For more precise notation we may define formally for any topological vector space X, a subset K ⊂ X, and a point x ∈ K the nth “Bernstein factor” as
Bn (K,x) :=
D p(x)
1 : deg p ≤ n, p(x) < pC(K) , sup n p2C(K) − p2 (x)
(1.3)
where D p(x) is the derivative of p at x, and even for an arbitrary unit vector y ∈ X
D p(x),y 1 : deg p ≤ n, p(x) < pC(K) . Bn (K,x,y) := sup n p2C(K) − p2 (x)
(1.4)
In the present paper, first we study the standard simplex of the d-dimensional Euclidean space Rd . We find the exact yield of the possibly nicest available method—the method of inscribed ellipses, introduced into the subject by S
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