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Unconditional constants and polynomial inequalities

Grecu, B.C. and Muñoz Fernández, Gustavo Adolfo and Seoane Sepúlveda, Juan Benigno (2009) Unconditional constants and polynomial inequalities. Journal of Approximation Theory, 161 (2). pp. 706-722. ISSN 0021-9045

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Abstract

If P is a polynomial on R of degree at most n, given by P(x) = Sigma(alpha is an element of Nm,vertical bar alpha vertical bar <= n) a(alpha)x(alpha), and P(n)(R(m)) is the space of such polynomials, then we define the polynomial vertical bar P vertical bar by vertical bar P vertical bar(x) = Sigma(alpha is an element of Nm,vertical bar alpha vertical bar <= n) vertical bar a(alpha vertical bar)x(alpha). Now if B subset of R(m) is a convex set, we define the norm parallel to P parallel to(B) := sup{vertical bar(x)vertical bar : x is an element of B} on P(n)(R(m)), and then we investigate the inequality vertical bar vertical bar vertical bar P vertical bar vertical bar vertical bar(B) <= C(B)vertical bar vertical bar vertical bar P vertical bar vertical bar vertical bar(B), providing sharp estimates on C(B) for some specific spaces of polynomials. These C(B)'s happen to be the unconditional constants of the canonical bases of the considered spaces.


Item Type:Article
Uncontrolled Keywords:Unconditional constant; Polynomial inequalities; Trinomials; Homogeneous polynomials; Extreme points
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:16967
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