International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 4, Pages 625-638
doi:10.1155/S0161171291000868
Finite-infinite range inequalities in the complex plane
Department of Mathematics, California State University, Los Angeles 90032, California, USA
Received 23 April 1990
Copyright © 1991 H. N. Mhaskar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let E⫅C be closed, ω be a suitable weight function on E, σ be a positive
Borel measure on E. We discuss the conditions on ω and σ which ensure the existence of
a fixed compact subset K of E with the following property. For any p, 0<P≤∞, there
exist positive constants c1, c2 depending only on E, ω, σ and p such that for every integer
n≥1 and every polynomial P of degree at most n,
∫E\K|ωnP|pdσ≤c1exp(−c2n)∫K|ωnP|pdσ.
In particular, we shall show that the support of a certain extremal measure is, in some
sense, the smallest set K which works. The conditions on σ are formulated in terms of
certain localized Christoffel functions related to σ.