Balayage Properties Related to Rational Interpolation

Distribution of critical points of polynomials

“…It may happen that this S Γ contains only the equilibrium measure ω Γ . This is the case for example, for the curve in part c) of Theorem 1.2 (which is taken from the paper [4] by Gardiner and Pommerenke). Theorem 1.4 Let Γ be a Jordan curve, and σ a probability measure supported on G Γ .…”

“…Γ is a Jordan curve consisting of the two segments connecting the origin with the points e −iπ/4 and −e iπ/4 , plus the longer arc of the unit circle connecting these two points. If we set ε(t) = 1 in (i) of Theorem 3 in [4], then this Γ is precisely the boundary of the domain described in that theorem. Suppose now hat ν Pn → ω Γ , and let ν be a weak * -limit point of ν P ′ n , say along a subsequence N of the natural numbers.…”

On the zeros of asymptotically extremal polynomial sequences in the plane

“…We remark that, for the case of a Jordan region, the hypothesis of Theorem 3 of [3] implies the existence of an NCS point. We also note that the assumption dist…”

“…Without loss of generality, we make the following simplifications regarding the two conditions in Definition 2.1: By performing a translation and scaling we take z 0 to be the origin and by rotation we take ζ 0 = iγ, for some γ ∈ (0, 1). These claims follow as in [3], utilizing in the justification of Claim (b) the essential condition that the origin is an NCS point so that Using Claims (a) and (b) and the fact that S(0) = 0 (since the origin is a regular point of Ω), it is easy to arrive at a relation that contradicts the mean value inequality for superharmonic functions (see also [3, This establishes the theorem for the case E 0 = E.…”

“…For example, this is the case whenever E is a non-convex polygonal region, a simple fact that has thus far not been sufficiently emphasized in the literature. Here we introduce more general sufficient conditions based on arguments of Gardiner and Pommerenke [3] dealing with the balayage of measures.…”