On the size of lemniscates of polynomials in one and several variables

Abstract

Abstract. In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set E := z : |z| ≤ r and |P (z)| ≤ n , for a polynomial P of degree ≤ n. Usually, P is taken to be monic, and either Cartan's Lemma or potential theory is used to estimate the size of E, in terms of Hausdorff contents, planar Lebesgue measure m 2 , or logarithmic capacity cap. Here we normalize P L∞ |z|≤r = 1 and show that cap(E) ≤ 2r and m 2 (E) ≤ π(2r ) 2 are the sharp estimates for the size of E. Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on C n or product capacity and Favarov's capacity. Several of our estimates are sharp with respect to order in r and . §1. IntroductionIn the convergence theory of Padé approximation, and more generally rational interpolation, an essential ingredient is an estimate on the size of the lemniscatewhere P is a polynomial of degree ≤ n. There are several ways to provide this estimate. Cartan's Lemma shows that if P is normalized to be monic of degree n, then we can cover this set by a union of ≤ n balls B j , 1 ≤ j ≤ , whose diameters d(B j ) satisfy, for a given α > 0,The remarkable thing about the estimate is its independence of the degree of P . See [1, p. 194

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“…This lemma has connections to estimating the size of lemniscates of multivariate polynomials, studied in[22,23]. However, here we prove a different form of upper bound which suits our framework to prove the finiteness of the noise variance in(20).…”

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“…then the inequality |p(z)| > M holds for z outside discs, the sum of their radii being at most 2eM 1/N . There is also a formulation in terms of Hausdorff measures, which we formulate in the following way [9]: for positive and α, the set E(p,…”

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On the Maximum and Minimum Modulus of Rational Functions

“…for entire or meromorphic functions f , plays a role in complex function theory in topics ranging from distribution of zeros and deficient values, to gap series. More specifically, the ratio of maximum modulus to minimum modulus for polynomials P plays a crucial role in several questions in rational approximation [1], [4], [10].…”