Weighted Maximum over Minimum Modulus of Polynomials, Applied to Ray Sequences of Padé Approximants
نویسنده
چکیده
Let a 0; " > 0. We use potential theory to obtain a sharp lower bound for the linear Lebesgue measure of the set ( r 2 [0; 1] : r maxjtj=1 jP (t)j minjtj=r jP (t)j " ) : Here P is an arbitrary polynomial of degree n. We then apply this to diagonal and ray Padé sequences for functions analytic (or meromorphic) in the unit ball. For example, we show that the diagonal f[n=n]gn=1 sequence provides good approximation on almost 1 8 of the circles centre 0, and the f[2n=n]gn=1 sequence on almost 1 4 of such circles. 1. Introduction Let f be a function analytic at 0, and hence possessing a Maclaurin series there. Recall that if m;n 0, the (m;n) Padé approximant to f is a rational function [m=n] (z) = (p=q) (z) ; where p; q are polynomials of degree m;n respectively, with q not identically zero, and (fq p) (z) = O z : The order relation indicates that the coe¢ cients of 1; z; z; :::z in the Maclaurin series of the left-hand side vanish. For an introduction to the subject, see [1]. The convergence theory of Padé approximation is rich and complex. It is known that if f is analytic at 0, and meromorphic in the whole plane, then f[n=n]gn=1 converges in measure, and in capacity the Nuttall-Pommerenke Theorem [13], [14]. More generally, given sequences of positive integers fmkgk=1 ; fnkg 1 k=1 that tend to 1 in such a way that for some xed 1; 1 mk nk ; k 1; and given r; " > 0; m2 fz : jzj r and jf [mk=nk]j (z) > "kg ! 0; k !1: Here m2 denotes planar measure, and it may be replaced by capacity. There are deeper analogues for functions with branchpoints [20], [21]. One unfortunate feature of the theorem is that it really requires f to be meromorphic in C. There are functions analytic in the unit ball for which f[n=n]gn=1 does Date : May 18, 2001 1991 Mathematics Subject Classi cation : 30E10, 30C15, 31A15, 41A21. 1
منابع مشابه
Szegő-type Asymptotics for Ray Sequences of Frobenius-padé Approximants
Let σ̂ be a Cauchy transform of a possibly complex-valued Borel measure σ and {pn} be a system of orthonormal polynomials with respect to a measure μ, supp(μ)∩ supp(σ) = ∅. An (m,n)-th Frobenius-Padé approximant to σ̂ is a rational function P/Q, deg(P) 6m, deg(Q) 6 n, such that the first m+n+ 1 Fourier coefficients of the linear form Qσ̂−P vanish when the form is developed into a series with respe...
متن کاملChromatic Harmonic Indices and Chromatic Harmonic Polynomials of Certain Graphs
In the main this paper introduces the concept of chromatic harmonic polynomials denoted, $H^chi(G,x)$ and chromatic harmonic indices denoted, $H^chi(G)$ of a graph $G$. The new concept is then applied to finding explicit formula for the minimum (maximum) chromatic harmonic polynomials and the minimum (maximum) chromatic harmonic index of certain graphs. It is also applied to split graphs and ce...
متن کاملFrobenius-Padé approximants for d-orthogonal series: theory and computational aspects
This work is about Frobenius-Padé approximants for series of orthogonal polynomials of dimension d (d ∈ N). Concerning to the series, we give the projection property of partial sums, we show how to compute their coe cients, and how to get the coe cients of the product of a series by a polynomial. Concerning to the approximants we work essentially about their recursive computation. Also, we give...
متن کاملWeighted sums of orthogonal polynomials with positive zeros
We study the two sequences of polynomials which arise as denominators of the approximants of even and odd order, respectively, of a Stieltjes fraction, and which may be defined alternatively as a sequence of orthogonal polynomials with positive zeros and the associated sequence of kernel polynomials. Motivated by problems in the setting of birth-death processes, where these sequences play a maj...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2005