Computation of Chebyshev Polynomials for Union of Intervals
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Computation of Chebyshev Polynomials for Union of Intervals Simon Foucart1 · Jean Bernard Lasserre2 Received: 30 June 2018 / Revised: 9 March 2019 / Accepted: 20 June 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Chebyshev polynomials of the first and second kind for a set K are monic polynomials with minimal L ∞ - and L 1 -norm on K , respectively. This article presents numerical procedures based on semidefinite programming to compute these polynomials in case K is a finite union of compact intervals. For Chebyshev polynomials of the first kind, the procedure makes use of a characterization of polynomial non-negativity. It can incorporate additional constraints, e.g. that all the roots of the polynomial lie in K . For Chebyshev polynomials of the second kind, the procedure exploits the method of moments. Keywords Chebyshev polynomials of the first kind · Chebyshev polynomials of the second kind · Non-negative polynomials · Method of moments · Semidefinite programming Mathematics Subject Classification 31A15 · 41A50 · 90C22
1 Introduction The N th Chebyshev polynomial for a compact infinite subset K of C is defined as the monic polynomial of degree N with minimal max-norm on K . Its uniqueness is a straightforward consequence of the uniqueness of best polynomial approximants to a
Communicated by Thomas Ransford. The research of Simon Foucart is partially funded by the NSF Grants DMS-1622134 and DMS-1664803. The research of Jean Bernard Lasserre is funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant agreement 666981 TAMING).
B
Simon Foucart [email protected]
1
Texas A&M University, College Station, USA
2
LAAS-CNRS, Toulouse, France
123
S. Foucart, J. B. Lasserre
continuous function (here z → z N ) with respect to the max-norm, see e.g. [4, p. 72, Thm. 4.2]. We shall denote it as T NK , i.e. T NK = argmin P K ,
where P K = max |P(z)|. z∈K
P(z)=z N +···
(1)
We reserve the notation TNK for the Chebyshev polynomial normalized to have max-norm equal to one on K , i.e. TNK =
T NK T NK K
.
(2)
With this notation, the usual N th Chebyshev polynomial (of the first kind) satisfies TN = TN[−1,1] = 2 N −1 T N[−1,1] ,
N ≥ 1.
(3)
Chebyshev polynomials for a compact subset K of C play an important role in logarithmic potential theory. For instance, it is known that the capacity cap(K ) of K is related to the Chebyshev numbers t NK := T NK K via
t NK
1/N
−→ cap(K ),
N →∞
(4)
see [12, p.163, Thm. 3.1] for a weighted version of this statement. Some recent works [1,2] have studied in greater detail the asymptotics of the convergence (4) in case K is a subset of R. This being said, the capacity is in general hard to determine—it can be found explicitly in a few specific situations, e.g. when K is the inverse image of an interval by certain polynomials (see [8, Thm. 11]) and otherwise some numerical methods for computing the capacity have been proposed in [11], see also [10, Sect. 5.2]. As for
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