The Zeros of Faber Polynomials for an m-Cusped Hypocycloid

He, M. X.; Saff, Edward B.

doi:10.1006/jath.1994.1087

Cited by 50 publications

(42 citation statements)

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“…Some references on this subject are [1,2,10,26,31]. See also [13,18] for the related study of Faber polynomials associated with hypocycloidal domains and stars. Lemma 1.3 allows us to obtain the asymptotic properties of the polynomials P n,n through the analysis of a matrix-valued Riemann-Hilbert problem of size (d + 1) × (d + 1) that encodes the multiple orthogonality conditions (1.32).…”

Section: Orthogonality On (D + 1)-star and Multiple Orthogonalitymentioning

confidence: 99%

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Kuijlaars

López-García

2015

Nonlinearity

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We investigate the asymptotic behavior of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree d + 1. The polynomials that we investigate are multiple orthogonal with respect to a system of d analytic weights defined on a symmetric (d + 1)-star centered at the origin. In the first part we analyze in detail a vector equilibrium problem involving a system of d interacting measures (µ1, . . . , µ d ) supported on star-like sets in the plane. We show that in the subcritical regime, the first component µ * 1 of the solution to this problem is the asymptotic zero distribution of the multiple orthogonal polynomials. It also characterizes the domain where the eigenvalues in the normal matrix model accumulate, in the sense that the Schwarz function associated with the boundary of this domain can be expressed explicitly in terms of µ * 1 . The second part of the paper is devoted to the asymptotic analysis of the multiple orthogonal polynomials. The asymptotic results are obtained again in the subcritical regime, and they follow from the Deift/Zhou steepest descent analysis of a Riemann-Hilbert problem of size (d + 1) × (d + 1). The vector equilibrium problem and the Riemann-Hilbert problem that we investigate are generalizations of those studied recently by Bleher-Kuijlaars in the case d = 2.

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Section: Orthogonality On (D + 1)-star and Multiple Orthogonalitymentioning

confidence: 99%

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Kuijlaars

López-García

2015

Nonlinearity

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“…[10], Proposition 2.1) and that the associated Faber polynomials can be explicitly calculated (see e.g. [10], Proposition 2.3 and [11]).…”

Section: Applicationsmentioning

confidence: 99%

“…[10], Proposition 2.1) and that the associated Faber polynomials can be explicitly calculated (see e.g. [10], Proposition 2.3 and [11]). Without entering into details, in this case for the construction of the q-Stancu-Faber polynomials one relies on the calculation of the integrals in (2).…”

Section: Applicationsmentioning

confidence: 99%

Approximation in compact sets by q-Stancu–Faber polynomials, q>1

Gal

2011

Computers & Mathematics with Applications

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“…This result is important for applications because for some f and K a computation of F n is more simple than p n (see Ellacott, 1983;Johnston, 1987;He and Saff, 1994;He, 1994;Coleman and Myers, 1995, and references therein).…”

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confidence: 93%

n-Widths, Faber Expansion, and Computation of Analytic Functions

Farkov

1996

Journal of Complexity

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The Zeros of Faber Polynomials for an m-Cusped Hypocycloid

Cited by 50 publications

References 10 publications

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Approximation in compact sets by q-Stancu–Faber polynomials, q>1

n-Widths, Faber Expansion, and Computation of Analytic Functions

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