## Green’s functions for multiply connected domains via conformal mapping (1999)

Venue: | SIAM Review |

Citations: | 8 - 2 self |

### BibTeX

@ARTICLE{Embree99green’sfunctions,

author = {Mark Embree and Lloyd N. Trefethen},

title = {Green’s functions for multiply connected domains via conformal mapping},

journal = {SIAM Review},

year = {1999},

pages = {745--761}

}

### OpenURL

### Abstract

Abstract. A method is described for the computation of the Green’s function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz–Christoffel conformal map of the part of the upper half-plane exterior to the problem domain onto a semi-infinite strip whose end contains K − 1 slits. From the Green’s function one can obtain a great deal of information about polynomial approximations, with applications in digital filters and matrix iterations. By making the end of the strip jagged, the method can be generalized to weighted Green’s functions and weighted approximations. Key words. Green’s function, conformal mapping, Schwarz–Christoffel formula, polynomial approximation,

### Citations

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Citation Context ...asic tools for the analysis of real and complex polynomial approximations [10,21,24,30,32], which are of central importance in the fields of digital signal processing [16,17,19] and matrix iterations =-=[5,6,11,20,28]-=-. The aim of this article is to show that when the domain of approximation is a collection of real intervals, or more generally symmetric polygons along the real axis, the Green's function can be comp... |

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Citation Context ...asic tools for the analysis of real and complex polynomial approximations [10,21,24,30,32], which are of central importance in the fields of digital signal processing [16,17,19] and matrix iterations =-=[5,6,11,20,28]-=-. The aim of this article is to show that when the domain of approximation is a collection of real intervals, or more generally symmetric polygons along the real axis, the Green's function can be comp... |

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Citation Context ...teration AMS subject classifications. 30E10, 31A99, 41A10, 65F10 1. Introduction. Green's functions in the complex plane are basic tools for the analysis of real and complex polynomial approximations =-=[10,21,24,30,32]-=-, which are of central importance in the fields of digital signal processing [16,17,19] and matrix iterations [5,6,11,20,28]. The aim of this article is to show that when the domain of approximation i... |

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Citation Context ...rtran package SCPACK [26]. The Green's function for a single interval can be obtained by a Joukowsky conformal map, and related polynomial approximation problems were solved by Chebyshev in the 1850s =-=[3]-=-. For two disjoint intervals, the Green's function can be expressed using elliptic functions, and approximation problems were investigated by Akhiezer in the 1930s [2]. For K ? 2 intervals, the Green'... |

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Citation Context ...teration AMS subject classifications. 30E10, 31A99, 41A10, 65F10 1. Introduction. Green's functions in the complex plane are basic tools for the analysis of real and complex polynomial approximations =-=[10,21,24,30,32]-=-, which are of central importance in the fields of digital signal processing [16,17,19] and matrix iterations [5,6,11,20,28]. The aim of this article is to show that when the domain of approximation i... |

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Citation Context ...rator, and thus g is harmonic throughout the complex plane exterior to the polygons P j . Standard results of potential theory ensure that there exists a unique function g satisfying these conditions =-=[12,13,29,32]-=-. The solution to (1) can be constructed by conformal mapping. What makes this possible is that the problem is symmetric with respect to the real axis, so it is enough to find g(z) for the part of the... |

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Citation Context ...ane. Harmonic measure. Another scale-independent quantity is the proportionsj of the total charge on each polygon P j , which is known as the harmonic measure of P j (with respect to the point z = 1) =-=[1,7,13]-=-. This quantity is equal tos\Gamma1 times the distance between the appropriate two slits in the strip domain (or a slit and one of the semi-infinite boundary lines), or equivalently tos\Gamma1 times t... |

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Citation Context ...rator, and thus g is harmonic throughout the complex plane exterior to the polygons P j . Standard results of potential theory ensure that there exists a unique function g satisfying these conditions =-=[12,13,29,32]-=-. The solution to (1) can be constructed by conformal mapping. What makes this possible is that the problem is symmetric with respect to the real axis, so it is enough to find g(z) for the part of the... |

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Citation Context ...rator, and thus g is harmonic throughout the complex plane exterior to the polygons P j . Standard results of potential theory ensure that there exists a unique function g satisfying these conditions =-=[12,13,29,32]-=-. The solution to (1) can be constructed by conformal mapping. What makes this possible is that the problem is symmetric with respect to the real axis, so it is enough to find g(z) for the part of the... |

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Citation Context ...ated in a landmark article by Widom in 1969 [32]. Polynomial approximations can be readily computed in this case by the Remes algorithm, which was adapted for digital filtering by Parks and McClellan =-=[3,18]-=-. By a second conformal map, these ideas for intervals can be transplanted to the more general problem of the Green's function for the region exterior to a string of symmetric domains along the real a... |

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Citation Context ...ns in the complex plane are basic tools for the analysis of real and complex polynomial approximations [10, 21, 24, 30, 32], which are of central importance in the fields of digital signal processing =-=[16, 17, 19]-=- and matrix iterations [5, 6, 11, 20, 28]. The aim of this article is to show that when the domain of approximation is a collection of real intervals, or more generally symmetric polygons along the re... |

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Citation Context ...asic tools for the analysis of real and complex polynomial approximations [10,21,24,30,32], which are of central importance in the fields of digital signal processing [16,17,19] and matrix iterations =-=[5,6,11,20,28]-=-. The aim of this article is to show that when the domain of approximation is a collection of real intervals, or more generally symmetric polygons along the real axis, the Green's function can be comp... |

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Citation Context ...that the upper and lower sides of each slit have equal length. Details can be found in [23] and [32]. A related linear Schwarz--Christoffel problem involving slits in the complex plane is implicit in =-=[14]-=-. By composing a third conformal map with the first two, we obtain a picture that is even more revealing than Fig. 1. Figure 2 depicts the image of the slit strip under the complex exponential: w = \P... |

43 |
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Citation Context ...teration AMS subject classifications. 30E10, 31A99, 41A10, 65F10 1. Introduction. Green's functions in the complex plane are basic tools for the analysis of real and complex polynomial approximations =-=[10,21,24,30,32]-=-, which are of central importance in the fields of digital signal processing [16,17,19] and matrix iterations [5,6,11,20,28]. The aim of this article is to show that when the domain of approximation i... |

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(Show Context)
Citation Context ...warz--Christoffel maps has become routine in recent years with the introduction of Driscoll's Matlab R fl Schwarz-- Christoffel Toolbox [4], a descendant of the second author's Fortran package SCPACK =-=[26]-=-. The Green's function for a single interval can be obtained by a Joukowsky conformal map, and related polynomial approximation problems were solved by Chebyshev in the 1850s [3]. For two disjoint int... |

20 |
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(Show Context)
Citation Context ...solved by Chebyshev in the 1850s [3]. For two disjoint intervals, the Green's function can be expressed using elliptic functions, and approximation problems were investigated by Akhiezer in the 1930s =-=[2]-=-. For K ? 2 intervals, the Green's function can be derived from a more general Schwarz--Christoffel conformal map, and the formulas that result were stated in a landmark article by Widom in 1969 [32].... |

13 |
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(Show Context)
Citation Context ...ize approximately (10:292969) n . Other related matters, such as generalized Faber polynomials [32], can also be pursued. GREEN'S FUNCTIONS FOR MULTIPLY CONNECTED DOMAINS 11 Theorem 1 is due to Szego =-=[25]-=-, who extended earlier work of Fekete; a proof can be found for example in [29]. For the case in which E is a smooth Jordan domain, Faber showed that in fact kTnk=C n ! 1 as n ! 1. If E consists of tw... |

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Citation Context ... Schwarz-- Christoffel conformal mapping. The computation of Schwarz--Christoffel maps has become routine in recent years with the introduction of Driscoll's Matlab R fl Schwarz-- Christoffel Toolbox =-=[4]-=-, a descendant of the second author's Fortran package SCPACK [26]. The Green's function for a single interval can be obtained by a Joukowsky conformal map, and related polynomial approximation problem... |

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Citation Context ...ft-hand edges that would achieve the uniform critical value. (This is an example of a generalized Schwarz--Christoffel parameter problem, in which geometric constraints from various domains are mixed =-=[27]-=-.) The locations that satisfy the conditions are 10:948290, 20:326250, and 31:191359, the critical potential value is g c = 0:0698122, and the capacity is C = 10:292969. 6. Applications to polynomial ... |

7 |
Topics in the Theory of Functions of One Complex Variable
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Citation Context ...ane. Harmonic measure. Another scale-independent quantity is the proportionsj of the total charge on each polygon P j , which is known as the harmonic measure of P j (with respect to the point z = 1) =-=[1,7,13]-=-. This quantity is equal tos\Gamma1 times the distance between the appropriate two slits in the strip domain (or a slit and one of the semi-infinite boundary lines), or equivalently tos\Gamma1 times t... |

5 |
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(Show Context)
Citation Context ...sh, Widom, and Fuchs, among others. In particular, Walsh, Russell, and Fuchs obtained theorems concerning simultaneous approximation of distinct entire functions on disjoint sets in the complex plane =-=[8,9,30]-=-, which we illustrate here in Section 6. Wolfgang Fuchs was for many years a leading figure at Cornell University until his unfortunate death in 1997. 2. Description of the algorithm. Let E be a compa... |

4 |
On Chebyshev approximation on several disjoint intervals
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(Show Context)
Citation Context ...sh, Widom, and Fuchs, among others. In particular, Walsh, Russell, and Fuchs obtained theorems concerning simultaneous approximation of distinct entire functions on disjoint sets in the complex plane =-=[8,9,30]-=-, which we illustrate here in Section 6. Wolfgang Fuchs was for many years a leading figure at Cornell University until his unfortunate death in 1997. 2. Description of the algorithm. Let E be a compa... |

3 | The asymptotics of optimal (equiripple) filters
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- 1999
(Show Context)
Citation Context ...grateful to him for many suggestions. The contributions of Jianhong Shen and Gilbert Strang at MIT were also a crucial help to us. Shen and Strang have studied the accuracy of lowpass digital filters =-=[22,23]-=-, and their asymptotic formulas are directly connected to these Schwarz--Christoffel methods. In addition we thank Toby Driscoll for his advice and assistance. Our algorithm makes possible the computa... |

3 |
General Orthogonal Polynomials, Cambridge U
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- 1992
(Show Context)
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2 |
On the convergence and overconvergence of sequences of polynomials of best simultaneous approximation to several functions analytic in distinct regions
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Citation Context ...s E with multiple components. Instead of discussing Chebyshev polynomials further, we shall consider a different, related approximation problem investigated by Walsh, Russell, and Fuchs, among others =-=[8,9,30,31]-=-. Let h 1 ; h 2 ; : : : ; h K be entire functions, i.e., each h j is analytic throughout the complex plane, and to keep the formulations simple, assume that these functions are distinct. The following... |

1 |
The potential theory of several intervals and its
- Shen, Strang, et al.
(Show Context)
Citation Context ...grateful to him for many suggestions. The contributions of Jianhong Shen and Gilbert Strang at MIT were also a crucial help to us. Shen and Strang have studied the accuracy of lowpass digital filters =-=[22,23]-=-, and their asymptotic formulas are directly connected to these Schwarz--Christoffel methods. In addition we thank Toby Driscoll for his advice and assistance. Our algorithm makes possible the computa... |

1 |
From potential theory to matrix iterations in six steps
- Trefethen
- 1998
(Show Context)
Citation Context ...ools for the analysis of real and complex polynomial approximations [10, 21, 24, 30, 32], which are of central importance in the fields of digital signal processing [16, 17, 19] and matrix iterations =-=[5, 6, 11, 20, 28]-=-. The aim of this article is to show that when the domain of approximation is a collection of real intervals, or more generally symmetric polygons along the real axis, the Green’s function can be comp... |

1 |
Saff andV. Totik, Logarithmic Potentials with External Fields
- B
- 1997
(Show Context)
Citation Context ...assifications. 30E10, 31A99, 41A10, 65F10 PII. S0036144598349277 1. Introduction. Green’s functions in the complex plane are basic tools for the analysis of real and complex polynomial approximations =-=[10, 21, 24, 30, 32]-=-, which are of central importance in the fields of digital signal processing [16, 17, 19] and matrix iterations [5, 6, 11, 20, 28]. The aim of this article is to show that when the domain of approxima... |

1 |
The potential theory of several intervals and its
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(Show Context)
Citation Context ...n the reverse, more trivial direction, with only a linear parameter problem to be solved to impose the condition that the upper and lower sides of each slit have equal length. Details can be found in =-=[23]-=- and [32]. A related linear Schwarz–Christoffel problem involvingslits in the complex plane is implicit in [14]. By composinga third conformal map with the first two, we obtain a picture that is even ... |