Green's Functions for Multiply Connected Domains via Conformal Mapping

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.

[1]  P. Strevens.  Iii , 1985 .

[2]  Gilbert Strang,et al.  The asymptotics of optimal (equiripple) filters , 1999, IEEE Trans. Signal Process..

[3]  Gilbert Strang,et al.  The Potential Theory of Several Intervals and Its Applications , 2001 .

[4]  Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten , .

[5]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[6]  L. Ahlfors.  Conformal Invariants: Topics in Geometric Function Theory , 1973 .

[7]  E. Cheney.  Introduction to approximation theory , 1966 .

[8]  J. Walsh,et al.  On the convergence and overconvergence of sequences of polynomials of best simultaneous approximation to several functions analytic in distinct regions , 1934 .

[9]  Wolfgang H. J. Fuchs,et al.  The Theory of Functions of One Complex Variable , 1967 .

[10]  Vilmos Totik,et al.  General Orthogonal Polynomials , 1992 .

[11]  R. Langer.  Interpolation and Approximation by Rational Functions in the Complex Domain , 1937 .

[12]  H. Widom.  Extremal polynomials associated with a system of curves in the complex plane , 1969 .

[13]  G. Szegö.  Bemerkungen zu einer Arbeit von Herrn M. Fekete: Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten , 1924 .

[14]  On Chebychev Approximation on Several Disjoint Intervals , 1980 .

[15]  辻 正次,et al.  Potential theory in modern function theory , 1959 .

[16]  D. Gaier,et al.  Lectures on complex approximation , 1987 .

[17]  J. Cooper,et al.  Theory of Approximation , 1960, Mathematical Gazette.

[18]  E. Hille.  Analytic Function Theory , 1961 .

[19]  C. K. Yuen,et al.  Theory and Application of Digital Signal Processing , 1978, IEEE Transactions on Systems, Man, and Cybernetics.

[20]  B. Fischer.  Polynomial Based Iteration Methods for Symmetric Linear Systems , 1996 .

[21]  L. Trefethen.  Numerical computation of the Schwarz-Christoffel transformation , 1979 .

[22]  T. W. Parks,et al.  Digital Filter Design , 1987 .

[23]  Tobin A. Driscoll,et al.  Algorithm 756: a MATLAB toolbox for Schwarz-Christoffel mapping , 1996, TOMS.

[24]  Lloyd N. Trefethen,et al.  Analysis and design of polygonal resistors by conformal mapping , 1984 .

[25]  C. David Levermore,et al.  The Small Dispersion Limit of the Korteweg-deVries Equation. I , 1982 .

[26]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[27]  Stephanos Venakides,et al.  The Small Dispersion Limit of the Korteweg-De Vries Equation , 1987 .

[28]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[29]  J. McClellan,et al.  Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase , 1972 .

[30]  Tobin A. Driscoll,et al.  From Potential Theory to Matrix Iterations in Six Steps , 1998, SIAM Rev..

[31]  E. Saff,et al.  Logarithmic Potentials with External Fields , 1997 .