{"title":"多项式预像的lemnisctic域上的Walsh保角映射II","authors":"Klaus Schiefermayr, Olivier Sète","doi":"10.1007/s40315-023-00492-6","DOIUrl":null,"url":null,"abstract":"Abstract We consider Walsh’s conformal map from the exterior of a set $$E=\\bigcup _{j=1}^{\\ell }E_j$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mo>⋃</mml:mo> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>ℓ</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:math> consisting of $$\\ell $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> compact disjoint components onto a lemniscatic domain. In particular, we are interested in the case when E is a polynomial preimage of $$[-1,1]$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , i.e., when $$E=P^{-1}([-1,1])$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where P is an algebraic polynomial of degree n . Of special interest are the exponents and the centers of the lemniscatic domain. In the first part of this series of papers, a very simple formula for the exponents has been derived. In this paper, based on general results of the first part, we give an iterative method for computing the centers when E is the union of $$\\ell $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> intervals. Once the centers are known, the corresponding Walsh map can be computed numerically. In addition, if E consists of $$\\ell =2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> or $$\\ell =3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> components satisfying certain symmetry relations then the centers and the corresponding Walsh map are given by explicit formulas. All our theorems are illustrated with analytical or numerical examples.","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"64 1","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Walsh’s Conformal Map Onto Lemniscatic Domains for Polynomial Pre-images II\",\"authors\":\"Klaus Schiefermayr, Olivier Sète\",\"doi\":\"10.1007/s40315-023-00492-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We consider Walsh’s conformal map from the exterior of a set $$E=\\\\bigcup _{j=1}^{\\\\ell }E_j$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mo>⋃</mml:mo> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>ℓ</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:math> consisting of $$\\\\ell $$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>ℓ</mml:mi> </mml:math> compact disjoint components onto a lemniscatic domain. In particular, we are interested in the case when E is a polynomial preimage of $$[-1,1]$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , i.e., when $$E=P^{-1}([-1,1])$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where P is an algebraic polynomial of degree n . Of special interest are the exponents and the centers of the lemniscatic domain. In the first part of this series of papers, a very simple formula for the exponents has been derived. In this paper, based on general results of the first part, we give an iterative method for computing the centers when E is the union of $$\\\\ell $$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>ℓ</mml:mi> </mml:math> intervals. Once the centers are known, the corresponding Walsh map can be computed numerically. In addition, if E consists of $$\\\\ell =2$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> or $$\\\\ell =3$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> components satisfying certain symmetry relations then the centers and the corresponding Walsh map are given by explicit formulas. 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引用次数: 0
摘要
摘要考虑了由$$\ell $$个紧不相交分量组成的集合$$E=\bigcup _{j=1}^{\ell }E_j$$ E =∑j = 1∑E j的外部到一个模域上的Walsh共形映射。特别地,我们对E是$$[-1,1]$$[- 1,1]的多项式原像的情况感兴趣,即$$E=P^{-1}([-1,1])$$ E = P - 1([- 1,1]),其中P是n次的代数多项式。我们特别感兴趣的是几何域的指数和中心。在本系列文章的第一部分,我们推导了一个非常简单的指数公式。本文在第一部分的一般结果的基础上,给出了E为$$\ell $$ - r区间的并时计算中心的迭代方法。一旦知道了这些中心,相应的沃尔什图就可以用数值方法计算出来。另外,如果E由满足一定对称关系的$$\ell =2$$ r = 2或$$\ell =3$$ r = 3分量组成,则用显式公式给出其中心和对应的Walsh映射。我们所有的定理都用解析或数值例子加以说明。
Walsh’s Conformal Map Onto Lemniscatic Domains for Polynomial Pre-images II
Abstract We consider Walsh’s conformal map from the exterior of a set $$E=\bigcup _{j=1}^{\ell }E_j$$ E=⋃j=1ℓEj consisting of $$\ell $$ ℓ compact disjoint components onto a lemniscatic domain. In particular, we are interested in the case when E is a polynomial preimage of $$[-1,1]$$ [-1,1] , i.e., when $$E=P^{-1}([-1,1])$$ E=P-1([-1,1]) , where P is an algebraic polynomial of degree n . Of special interest are the exponents and the centers of the lemniscatic domain. In the first part of this series of papers, a very simple formula for the exponents has been derived. In this paper, based on general results of the first part, we give an iterative method for computing the centers when E is the union of $$\ell $$ ℓ intervals. Once the centers are known, the corresponding Walsh map can be computed numerically. In addition, if E consists of $$\ell =2$$ ℓ=2 or $$\ell =3$$ ℓ=3 components satisfying certain symmetry relations then the centers and the corresponding Walsh map are given by explicit formulas. All our theorems are illustrated with analytical or numerical examples.
期刊介绍:
CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.