Pub Date : 2023-08-07DOI: 10.1007/s40315-023-00492-6
Klaus Schiefermayr, Olivier Sète
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.
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。
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Pub Date : 2023-07-22DOI: 10.1007/s40315-023-00491-7
A. Solynin
{"title":"Canonical Embeddings of Pairs of Arcs and Extremal Problems on Ring Domains","authors":"A. Solynin","doi":"10.1007/s40315-023-00491-7","DOIUrl":"https://doi.org/10.1007/s40315-023-00491-7","url":null,"abstract":"","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"85 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88072065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-10DOI: 10.1007/s40315-023-00488-2
B. Bilalov, S. Sadigova, V. G. Alili
{"title":"The Method of Boundary Value Problems in the Study of the Basis Properties of Perturbed System of Exponents in Banach Function Spaces","authors":"B. Bilalov, S. Sadigova, V. G. Alili","doi":"10.1007/s40315-023-00488-2","DOIUrl":"https://doi.org/10.1007/s40315-023-00488-2","url":null,"abstract":"","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"79 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80279756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.1007/s40315-023-00484-6
A. Menovschikov, A. Ukhlov
{"title":"Composition Operators on Sobolev Spaces and Q-Homeomorphisms","authors":"A. Menovschikov, A. Ukhlov","doi":"10.1007/s40315-023-00484-6","DOIUrl":"https://doi.org/10.1007/s40315-023-00484-6","url":null,"abstract":"","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80725346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-30DOI: 10.1007/s40315-023-00485-5
Alessandro Perotti
Abstract We prove a local Cauchy-type integral formula for slice-regular functions. The formula is obtained as a corollary of a general integral representation formula where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two axially monogenic functions.
{"title":"A Local Cauchy Integral Formula for Slice-Regular Functions","authors":"Alessandro Perotti","doi":"10.1007/s40315-023-00485-5","DOIUrl":"https://doi.org/10.1007/s40315-023-00485-5","url":null,"abstract":"Abstract We prove a local Cauchy-type integral formula for slice-regular functions. The formula is obtained as a corollary of a general integral representation formula where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two axially monogenic functions.","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135478840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-03DOI: 10.1007/s40315-023-00483-7
Amina Bibi, Xiao-Min Li, H. Yi
{"title":"On the Difference Independence of the Euler Gamma Function and the Riemann Zeta Function","authors":"Amina Bibi, Xiao-Min Li, H. Yi","doi":"10.1007/s40315-023-00483-7","DOIUrl":"https://doi.org/10.1007/s40315-023-00483-7","url":null,"abstract":"","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"45 1","pages":"771 - 788"},"PeriodicalIF":2.1,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85933627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-24DOI: 10.1007/s40315-023-00482-8
M. B. Ahamed, Sabir Ahammed
{"title":"Bohr Type Inequalities for the Class of Self-Analytic Maps on the Unit Disk","authors":"M. B. Ahamed, Sabir Ahammed","doi":"10.1007/s40315-023-00482-8","DOIUrl":"https://doi.org/10.1007/s40315-023-00482-8","url":null,"abstract":"","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"6 1","pages":"789 - 806"},"PeriodicalIF":2.1,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78690166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-10DOI: 10.1007/s40315-023-00479-3
Elias Wegert
Honoring Lawrence Zalcman’s work, the cover of this volume shows the phase plot of a function that appears in the construction of a special non-normal family of meromorphic functions. Neglecting all technicalities, we summarize the basic ideas of this construction and illustrate them by phase plots.
{"title":"About the Cover: Non-normal Families and Attracting Fixed Points","authors":"Elias Wegert","doi":"10.1007/s40315-023-00479-3","DOIUrl":"https://doi.org/10.1007/s40315-023-00479-3","url":null,"abstract":"Honoring Lawrence Zalcman’s work, the cover of this volume shows the phase plot of a function that appears in the construction of a special non-normal family of meromorphic functions. Neglecting all technicalities, we summarize the basic ideas of this construction and illustrate them by phase plots.","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136097024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-09DOI: 10.1007/s40315-023-00481-9
F. Wang, Kai Liu
{"title":"Value Cross-Sharing of Meromorphic Functions","authors":"F. Wang, Kai Liu","doi":"10.1007/s40315-023-00481-9","DOIUrl":"https://doi.org/10.1007/s40315-023-00481-9","url":null,"abstract":"","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"6 1","pages":"1-22"},"PeriodicalIF":2.1,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73150394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-08DOI: 10.1007/s40315-023-00480-w
Doron Lubinsky, Pekka Koskela
{"title":"Editorial Note","authors":"Doron Lubinsky, Pekka Koskela","doi":"10.1007/s40315-023-00480-w","DOIUrl":"https://doi.org/10.1007/s40315-023-00480-w","url":null,"abstract":"","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"87 1","pages":"1"},"PeriodicalIF":2.1,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72972202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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