블라쉬케 곱 (Blaschke product)

수학노트
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개요

  • 다음과 같은 꼴의 뫼비우스 변환들은 단위원을 단위원으로 보내는 전단사 해석함수이다\[B(a,z)=\frac{|a|}{a}\frac{z-a}{1-\bar{a}z}\]
  • Blaschke product는 이러한 꼴의 함수들의 유한 또는 무한곱으로 쓰여짐.\[B(z)=\prod_n B(a_n,z)\]
  • 단위원에서 정의된 함수로 주어진 점에서 zero 를 갖는 해석함수를 만들기 위해 사용됨




타원과 3차 블라쉬케 곱

  • 다음과 같은 3차의 블라쉬케 곱을 생각하자\[B(z)=z\frac{z-a}{1-\bar{a}z}\frac{z-b}{1-\bar{b}z}\]
  • 단위원 위의 점 \(\lambda\) 에 대하여, \(B(z)=\lambda\) 의 세 해를 \(z_ 1,z_ 2,z_ 3\) 로 두면, 세 직선 \(\overline{z_ 1z_ 2},\overline{z_ 2 z_ 3},\overline{z_ 1 z_ 3}\) 은 다음 타원에 접한다\[|w-a|+|w-b|=|1-\bar{a}b|\]
  • \(a=0.5,b=-0.4+0.4 i\) 로 두고, 다양한 \(\lambda\) 에 대하여 위의 결과를 적용하여 얻은 그림

블라쉬케 곱(Blaschke product)1.gif

  • [DPR2002] 참조




역사



메모



관련된 항목들

매스매티카 파일 및 계산 리소스


수학용어번역






사전 형태의 자료



리뷰논문, 에세이, 강의노트

  • Garcia, Stephan Ramon, Javad Mashreghi, and William T. Ross. “Finite Blaschke Products: A Survey.” arXiv:1512.05444 [math], December 16, 2015. http://arxiv.org/abs/1512.05444.

관련논문

  • Fletcher, Alastair. “Blaschke Products and Domains of Ellipticity.” arXiv:1408.2418 [math], August 11, 2014. http://arxiv.org/abs/1408.2418.
  • [DPR2002]Daepp, Ulrich, Pamela Gorkin, and Raymond Mortini. 2002. Ellipses and Finite Blaschke Products. The American Mathematical Monthly 109 (9) (November 1): 785-795. doi:10.2307/3072367.


노트

위키데이터

말뭉치

  1. Thus, Blaschke's theorem describes the sequences of zeros of all possible Blaschke products.[1]
  2. P.M. Tamrazov, "Conformal-metric theory of doubly connected domains and the generalized Blaschke product" Soviet Math.[1]
  3. We present four algorithms to determine whether or not a Blaschke product is a composition of two non-trivial Blaschke products and, if it is, the algorithms suggest what the composition must be.[2]
  4. The final algorithm looks at inverse images under the Blaschke product.[2]
  5. Blaschke products were introduced by Wilhelm Blaschke (1915).[3]
  6. This monograph offers an introduction to finite Blaschke products and their connections to complex analysis, linear algebra, operator theory, matrix analysis, and other fields.[4]
  7. Deep connections to hyperbolic geometry are explored, as are the mapping properties, zeros, residues, and critical points of finite Blaschke products.[4]
  8. This book gathers the principal results about Blaschke products heretofore scattered in research papers over the past 70 years and provides an extensive bibliography of over 300 items.[5]
  9. It is hoped that research workers in and students of function theory will find the book a useful guide and reference to the subject of Blaschke products.[5]
  10. All examples of Blaschke products constructed so far to prove this result have their zeros located on a ray.[6]
  11. A Blaschke product always belongs to the set I of inner functions; it has norm 1 and radial limits of modulus 1 almost everywhere.[7]

소스

  1. ↑ 1.0 1.1 Encyclopedia of Mathematics
  2. ↑ 2.0 2.1 Decomposing finite Blaschke products
  3. ↑ Blaschke product
  4. ↑ 4.0 4.1 Finite Blaschke Products and Their Connections
  5. ↑ 5.0 5.1 Blaschke Products
  6. ↑ Girela , Peláez : On the derivative of infinite Blaschke products
  7. ↑ Some Lemmas on Interpolating Blaschke Products and a Correction

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