블라쉬케 곱 (Blaschke product)

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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
• [DPR2002] 참조

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

말뭉치

1. Thus, Blaschke's theorem describes the sequences of zeros of all possible Blaschke products.
2. P.M. Tamrazov, "Conformal-metric theory of doubly connected domains and the generalized Blaschke product" Soviet Math.
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.
4. The final algorithm looks at inverse images under the Blaschke product.
5. Blaschke products were introduced by Wilhelm Blaschke (1915).
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.
7. Deep connections to hyperbolic geometry are explored, as are the mapping properties, zeros, residues, and critical points of finite Blaschke products.
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.
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.
10. All examples of Blaschke products constructed so far to prove this result have their zeros located on a ray.
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.

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Spacy 패턴 목록

• [{'LOWER': 'blaschke'}, {'LEMMA': 'product'}]