131,546 research outputs found
Zero Tracts of Blaschke Products
No full textLet ﹛an﹜ be a sequence of complex numbers such thatandThen {an} is called a Blaschke sequence. For each Blaschke sequence {an} a Blaschke product is defined asThus a Blaschke product B(z, ﹛an﹜) is a function regular in the open unit disk D = {z: |z| < 1﹜ and having a zero at each point of the sequence ﹛an﹜.</jats:p
Ein Referenzrahmen familienpolitischer Reflexion: das Projekt "Wirkungen öffentlicher Sozialleistungen auf den Sozialisationsprozeß" im Spannungsfeld von Wissenschaft und Politik
No full textKaufmann F-X. Ein Referenzrahmen familienpolitischer Reflexion: das Projekt "Wirkungen öffentlicher Sozialleistungen auf den Sozialisationsprozeß" im Spannungsfeld von Wissenschaft und Politik. In: Blaschke D, ed. Sozialwissenschaftliche Forschung, Entwicklungen und Praxisorientierungen: Festgabe für Gerhard Wurzbacher zum 65. Geburtstag. Nürnberger Forschungsberichte ; Sonderband. Nürnberg: Verl. d. Nürnberger Forschungsvereinigung; 1977: 367-397
Blaschke Decompositions on Weighted Hardy Spaces and the Unwinding Series
No full textRecently, several authors have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either the unwinding series or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of the boundary value of an analytic function F : ∂D → C, and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be efficiently approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was initiated by Coifman and Steinerberger, under the assumption that the complex valued function F has an analytic extension to D_1+ε for some ε \u3e 0. This provided bounds on weighted Hardy norms involving a single zero, α ∈ D, of F and its Blaschke decomposition. That work also noted that in many specific examples, the unwinding series of F converges at an exponential rate to F, which when coupled with an efficient algorithm to compute a Blaschke decomposition, has led to the hope that this process will provide a new and efficient way to approximate signals.
In this work, we accomplish three things. Firstly, we continue the study of Blaschke decompositions on weighted Hardy Spaces for functions in the larger space H^2(D) under the assumption that the function has finitely many roots in D. This is meaningful, since there are many functions that meet this criterion but do not extend analytically to D_1+ε for any ε \u3e 0, for example F(z) = log(1−z). By studying the growth rate of the weights, we improve the bounds provided by Coifman and Steinerberger to obtain new estimates containing all roots of F in D. This provides us with new insights into Blaschke decompositions on classical function spaces including the Hardy-Sobolev spaces and weighted Bergman spaces, which
correspond to making specific choices for the aforementioned weights. Further, we state a sufficient condition on the weights for our improved bounds to hold for any function in the Hardy space, H^2(D), in particular functions with an infinite number of roots in D. Second, we compare the Fourier series and the unwinding series: we show that there are many examples of functions whose unwinding series converges much faster than the Fourier series, but there are also functions for which the Fourier and unwinding series are term wise equal. From the latter, we show the existence of functions that have unwinding series that do not converge exponentially. Lastly, we discuss an efficient algorithm for computing Blaschke decompositions, and apply this algorithm to verify our theoretical results and to gain a better understanding of the underlying mechanics of the unwinding series
Blaschke product generated covering surfaces
No full textsummary:It is known that, under very general conditions, Blaschke products generate branched covering surfaces of the Riemann sphere. We are presenting here a method of finding fundamental domains of such coverings and we are studying the corresponding groups of covering transformations
Width of the QCD transition in a Polyakov-loop DSE model
No full textHorvatić D, Blaschke D, Klabučar D, Kaczmarek O. Width of the QCD transition in a Polyakov-loop DSE model. In: Acta Physica Polonica B Proceedings Supplement. Acta Physica Polonica B Proceedings Supplement. Vol 5. Jagiellonian University; 2012
Shabat-Blaschke products
No full textThe aims of this thesis are to introduce a hyperbolic analogue of Grothendieck's dessins d'enfant, to give arithmetic properties of the coefficients of the Chebyshev-Blaschke products, and to prove some Landen-type identities for theta functions.
In 1979, Belyi proved that a connected compact Riemann surface is defined over the field of algebraic numbers if and only if there exists a Belyi map, i.e. a nonconstant holomorphic map with at most critical values in the Riemann sphere . Inspired by Belyi's theorem, Grothendieck introduced the theory of dessin d'enfant in the hope of a better understanding of the absolute Galois group . The dessins d'enfant and the related objects called the transitive monodromies are discrete combinatorial objects that determine uniquely the Belyi maps up to some equivalence. By introducing a Galois action on the dessins d'enfant, the structure of can then be revealed from the combinatorial properties of the dessins d'enfant. Special examples of dessins d'enfant are trees, and the Belyi maps that correspond to them are Shabat polynomials, i.e. polynomials with at most two finite critical values.
In chapter 2 of this thesis, we will establish a hyperbolic analogue of dessins d'enfant by replacing the Riemann sphere with three marked points by the open unit disk with two marked points. However, in this analogue the hyperbolic Belyi maps constructed from a given transitive monodromy are deformed, i.e. they depend on the hyperbolic distance between the two marked points under the Poincar\'{e} metric. Moreover, we will establish a hyperbolic analogue of Shabat's correspondence. In fact, the monodromies that correspond to a tree will also correspond to finite Blaschke products with at most two critical values in , and such finite Blaschke products will be referred to as Shabat-Blaschke products. We will also introduce and study the size of the hyperbolic dessin d'enfant of a Shabat-Blaschke product.
It is natural to ask if there is a hyperbolic analogue of Belyi's theorem when one replaces the Riemann sphere by the open unit disk . To formulate such a result, we need to know what should be used to replace . Since the Chebyshev polynomials are examples of Belyi maps and their coefficients are integers, we are motivated to prove a hyperbolic analogue of this statement in Chapter 3. Chebyshev-Blaschke products, which are hyperbolic analogues of Chebyshev polynomials, were studied by Ng, Tsang and Wang recently. The Chebyshev-Blaschke products are examples of Shabat-Blaschke products. We prove that the Chebyshev-Blaschke products are defined over
where is the degree of the Chebyshev-Blaschke product, is the scaling by , and are defined in terms of Jacobi theta functions, and is the -invariant. We also prove that the Chebyshev-Blaschke products are defined over , the ring of power series in over , where . Finally, we also obtain a family of Landen-type identities for theta functions as byproducts, which degenerates to a family of trigonometric identities.published_or_final_versionMathematicsMasterMaster of Philosoph
A uniqueness theorem for monic Blaschke products
No full textIf two monic Blaschke products of order
n
n
agree at
n
n
points of the open unit disc
D
{\mathbf {D}}
, then they must be identical.</p
Dynamics of finite Blaschke products
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2024, Director: Núria Fagella Rabionet[en] The aim of this project is to characterise the dynamics of finite Blaschke products, which are precisely the proper maps of the unit disk. It is proven that, inside the unit disk, all points converge to a unique point, the Wolff-Denjoy point. We build a classification of finite Blaschke products according to the position of the Wolff-Denjoy point and the dynamics around it.
Finally, we study the restriction of finite Blaschke products to the unit circle and calculate explicitly a conjugacy to . We end this work by showing a brief example of generalised Blaschke products, a nuanced variation of the previous family that presents rich dynamical phenomena, such as the emergence of Herman rings
On the generation of h∞ by blaschke products
No full text碩士布拉施克積的有限多個線性組合在單位圓盤上是均勻稠密在有界的可解析函數代數上。The finite linear combinations of Blaschke products are uniformly dense in the algebra of bounded analytic functions on the open unit disc.目錄
0. 前言 ………………………………………………………P.1
1. H^p 空間的定義和基本的性質……………………………P.1
1.1 布拉施克積………………………………………………P.3
1.2 某些位勢論………………………………………………P.9
2. 布拉施克積的有限的線性組合生成 H^infinity………P.11
2.1 Frostman定理……………………………………………P.11
2.2 商布拉施克積……………………………………………P.14
2.3 ㄧ些極值問題……………………………………………P.16
2.4 Marshall定理……………………………………………P.23
參考文獻………………………………………………………P.26
Contents
0. Introduction ……………………………………………P.27
1. Definitions and basic properties of H^p spaces…P.27
1.1 Blaschke products ……………………………………P.29
1.2 Some Potential theory ………………………………P.36
2. Blaschke products as generators of H^infinity…P.38
2.1 Frostman''s theorem …………………………………P.38
2.2 On the quotients of Blaschke products …………P.41
2.3 Some Extremal problem ………………………………P.44
2.4 Marshall''s theorem …………………………………P.51
References……………………………………………………P.54學號: 693150228, 學年度: 9
Decomposing Blaschke Products And Polynomials
No full textThe goal of this paper is to contribute to the understanding of complex polynomials and Blaschke products, two very important function classes in mathematics. For a polynomial, of degree we study when it is possible to write as a composition , where and are polynomials, each of degree less than A polynomial is defined to be \emph{decomposable }if such an and exist, and a polynomial is said to be \emph{indecomposable} if no such and exist. We apply the results of Rickards in \cite{key-2}. We show that C_{n}=\{(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}\,|\,(z-z_{1})(z-z_{2})...(z-z_{n})\,\mbox{is decomposable}\}, has measure when considered a subset of Using this we prove the stronger result that D_{n}=\{(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}\,|\,\mbox{There exists\,}a\in\mathbb{C}\,\,\mbox{with}\,\,(z-z_{1})(z-z_{2})...(z-z_{n})(z-a)\,\mbox{decomposable}\}, also has measure zero when considered a subset of We show that for any polynomial , there exists an such that is indecomposable, and we also examine the case of in detail. The main work of this paper studies finite Blaschke products, analytic functions on that map to In analogy with polynomials, we discuss when a degree Blaschke product, can be written as a composition , where and are finite Blaschke products, each of degree less than Decomposable and indecomposable are defined analogously. Our main results are divided into two sections. First, we equate a condition on the zeros of the Blaschke product with the existence of a decomposition where the right-hand factor, has degree We also equate decomposability of a Blaschke product, with the existence of a Poncelet curve, whose foci are a subset of the zeros of such that the Poncelet curve satisfies certain tangency conditions. This result is hard to apply in general, but has a very nice geometric interpretation when we desire a composition where the right-hand factor is degree 2 or 3. Our second section of finite Blaschke product results builds off of the work of Cowen in \cite{key-3}. For a finite Blaschke product Cowen defines the so-called monodromy group, of the finite Blaschke product. He then equates the decomposability of a finite Blaschke product, with the existence of a nontrivial partition, of the branches of such that respects . We present an in-depth analysis of how to calculate , extending Cowen\u27s description. These methods allow us to equate the existence of a decomposition where the left-hand factor has degree 2, with a simple condition on the critical points of the Blaschke product. In addition we are able to put a condition of the structure of for any decomposable Blaschke product satisfying certain normalization conditions. The final section of this paper discusses how one can put the results of the paper into practice to determine, if a particular Blaschke product is decomposable. We compare three major algorithms. The first is a brute force technique where one searches through the zero set of for subsets which could be the zero set of , exhaustively searching for a successful decomposition The second algorithm involves simply examining the cardinality of the image, under of the set of critical points of For a degree Blaschke product, if this cardinality is greater than , the Blaschke product is indecomposable. The final algorithm attempts to apply the geometric interpretation of decomposability given by our theorem concerning the existence of a particular Poncelet curve. The final two algorithms can be implemented easily with the use of an HTM
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