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article

Analytic Toeplitz Operators, Distinguished Varieties and Boundary Behavior of Symbols

Hui Dan, Kunyu Guo, Hansong Huang

Integral Equations and Operator Theory > 2019 > 91 > 5 > 1-15

Based on our paper (Dan et al. in J Funct Anal 273:559–597, 2017), we continue the study on totally Abelian analytic Toeplitz operators on the Hardy space. However, rather than to focus on the geometry of the symbol curves associated with these operators as done before, this paper is to consider the Riemann surfaces in $${\mathbb {C}}^{2}$$ C2 defined by these symbols. In this way, we reveal the...

article

Finite Blaschke products and decomposition

Sümeyra Uçar, Nihal Yılmaz Ӧzgür

Journal of Mathematics and Applications > 2019 > Vol. 42 > 187--202

Let B(z) be a finite Blaschke product of degree n. We consider the problem when a finite Blaschke product can be written as a composition of two nontrivial Blaschke products of lower degree related to the condition B ◦ M = B where M is a Mobius transformation from the unit disk onto itself.

article

Characterizing Meromorphic Pseudo-lemniscates

Trevor Richards

Computational Methods and Function Theory > 2018 > 18 > 4 > 609-616

Let f be a meromorphic function with simply connected domain $$G\subset \mathbb {C}$$ G⊂C , and let $$\Gamma \subset \mathbb {C}$$ Γ⊂C be a smooth Jordan curve. We call a component of $$f^{-1}(\Gamma )$$ f-1(Γ) in G a $$\Gamma $$ Γ -pseudo-lemniscate of f. In this note, we give criteria for a smooth Jordan curve $$\mathcal {S}$$ S in G (with bounded face D) to be a $$\Gamma $$ Γ -pseudo-lemniscate...

article

Smale’s mean value conjecture for finite Blaschke products

Tuen-Wai Ng, Yongquan Zhang

The Journal of Analysis > 2016 > 24 > 2 > 331-345

Motivated by a dictionary between polynomials and finite Blaschke products, we study both Smale’s mean value conjecture and its dual conjecture for finite Blaschke products in this paper. Our result on the dual conjecture for finite Blaschke products allows us to improve a bound obtained by Dubinin and Sugawa for the dual mean value conjecture for polynomials.

article

Reducing Subspaces of Multiplication Operators on the Dirichlet Space

Shuaibing Luo

Integral Equations and Operator Theory > 2016 > 85 > 4 > 539-554

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $${\phi}$$ ϕ on the Dirichlet space D. We prove that any two distinct nontrivial minimal reducing subspaces of $${M_\phi}$$ M ϕ are orthogonal. When the order n of $${\phi}$$ ϕ is 2 or 3, we show that $${M_\phi}$$ M ϕ is reducible on D if and only if ...

article

Chebyshev–Blaschke products: Solutions to certain approximation problems and differential equations

Tuen Wai Ng, Chiu Yin Tsang

Journal of Computational and Applied Mathematics > 2015 > 277 > C > 106-114

In this paper, we study a special kind of finite Blaschke products called Chebyshev–Blaschke products fn,τ which can be defined by the Jacobi cosine function cd(u,τ), where τ∈R+i. We will show that Chebyshev–Blaschke products solve a number of approximation problems, which are related to Zolotarev’s 3rd and 4th problems. More importantly, such a Chebyshev–Blaschke product fn,τ will be shown to be...

article

Approximation by the nonlinear Fourier basis

Chao Huang, LiHua Yang

Science China Mathematics > 2011 > 54 > 6 > 1207-1214

As a typical family of mono-component signals, the nonlinear Fourier basis $\left\{ {e^{ik\theta _a \left( t \right)} } \right\}_{k \in \mathbb{Z}}$ , defined by the nontangential boundary value of the Möbius transformation, has attracted much attention in the field of nonlinear and nonstationary signal processing in recent years. In this paper, we establish the Jackson’s and Bernstein’s theorems...

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Search results

Analytic Toeplitz Operators, Distinguished Varieties and Boundary Behavior of Symbols

Finite Blaschke products and decomposition

Characterizing Meromorphic Pseudo-lemniscates

Smale’s mean value conjecture for finite Blaschke products

Reducing Subspaces of Multiplication Operators on the Dirichlet Space

Chebyshev–Blaschke products: Solutions to certain approximation problems and differential equations

Approximation by the nonlinear Fourier basis

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