scholarly journals Distortion and Critical Values of the Finite Blaschke Product

2020 ◽  
Author(s):  
V. N. Dubinin
Keyword(s):  
Blaschke Product ◽  
Critical Values ◽  
Finite Blaschke Product

Approximation by Unimodular Functions

1971 ◽  
Vol 23 (2) ◽  
pp. 257-269 ◽  
Author(s):  
Stephen Fisher
Keyword(s):  
Blaschke Product ◽  
Uniform Approximation ◽  
Unit Disc ◽  
Open Unit ◽  
Blaschke Products ◽  
Open Unit Disc ◽  
Pointwise Approximation ◽  
Finite Blaschke Product ◽  
Image Position ◽  
Restricted Zeros

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

scholarly journals Fatou’s Associates

2020 ◽  
Vol 6 (3-4) ◽  
pp. 459-493
Author(s):  
Vasiliki Evdoridou ◽  
Lasse Rempe ◽  
David J. Sixsmith
Keyword(s):  
Entire Function ◽  
Blaschke Product ◽  
Entire Functions ◽  
Julia Set ◽  
Singular Values ◽  
Strong Relationship ◽  
Inner Function ◽  
Transcendental Entire Function ◽  
Fatou Component ◽  
Finite Blaschke Product

AbstractSuppose that f is a transcendental entire function, $$V \subsetneq {\mathbb {C}}$$ V ⊊ C is a simply connected domain, and U is a connected component of $$f^{-1}(V)$$ f - 1 ( V ) . Using Riemann maps, we associate the map $$f :U \rightarrow V$$ f : U → V to an inner function $$g :{\mathbb {D}}\rightarrow {\mathbb {D}}$$ g : D → D . It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain.

scholarly journals Representation with majorant of the Schwarz lemma at the boundary

2017 ◽  
Vol 101 (115) ◽  
pp. 191-196
Author(s):  
Bülent Örnek ◽  
Tuğba Akyel
Keyword(s):  
Holomorphic Function ◽  
Blaschke Product ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Schwarz Lemma ◽  
Local Behavior ◽  
Finite Blaschke Product ◽  
Boundary Points ◽  
Finite Set ◽  
The Schwarz Lemma

Let f be a holomorphic function in the unit disc and |f(z)?1| < 1 for |z| < 1. We generalize the uniqueness portion of Schwarz?s lemma and provide sufficient conditions on the local behavior of f near a finite set of boundary points that needed for f to be a finite Blaschke product.

scholarly journals COMPOSITION OPERATORS ON FINITE RANK MODEL SUBSPACES

2012 ◽  
Vol 55 (1) ◽  
pp. 69-83 ◽  
Author(s):  
JAVAD MASHREGHI ◽  
MAHMOOD SHABANKHAH
Keyword(s):  
Blaschke Product ◽  
Finite Rank ◽  
Composition Operators ◽  
Group Structure ◽  
Finite Blaschke Product ◽  
Model Subspaces

AbstractWe give a complete description of bounded composition operators on model subspaces KB, where B is a finite Blaschke product. In particular, if B has at least one finite pole, we show that the collection of all bounded composition operators on KB has a group structure. Moreover, if B has at least two distinct finite poles, this group is finite and cyclic.

On Quasisimilarity for Analytic Toeplitz Operators

1988 ◽  
Vol 31 (1) ◽  
pp. 111-116 ◽  
Author(s):  
Katsutoshi Takahashi
Keyword(s):  
Toeplitz Operator ◽  
Blaschke Product ◽  
Toeplitz Operators ◽  
Finite Blaschke Product ◽  
Analytic Toeplitz Operator

AbstractLet f be a function in H∞. We show that if f is inner or if the commutant of the analytic Toeplitz operator Tf is equal to that of Tb for some finite Blaschke product b, then any analytic Toeplitz operator quasisimilar to Tf is unitarily equivalent to Tf.

Multiplication Invariant Subspaces of Hardy Spaces

1997 ◽  
Vol 49 (1) ◽  
pp. 100-118 ◽  
Author(s):  
T. L. Lance ◽  
M. I. Stessin
Keyword(s):  
Hardy Spaces ◽  
Blaschke Product ◽  
Invariant Subspaces ◽  
Decomposition Theorem ◽  
Canonical Decomposition ◽  
Additional Property ◽  
Inner Functions ◽  
Finite Blaschke Product ◽  
Direct Sum

AbstractThis paper studies closed subspaces L of the Hardy spaces Hp which are g-invariant (i.e., g. L ⊆ L) where g is inner, g ≠ 1. If p = 2, theWold decomposition theorem implies that there is a countable “g-basis” f1, f2, . . . of L in the sense that L is a direct sum of spaces fj . H2[g] where H2[g] = {f o g | f ∈ H2}. The basis elements fj satisfy the additional property that ∫T |fj|2gk = 0, k = 1, 2, . . . . We call such functions g-2-inner. It also follows that any f ∈ H2 can be factored f = hf ,2 . (F2 o g) where hf,2 is g-2-inner and F is outer, generalizing the classical Riesz factorization. Using Lp estimates for the canonical decomposition of H2,we find a factorization f = hf ,p.(Fpog) for f ∈ Hp. If p ≤ 1 and g is a finite Blaschke product we obtain, for any g-invariant L ⊆ Hp, a finite g-basis of g-p-inner functions.

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