scholarly journals On the decay of singular inner functions

2020 ◽  
pp. 1-4
Author(s):  
Thomas Ransford
Keyword(s):  
Unit Disk ◽  
Inner Function ◽  
Inner Functions

Abstract It is known that if $S(z)$ is a non-constant singular inner function defined on the unit disk, then $\min _{|z|\le r}|S(z)|\to 0$ as $r\to 1^-$ . We show that the convergence can be arbitrarily slow.

The Bernoulli property of inner functions

1992 ◽  
Vol 12 (2) ◽  
pp. 209-215
Author(s):  
Marcos Craizer
Keyword(s):  
Fixed Point ◽  
Unit Circle ◽  
Harmonic Measure ◽  
Bernoulli Shift ◽  
Inner Function ◽  
Inner Functions

AbstractLet f: D → D be an inner function with a fixed point p ∈ D, and f*: S1 → S1 be its extension to the unit circle. We prove in this paper that the Rohlin invertible extension of the system (f*, λp) is equivalent to a generalized Bernoulli shift, where λp is the harmonic measure associated with p.

scholarly journals Continuations of Analytic Functions of Class S and Class U

1970 ◽  
Vol 40 ◽  
pp. 33-37
Author(s):  
Shinji Yamashita
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Analytic Functions ◽  
Inner Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Radial Limit

Let f be of class U in Seidel’s sense ([4, p. 32], = “inner function” in [3, p. 62]) in the open unit disk D. Then f has, by definition, the radial limit f(eiθ) of modulus one a.e. on the unit circle K. As a consequence of Smirnov’s theorem [5, p. 64] we know that the function

Universal Singular Inner Functions

2004 ◽  
Vol 47 (1) ◽  
pp. 17-21 ◽  
Author(s):  
Pamela Gorkin ◽  
Raymond Mortini
Keyword(s):  
Holomorphic Functions ◽  
Inner Function ◽  
Inner Functions

AbstractWe show that there exists a singular inner function S which is universal for noneuclidean translates; that is one for which the set is locally uniformly dense in the set of all zero-free holomorphic functions in bounded by one.

Tractable Relaxations of Composite Functions

2021 ◽  
Author(s):  
Taotao He ◽  
Mohit Tawarmalani
Keyword(s):  
Optimal Transport ◽  
Integral Formula ◽  
Transport Problem ◽  
Inner Function ◽  
Outer Function ◽  
Inner Functions ◽  
Separation Algorithm ◽  
Large Classes ◽  
Composite Functions ◽  
Factorable Programming

In this paper, we introduce new relaxations for the hypograph of composite functions assuming that the outer function is supermodular and concave extendable. Relying on a recently introduced relaxation framework, we devise a separation algorithm for the graph of the outer function over P, where P is a special polytope to capture the structure of each inner function using its finitely many bounded estimators. The separation algorithm takes [Formula: see text] time, where d is the number of inner functions and n is the number of estimators for each inner function. Consequently, we derive large classes of inequalities that tighten prevalent factorable programming relaxations. We also generalize a decomposition result and devise techniques to simultaneously separate hypographs of various supermodular, concave-extendable functions using facet-defining inequalities. Assuming that the outer function is convex in each argument, we characterize the limiting relaxation obtained with infinitely many estimators as the solution of an optimal transport problem. When the outer function is also supermodular, we obtain an explicit integral formula for this relaxation.

scholarly journals Multipliers of invariant subspaces in the bidisc

1994 ◽  
Vol 37 (2) ◽  
pp. 193-199 ◽  
Author(s):  
Takahiko Nakazi
Keyword(s):  
Invariant Subspace ◽  
Hankel Operator ◽  
Invariant Subspaces ◽  
Inner Function ◽  
Finite Codimension ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inner Functions ◽  
Necessary And Sufficient

For any nonzero invariant subspace M in H2(T2), set . Then Mx is also an invariant subspace of H2(T2) that contains M. If M is of finite codimension in H2(T2) then Mx = H2(T2) and if M = qH2(T2) for some inner function q then Mx = M. In this paper invariant subspaces with Mx = M are studied. If M = q1H2(T2) ∩ q2H2(T2) and q1, q2 are inner functions then Mx = M. However in general this invariant subspace may not be of the form: qH2(T2) for some inner function q. Put (M) = {ø ∈ L ∞: ø M ⊆ H2(T2)}; then (M) is described and (M) = (Mx) is shown. This is the set of all multipliers of M in the title. A necessary and sufficient condition for (M) = H∞(T2) is given. It is noted that the kernel of a Hankel operator is an invariant subspace M with Mx = M. The argument applies to the polydisc case.

Composition Operators

1968 ◽  
Vol 20 ◽  
pp. 442-449 ◽  
Author(s):  
Eric A. Nordgren
Keyword(s):  
Holomorphic Function ◽  
Unit Circle ◽  
Composition Operators ◽  
Complex Function ◽  
Inner Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Inner Functions ◽  
Almost Everywhere ◽  
Image Position

The object of this note is to report on some of the properties of a class of operators induced by inner functions. If m is normalized Lebesgue measure on the unit circle X in the complex plane and Cϕ is an inner function (a complex function on X of unit modulus almost everywhere whose Poisson integral is a non-constant holomorphic function in the open unit disk), then an operator Cϕ on L2(m) is defined by

scholarly journals Inner Functions in Lipschitz, Besov, and Sobolev Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Daniel Girela ◽  
Cristóbal González ◽  
Miroljub Jevtić
Keyword(s):  
Sobolev Spaces ◽  
Besov Spaces ◽  
Inner Function ◽  
Blaschke Products ◽  
Inner Functions ◽  
Finite Blaschke Products ◽  
The Subject

We study the membership of inner functions in Besov, Lipschitz, and Hardy-Sobolev spaces, finding conditions that enable an inner function to be in one of these spaces. Several results in this direction are given that complement or extend previous works on the subject from different authors. In particular, we prove that the only inner functions in either any of the Hardy-Sobolev spacesHαpwith1/p≤α<∞or any of the Besov spacesBαp,  qwith0<p,q≤∞andα≥1/p, except whenp=∞,α=0, and2<q≤∞or when0<p<∞,q=∞, andα=1/pare finite Blaschke products. Our assertion for the spacesB0∞,q,0<q≤2, follows from the fact that they are included in the spaceVMOA. We prove also that for2<q<∞,VMOAis not contained inB0∞,qand that this space contains infinite Blaschke products. Furthermore, we obtain distinct results for other values ofαrelating the membership of an inner functionIin the spaces under consideration with the distribution of the sequences of preimages{I-1(a)},|a|<1. In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.

Some Lemmas on Interpolating Blaschke Products and a Correction

1969 ◽  
Vol 21 ◽  
pp. 531-534 ◽  
Author(s):  
A. Kerr-Lawson
Keyword(s):  
Blaschke Product ◽  
Unit Disc ◽  
Inner Function ◽  
Open Unit ◽  
Inner Functions ◽  
Singular Measure ◽  
Radial Limits ◽  
Almost Everywhere ◽  
Image Position ◽  
Interpolating Blaschke Products

A Blaschke product on the unit disc,where |c|= 1 and kis a non-negative integer, is said to be interpolatingif the conditionCis satisfied for a constant δ independent of m.A Blaschke product always belongs to the set I of inner functions; it has norm 1 and radial limits of modulus 1 almost everywhere. The most general inner function can be uniquely factored into a product BS,where Bis a Blaschke product andfor some positive singular measure μ(θ) on the unit circle. The discussion will be carried out in terms of the hyperbolic geometry on the open unit disc D,its metricand its neighbourhoods N(x, ∈) = ﹛z′ ∈ D: Ψ(z, z′) < ∈ ﹜

scholarly journals A note on a result of A. J. Lohwater and George Piranian

1975 ◽  
Vol 56 ◽  
pp. 171-174
Author(s):  
G. L. Csordas
Keyword(s):  
Unit Disk ◽  
Boundary Behavior ◽  
Single Point ◽  
Open Unit ◽  
Outer Function ◽  
Open Unit Disk ◽  
Inner Functions ◽  
Bounded Analytic Functions ◽  
Closed Unit Disk ◽  
Cluster Set

Let I denote the set of all inner functions in H∞, where H∞ is the Banach algebra of all bounded analytic functions on the open unit disk D. Let I* denote the set of all functions f(z) in H∞ for which the cluster set C(f,α) at any point α on the circumference C = {α| |α| = 1} is either the closed unit disk |w| ≤ 1 or else a single point of modulus one. Clearly, I is a subset of I*. In [3] the author has proved that I is properly contained in I*. Recently, Lohwater and Piranian [7] have shown that there is an outer function in I*. The purpose of this note is to point out some applications of this result. In particular we shall show in Theorem 2.3 that there exist outer functions whose boundary behavior is similar to that of inner functions.

Eigenfunctions of Operator-Valued Analytic Functions

1970 ◽  
Vol 22 (3) ◽  
pp. 686-690
Author(s):  
Malcolm J. Sherman
Keyword(s):  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Complete Characterization ◽  
Inner Function ◽  
Linear Measure ◽  
Inner Functions ◽  
Almost Everywhere ◽  
Normal Operators ◽  
Functions Of Normal Operators

This paper is a sequel to [2], whose primary purposes are to clarify and generalize the concept introduced there of an eigenfunction of an inner function, and to answer questions raised there concerning the equivalence of several possible forms of the definition. A new definition, proposed here, leads to a complete characterization of the eigenfunctions of Potapov inner functions of normal operators, and the result is more satisfactory than [2, Theorem 3.4], although the latter is used strongly in the proof.Let be an inner function in the sense of Lax; i.e., is almost everywhere (a.e.) a unitary operator on a separable Hilbert space and belongs weakly to the Hardy class H2. An analytic function q (which will have to be a scalar inner function) was defined to be an eigenfunction of if the set of z in the disk {z: |z| ≦ 1} for which is invertible is a set of linear measure 0 on the circle {z: |z| = 1}.

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