Discrete orthogonality of the analytic wavelets in the Hardy space of the upper half plane

2017 ◽  
Vol 15 (03) ◽  
pp. 1750024
Author(s):  
Tímea Eisner ◽  
Margit Pap
Keyword(s):  
New York ◽  
Hardy Space ◽  
Half Plane ◽  
Reproducing Kernel ◽  
Kernel Functions ◽  
Blaschke Products ◽  
Orthogonal Wavelet ◽  
Discrete Orthogonal ◽  
Upper Half Plane ◽  
Hyperbolic Wavelets

We will prove that the analytic orthogonal wavelet-system, which was introduced by Feichtinger and Pap in [Hyperbolic wavelets and multiresolution in the Hardy space of the upper half plane, in Blaschke Products and Their Applications: Fields Institute Communications, Vol. 65 (Springer, New York, 2013), pp. 193–208] is discrete orthogonal too. We will discuss the discrete orthogonality and the properties of the reproducing kernel functions of the introduced wavelet-spaces.

scholarly journals GENERALIZED EIGENVECTORS FOR RESONANCES IN THE FRIEDRICHS MODEL AND THEIR ASSOCIATED GAMOV VECTORS

2006 ◽  
Vol 18 (01) ◽  
pp. 61-78 ◽  
Author(s):  
HELLMUT BAUMGÄRTEL
Keyword(s):  
Hardy Space ◽  
Half Plane ◽  
Lower Half Plane ◽  
Linear Forms ◽  
Generalized Eigenvalues ◽  
Dirac Type ◽  
Generalized Eigenvectors ◽  
Upper Half Plane ◽  
Friedrichs Model ◽  
Generalized Eigenvector

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on ℝ with multiplicity space [Formula: see text], [Formula: see text], is constructed such that exactly the resonances (poles of the inverse of the Livšic-matrix) are (generalized) eigenvalues of H. The corresponding eigen(anti)linear forms are calculated explicitly. Using the wave matrices for the wave (Möller) operators the corresponding eigen(anti)linear forms on the Schwartz space [Formula: see text] for the unperturbed Hamiltonian H0 are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector λ → k/(ζ0 - λ)-1, ζ0 resonance, [Formula: see text], which is uniquely determined by restriction of [Formula: see text] to [Formula: see text], where [Formula: see text] denotes the Hardy space of the upper half-plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup for t ≥ 0 of the Toeplitz type on [Formula: see text]. That is: Exactly those pre-Gamov vectors λ → k/(ζ - λ)-1, ζ from the lower half-plane, [Formula: see text], have an extension to a generalized eigenvector of H if ζ is a resonance and if k is from that subspace of [Formula: see text] which is uniquely determined by its corresponding Dirac type antilinear form.

Admissible Majorants for Model Subspaces of H2, Part I: Slow Winding of the Generating Inner Function

2003 ◽  
Vol 55 (6) ◽  
pp. 1231-1263 ◽  
Author(s):  
Victor Havin ◽  
Javad Mashreghi
Keyword(s):  
Hardy Space ◽  
Half Plane ◽  
Blaschke Product ◽  
Inner Function ◽  
Rate Of Growth ◽  
Model Subspaces ◽  
Almost Everywhere ◽  
Model Subspace ◽  
Upper Half Plane

AbstractA model subspace Kϴ of the Hardy space H2 = H2(ℂ+) for the upper half plane ℂ+ is H2(ℂ+) ϴ ϴH2(ℂ+) where ϴ is an inner function in ℂ+. A function ω: ⟼ [0,∞) is called an admissible majorant for Kϴ if there exists an f ∈ Kϴ, f ≢ 0, |f(x)| ≤ ω(x) almost everywhere on ℝ. For some (mainly meromorphic) ϴ's some parts of Adm ϴ (the set of all admissible majorants for Kϴ) are explicitly described. These descriptions depend on the rate of growth of argϴ along ℝ. This paper is about slowly growing arguments (slower than x). Our results exhibit the dependence of Adm B on the geometry of the zeros of the Blaschke product B. A complete description of Adm B is obtained for B's with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.

scholarly journals A Density problem for Hardy spaces of almost periodic functions

1984 ◽  
Vol 29 (3) ◽  
pp. 315-327 ◽  
Author(s):  
Robyn Owens
Keyword(s):  
Hardy Space ◽  
Half Plane ◽  
Hardy Spaces ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Upper Half Plane ◽  
Density Problem

We construct a counterexample, for p = 1, to the conjecture posed by Milaszevitch in 1970: is the space of functions which are analytic in the upper half plane and uniformly almost periodic in its closure dense in the Hardy space Hp (0 < p ∞) of analytic almost periodic functions?

scholarly journals Composition Operators from the Hardy Space to the Zygmund-Type Space on the Upper Half-Plane

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Stevo Stević
Keyword(s):  
Hardy Space ◽  
Half Plane ◽  
Complex Plane ◽  
Composition Operator ◽  
Weighted Space ◽  
Composition Operators ◽  
Sufficient Conditions ◽  
Type Space ◽  
Upper Half Plane ◽  
Necessary And Sufficient

Here we introduce thenth weighted space on the upper half-planeΠ+={z∈ℂ:Im z>0}in the complex planeℂ. For the casen=2, we call it the Zygmund-type space, and denote it by&#x1D4B5;(Π+). The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operatorCφf(z)=f(φ(z))from the Hardy spaceHp(Π+)on the upper half-plane, to the Zygmund-type space, whereφis an analytic self-map of the upper half-plane.

The convergence properties of orthogonal rational functions on the extended real line and analytic on the upper half plane

2020 ◽  
Vol 18 (05) ◽  
pp. 2050032
Author(s):  
Xu Xu ◽  
Xiaoqiang Xu ◽  
Laiyi Zhu
Keyword(s):  
Half Plane ◽  
Reproducing Kernel ◽  
Rational Functions ◽  
Inner Product ◽  
Convergence Properties ◽  
Orthogonal Rational Functions ◽  
Convergence Results ◽  
Carathéodory Function ◽  
Weak Star ◽  
Upper Half Plane

We study the orthogonal rational functions with a given sequence of poles on the half plane. We give the weak-star convergence results for the rational measures which relate to the Poisson kernel and reproducing kernel. Those rational measures represent the same inner product as the normalized Nevanlinna measure in [Formula: see text]. Moreover, we consider the convergence properties of the interpolants which interpolate the Carathéodory function [Formula: see text].

The Upper Half Plane Model for Hyperbolic Geometry

1980 ◽  
Vol 87 (1) ◽  
pp. 48 ◽  
Author(s):  
Richard S. Millman
Keyword(s):  
Half Plane ◽  
Hyperbolic Geometry ◽  
Plane Model ◽  
Upper Half Plane
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