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].

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 Rational functions, cotangent sums and Eichler integrals

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Johann Franke
Keyword(s):  
Eisenstein Series ◽  
Half Plane ◽  
Modular Forms ◽  
Holomorphic Functions ◽  
Rational Functions ◽  
Close Connection ◽  
General Transformation ◽  
Fourier Expansions ◽  
Upper Half Plane ◽  
Eichler Integrals

AbstractWith the help of so called pre-weak functions, we formulate a very general transformation law for some holomorphic functions on the upper half plane and motivate the term of a generalized Eisenstein series with real-exponent Fourier expansions. Using the transformation law in the case of negative integers k, we verify a close connection between finite cotangent sums of a specific type and generalized L-functions at integer arguments. Finally, we expand this idea to Eichler integrals and period polynomials for some types of modular forms.

scholarly journals Rational period functions of the modular group II

1981 ◽  
Vol 22 (2) ◽  
pp. 185-197 ◽  
Author(s):  
Marvin I. Knopp
Keyword(s):  
Half Plane ◽  
Functional Equations ◽  
Modular Group ◽  
Fourier Expansion ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Linear Fractional Transformations ◽  
Upper Half Plane ◽  
Image Position ◽  
Rational Period Functions

In the earlier article [7], I began the study of rational period functions for the modular group Γ(l) = SL(2, Z) (regarded as a group of linear fractional transformations) acting on the Riemann sphere. These are rational functions q(z) which occur in functional equations of the formwhere k∈Z and F is a function meromorphic in the upper half-plane ℋ, restricted in growth at the parabolic cusp ∞. The growth restriction may be phrased in terms of the Fourier expansion of F(z) at ∞:with some μ∈Z. If (1.1) and (1.2) hold, then we call F a modular integral of weight 2k and q(z) the period of F.

scholarly journals Orthogonal rational functions and modified approximants

1996 ◽  
Vol 11 (1) ◽  
pp. 57-69 ◽  
Author(s):  
A. Bultheel ◽  
P. González-Vera ◽  
E. Hendriksen ◽  
O. Njåstad
Keyword(s):  
Rational Functions ◽  
Orthogonal Rational Functions
Sign in / Sign up

Export Citation Format

Share Document