On some spaces of holomorphic functions of exponential growth on a half-plane

Marco M. Peloso; Maura Salvatori

Concrete Operators (2016)

  • Volume: 3, Issue: 1, page 52-67
  • ISSN: 2299-3282

Abstract

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In this paper we study spaces of holomorphic functions on the right half-plane R, that we denote by Mpω, whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on [0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression on [0, +∞). We obtain a Paley–Wiener theorem for M2ω, and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that Mpω contains functions of order 1. Moreover, we prove that the orthogonal projection from Lp(R,dω) into Mpω is unbounded for p ≠ 2. Furthermore, we compare the spaces Mpω with the classical Hardy and Bergman spaces, and some other Hardy– Bergman-type spaces introduced more recently.

How to cite

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Marco M. Peloso, and Maura Salvatori. "On some spaces of holomorphic functions of exponential growth on a half-plane." Concrete Operators 3.1 (2016): 52-67. <http://eudml.org/doc/277079>.

@article{MarcoM2016,
abstract = {In this paper we study spaces of holomorphic functions on the right half-plane R, that we denote by Mpω, whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on [0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression on [0, +∞). We obtain a Paley–Wiener theorem for M2ω, and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that Mpω contains functions of order 1. Moreover, we prove that the orthogonal projection from Lp(R,dω) into Mpω is unbounded for p ≠ 2. Furthermore, we compare the spaces Mpω with the classical Hardy and Bergman spaces, and some other Hardy– Bergman-type spaces introduced more recently.},
author = {Marco M. Peloso, Maura Salvatori},
journal = {Concrete Operators},
keywords = {Holomorphic function on half-plane; Reproducing kernel Hilbert space; Hardy spaces, Bergman spaces; holomorphic function on the half-plane; Hilbert spaces with reproducing kernel; Hardy spaces; Bergman spaces},
language = {eng},
number = {1},
pages = {52-67},
title = {On some spaces of holomorphic functions of exponential growth on a half-plane},
url = {http://eudml.org/doc/277079},
volume = {3},
year = {2016},
}

TY - JOUR
AU - Marco M. Peloso
AU - Maura Salvatori
TI - On some spaces of holomorphic functions of exponential growth on a half-plane
JO - Concrete Operators
PY - 2016
VL - 3
IS - 1
SP - 52
EP - 67
AB - In this paper we study spaces of holomorphic functions on the right half-plane R, that we denote by Mpω, whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on [0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression on [0, +∞). We obtain a Paley–Wiener theorem for M2ω, and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that Mpω contains functions of order 1. Moreover, we prove that the orthogonal projection from Lp(R,dω) into Mpω is unbounded for p ≠ 2. Furthermore, we compare the spaces Mpω with the classical Hardy and Bergman spaces, and some other Hardy– Bergman-type spaces introduced more recently.
LA - eng
KW - Holomorphic function on half-plane; Reproducing kernel Hilbert space; Hardy spaces, Bergman spaces; holomorphic function on the half-plane; Hilbert spaces with reproducing kernel; Hardy spaces; Bergman spaces
UR - http://eudml.org/doc/277079
ER -

References

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