Explorations in harmonic analysis. With applications to complex function theory and the Heisenberg group
type: Item
instance of: scholarly article
title: Explorations in harmonic analysis. With applications to complex function theory and the Heisenberg group
zbMATH Open document ID: 1171.43001
DOI: 10.1007/978-0-8176-4669-1
author: Steven G. Krantz
published in: Applied and Numerical Harmonic Analysis
publication date: 2009-07-02T00:00:00Z
review text: Harmonic analysis is a venerable part of modern mathematics. Its roots began, perhaps, with the late eighteenth-century discussions of the wave equation. Using the method of separation of variables, it was realized that the equation could be solved with a data function of the form \(\varphi(x) = \sin jx\) for \(j\in\mathbb Z\). It was natural to ask, using the philosophy of superposition, whether the equation could then be solved with data on the interval \([0,\pi]\) consisting of a finite linear combination of the \(\sin jx\). With an affirmative answer to that question, one is led to ask about infinite linear combinations. This was an interesting venue in which physical reasoning interacted with mathematical reasoning. Physical intuition certainly suggests that any continuous function \(\varphi\) can be a data function for the wave equation. So one is led to ask whether any continuous \(\varphi\) can be expressed as an (infinite) superposition of sine functions. Thus the fundamental question of Fourier series was born. Fourier series live on the interval \([0,2\pi)\), or, even more naturally, on the circle group \(\mathbb T\). The Fourier analysis of the real line (i.e., the Fourier transform) was introduced at about the same time as Fourier series. But it was not until the mid-twentieth century that Fourier analysis on \(\mathbb R^N\) came to fruition. Meanwhile, abstract harmonic analysis (i.e., the harmonic analysis of locally compact abelian groups) had developed a life of its own and the theory of Lie group representations provided a natural crucible for noncommutative harmonic analysis. The point here is that the subject of harmonic analysis is a point of view and a collection of tools, and harmonic analysts continually seek new venues in which to ply their wares. In the 1970s, E.M.~Stein and his school introduced the idea of studying classical harmonic analysis -- fractional integrals and singular integrals -- on the Heisenberg group. This turned out to be …
Mathematics Subject Classification ID: 43-02
Mathematics Subject Classification ID: 32-XX
Mathematics Subject Classification ID: 42-XX
Mathematics Subject Classification ID: 44-XX
Mathematics Subject Classification ID: 43-XX
zbMATH DE Number: 5574611
zbMATH Keywords: Fourier series
zbMATH Keywords: Hilbert transform
zbMATH Keywords: Laplace equation
zbMATH Keywords: singular integrals
zbMATH Keywords: Fourier transform
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: So\-bo\-lev spaces
zbMATH Keywords: several complex variables
zbMATH Keywords: complex integral operators
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Heisenberg groups
zbMATH Keywords: domains of finite type
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-0-8176-4669-1
OpenAlex ID: W2498816272
description: scientific article; zbMATH DE number 5574611