Some topics in the history of harmonic analysis in the twentieth century
type: Item
instance of: scholarly article
title: Some topics in the history of harmonic analysis in the twentieth century
zbMATH Open document ID: 1391.42002
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2018-02-28T00:00:00Z
review text: The paper gives a sketch of the thematic developments of some of the main techniques and results of the 20th-century harmonic analysis. It is subdivided into the following sections: 1. Some notation and terminology, 2. Fourier series in 1900: pointwise convergence. 3. The first decade: 1900--1910, 4. Fourier series and complex analysis, 1915--1930, 5. Interpolation theorems, 6. The transition to \(\mathbb R^n\), 7. Singular integrals, 8. \(H^p\) spaces: the real-variable theory, 9. Singular integrals revisited, 10. Littlewood-Paley theory, 11. Harmonic analysis on groups, 12. Wavelets. The last section raises interest if for no other reason than for the fact that the latest Abel Prize is for the area of wavelets theory. The paper does offer some additional material not mentioned (or not treated satisfactorily) in earlier papers on the same subject. The author has made some effort to cite original sources. Some of the many interesting results that caught this reviewer's attention are as follows: Kolmogorov constructed (in 1923, 1926) an \(L^1(T)\) function whose Fourier series diverges everywhere (\(T\), as usual, denotes the circle group). A consequence of Peter and Weyl's 1927 result: The regular representation of a compact group \(G\) is a direct sum of irreducible subrepresentations, and, for every \(\pi\in\hat G\), the equivalence class of \(\pi\) occurs in this direct sum with multiplicity \(d_\pi\) (here \(d_\pi\) is the dimension of the Hilbert space \(\mathcal H_\pi\) on which \(\pi\) acts).
reviewed by: Radoslav M. Dimitrić
Mathematics Subject Classification ID: 42-03
Mathematics Subject Classification ID: 43-03
Mathematics Subject Classification ID: 01A60
zbMATH DE Number: 6844792
zbMATH Keywords: Fourier analysis
zbMATH Keywords: harmonic analysis
zbMATH Keywords: singular integral operators
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Littlewood-Paley theory
zbMATH Keywords: wavelets
author: Gerald B. Folland
MaRDI profile type: Publication
full work available at URL: s13226-016-0198-z
OpenAlex ID: W2522093030
cites work: A Maximal Function Characterization of the Class H p
cites work: A Class of Bounded Pseudo-Differential Operators
cites work: On the existence of certain singular integrals
cites work: Singular Integral Operators and Differential Equations
cites work: On convergence and growth of partial sums of Fourier series
cites work: Théorie de la dualité et analyse harmonique dans les groupes abéliens localement compacts
cites work: Singular and maximal Radon transforms: Analysis and geometry
cites work: A real variable characterization of $H^{p}$
cites work: Q2548398
cites work: Extensions of Hardy spaces and their use in analysis
cites work: Q3835900
cites work: Orthonormal bases of compactly supported wavelets
cites work: Ten Lectures on Wavelets
cites work: A boundedness criterion for generalized Calderón-Zygmund operators
cites work: Sur Les Représentations Unitaires Des Groupes De Lie Nilpotents. III
cites work: Characterizations of bounded mean oscillation
cites work: Pointwise convergence of Fourier series
cites work: \(H^p\) spaces of several variables
cites work: Singular integrals on product spaces
cites work: Applications of analysis on nilpotent groups to partial differential equations
cites work: Harmonic Analysis in Phase Space. (AM-122)
cites work: Q4858539
cites work: Q4255465
cites work: Q4888829
cites work: Estimates for the \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \partial \limits^ - _b $\end{document} complex and analysis on the heisenberg group
cites work: Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28
cites work: Der Maßbegriff in der Theorie der kontinuierlichen Gruppen
cites work: Q4423258
cites work: Q4896587
cites work: Pseudo‐differential operators
cites work: On functions of bounded mean oscillation
cites work: Series in Banach spaces: conditional and unconditional convergence. Transl. from the Russian by Andrei Iacob
cites work: SINGULAR INTEGRALS AND THE PRINCIPAL SERIES
cites work: Intertwining operators for semisimple groups
cites work: An algebra of pseudo‐differential operators
cites work: Q4042093
cites work: A characterization of $H^{p}(R^{n})$ in terms of atoms
cites work: Theorems on Fourier Series and Power Series
cites work: Theorems on Fourier Series and Power Series (II)†
cites work: A theorem of Stone and von Neumann
cites work: Imprimitivity for Representations of Locally Compact Groups I
cites work: Harmonic analysis as the exploitation of symmetry-a historical survey
cites work: Q5773234
cites work: Q3793160
cites work: Q3995454
cites work: Q3996824
cites work: Hilbert integrals, singular integrals, and Radon transforms. I
cites work: Singular Radon transforms and oscillatory integrals
cites work: Q5836056
cites work: Q5786103
cites work: Interpolation of Linear Operators
cites work: On the Functions of Littlewood-Paley, Lusin, and Marcinkiewicz
cites work: Q5607484
cites work: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. (AM-63)
cites work: The development of square functions in the work of A. Zygmund
cites work: Q3142876
cites work: Problems in harmonic analysis related to curvature
cites work: On the theory of harmonic functions of several variables

Some topics in the history of harmonic analysis in the twentieth century (cites work)
cites work: Q5640160

Some topics in the history of harmonic analysis in the twentieth century (cites work)
cites work: Multi-parameter Singular Integrals

Some topics in the history of harmonic analysis in the twentieth century (cites work)
cites work: Q3900552

Some topics in the history of harmonic analysis in the twentieth century (cites work)
cites work: Q5846810

Some topics in the history of harmonic analysis in the twentieth century (cites work)
cites work: On a theorem of Marcinkiewicz concerning interpolation of operations

Some topics in the history of harmonic analysis in the twentieth century (cites work)
cites work: On multipliers preserving convergence of trigonometric series almost everywhere

Some topics in the history of harmonic analysis in the twentieth century (DOI)
DOI: 10.1007/S13226-016-0198-Z