Some topics in the history of harmonic analysis in the twentieth century
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).
scientific article; zbMATH DE number 6844792
scientific article