study of superpositions in mathematics
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- The roots of harmonic analysis are in the study of harmonics, which are the basic sound waves whose frequencies are multiples of each other.
- The rapid development of quantum mechanics stimulated research in operator theory and group representation theory. Initiated during the mid-twenties, intensive study of topological groups and their representations led to Haar's discovery of the basic construction of invariant integration on a topological group. Bohr's theory of almost periodic functions influenced the work of Wiener, Bochner and many other analysts. They enriched the technical arsenal of harmonic analysis and the scope of its applications (statistical mechanics, ergodic theory, time series, etc.) The new notion of the generalized Fourier transform made it possible to consider Plancherel's theory simultaneously with Bohr's theory, the continuous spectrum with the discrete. The Pontrjagin-van Kampen duality opened the way for an unobstructed development of Fourier analysis on locally compact abelian groups, allowing Fourier series, Fourier integrals and expansions via numerical characters to be viewed as objects of the same kind. The Peter–Weyl theory made it possible for von Neumann to analyze almost periodic functions on groups by connecting them to group representation theory. Along with the many other discoveries of that period, this led to the inclusion of group theorethical methods into the tool kit of harmonic analysis.
- V. P. Havin; N. K. Nikolski (6 December 2012). Commutative Harmonic Analysis II: Group Methods in Commutative Harmonic Analysis. Springer Science & Business Media. p. 7. ISBN 978-3-642-58946-1.
- One of the many ways in which harmonic analysis distinguishes itself among other areas of modern mathematics is through the emphasis placed on algorithm development and the connections that it builds with applied sciences. This phenomenon goes back to Joseph Fourier, whose main motivation for introducing what we know as the Fourier transform, was his work on heat flow and thermal conduction. Other prominent examples of these interactions include the role of Radon transform in Magnetic Resonance Imaging, the impact of Kaczmarz algorithm on Computed Tomography, and the role played by the Phase Problem in X-ray crystallography—all rewarded with Nobel Prizes.
- Radu Balan; Matthew Begué; John J. Benedetto (2 June 2015). Excursions in Harmonic Analysis, Volume 3: The February Fourier Talks at the Norbert Wiener Center. Birkhäuser. p. 64. ISBN 978-3-319-13230-3.