Editing Harmonic analysis (section)

== Development of harmonic analysis ==
Historically, [[harmonic function]]s first referred to the solutions of [[Laplace's equation]].<ref><!-- DO NOT USE CITATION TEMPLATES FOR THIS REFERENCE! They are unable to handle the Fall 2020–2021 date specified in the reference. -->Burtscher, Annegret (Fall 2020–2021). "[https://www.math.ru.nl/~burtscher/lecturenotes/2021PDEnotes.pdf Introduction to Partial Differential Equations, Course module NWI-WB046B]" (PDF). Radboud University Nijmegen. Retrieved 2025-01-19.</ref> This terminology was extended to other [[special functions]] that solved related equations,<ref>{{cite book|title=Special functions and the theory of group representation|author=N. Vilenkin|year=1968}}</ref> then to [[Eigenfunction|eigenfunctions]] of general [[Elliptic operator|elliptic operators]],<ref>{{See also|Atiyah-Singer index theorem}}</ref> and nowadays harmonic functions are considered as a generalization of periodic functions<ref>{{cite web | url=https://www.britannica.com/science/harmonic-analysis | title=Harmonic analysis &#124; Mathematics, Fourier Series & Waveforms &#124; Britannica }}</ref> in [[function space]]s defined on [[manifold|manifolds]], for example as solutions of general, not necessarily [[elliptic partial differential equation|elliptic]], [[partial differential equations]] including some [[boundary conditions]] that may imply their symmetry or periodicity.<ref name=tao>{{citation|url=https://www.math.ucla.edu/~tao/247a.1.06f/notes0.pdf|first=Terence|last=Tao|author-link=Terence Tao|title=Harmonic Analysis|work=MATH 247A : Fourier analysis|publisher=University of California, Los Angeles|access-date=2025-01-19}}</ref>