Editing Paley–Wiener theorem (section)

{{Short description|Mathematical theorem}}
In [[mathematics]], a '''Paley–Wiener theorem''' is a theorem that relates decay properties of a function or [[distribution (mathematics)|distribution]] at infinity with [[analytic function|analyticity]] of its [[Fourier transform]].  It is named after [[Raymond Paley]] (1907–1933) and [[Norbert Wiener]] (1894–1964) who, in 1934, introduced various versions of the theorem.{{sfn | Paley | Wiener | 1934}} The original theorems did not use the language of [[generalized function|distributions]], and instead applied to [[Lp space|square-integrable functions]].  The first such theorem using distributions was due to [[Laurent Schwartz]]. These theorems heavily rely on the [[triangle inequality]] (to interchange the absolute value and integration).

The original work by Paley and Wiener is also used as a namesake in the fields of [[control theory]] and [[harmonic analysis]]; introducing the '''[[Jensen%27s_formula#Applications|Paley–Wiener condition]]''' for [[polynomial_matrix_spectral_factorization|spectral factorization]] and the '''[[Riesz_sequence#Paley-Wiener_criterion|Paley–Wiener criterion]]''' for [[Frame_(linear_algebra)#Non-harmonic_Fourier_series|non-harmonic Fourier series]] respectively.{{sfn | Paley | Wiener | 1934|pp=14-20,100}} These are related mathematical concepts that place the decay properties of a function in context of [[stability theory|stability problems]].