Editing Pi (section)

=== Fourier series ===
[[File:2-adic integers with dual colorings.svg|thumb|{{pi}} appears in characters of [[p-adic numbers]] (shown), which are elements of a [[Prüfer group]]. [[Tate's thesis]] makes heavy use of this machinery.<ref>{{cite conference |last1=Tate |first1=John T. |author-link=John Tate (mathematician) |editor1-first=J. W. S. |editor1-last=Cassels |editor2-first=A. |editor2-last=Fröhlich |title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) |publisher=Thompson, Washington, DC |isbn=978-0-9502734-2-6 |mr=0217026 |year=1950 |contribution=Fourier analysis in number fields, and Hecke's zeta-functions |pages=305–347}}</ref>|left]]
The constant {{pi}} also appears naturally in [[Fourier series]] of [[periodic function]]s.  Periodic functions are functions on the group {{math|'''T''' {{=}}'''R'''/'''Z'''}} of fractional parts of real numbers.  The Fourier decomposition shows that a complex-valued function {{math|''f''}} on {{math|'''T'''}} can be written as an infinite linear superposition of [[unitary character]]s of {{math|'''T'''}}.   That is, continuous [[group homomorphism]]s from {{math|'''T'''}} to the [[circle group]] {{math|''U''(1)}} of unit modulus complex numbers.  It is a theorem that every character of {{math|'''T'''}} is one of the complex exponentials <math>e_n(x)= e^{2\pi i n x}</math>.

There is a unique character on {{math|'''T'''}}, up to complex conjugation, that is a group isomorphism.  Using the [[Haar measure]] on the circle group, the constant {{pi}} is half the magnitude of the [[Radon–Nikodym derivative]] of this character.  The other characters have derivatives whose magnitudes are positive integral multiples of 2{{pi}}.{{r|Nicolas Bourbaki}} As a result, the constant {{pi}} is the unique number such that the group '''T''', equipped with its Haar measure, is [[Pontrjagin dual]] to the [[lattice (group)|lattice]] of integral multiples of 2{{pi}}.{{sfn|Dym|McKean|1972|loc=Chapter 4}}  This is a version of the one-dimensional [[Poisson summation formula]].