section_path: Role and characterizations in mathematics/Fourier series
section_name: Fourier series
paragraph_text: Fourier series
  
  The constant π also appears naturally in Fourier series of periodic functions. Periodic functions are functions on the group T = R / Z of fractional parts of real numbers. The Fourier decomposition shows that a complex-valued function f on T can be written as an infinite linear superposition of unitary characters of T. That is, continuous group homomorphisms from T to the circle group U (1) of unit modulus complex numbers. It is a theorem that every character of T is one of the complex exponentials e n (x) = e 2 π i n x {\displaystyle e_{n}(x)=e^{2\pi inx}}.
  
  There is a unique character on T, up to complex conjugation, that is a group isomorphism. Using the Haar measure on the circle group, the constant π 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 π. As a result, the constant π is the unique number such that the group T, equipped with its Haar measure, is Pontrjagin dual to the lattice of integral multiples of 2 π. This is a version of the one-dimensional Poisson summation formula.