Exponential form of the Fourier series

By application of Euler’s formula, the Fourier series of a square-integrable $𝐿 2 (- 𝑎, 𝑎)$ function may be expressed as

𝑓 (𝑥) = s p a n ℤ 𝐶 𝑛 𝑒 𝑖 𝑛 𝜋 𝑥 / 𝑎

where

𝐶 𝑛 = ⟨ 𝑒 𝑛 | 𝑓 ⟩ = 1 2 𝑎 \int 𝐿 - 𝑎 𝑒 𝑖 𝑛 𝜋 𝑥 / 𝑎 𝑓 (𝑥) 𝑑 𝑥

Hence the Hilbert space $𝐿 2 (- 𝑎, 𝑎)$ has the orthonormal dense basis

𝐿2(−𝑎,𝑎)=\span{|𝑒𝑛⟩=𝑥↦𝑒𝑖𝑛𝜋𝑥/𝑎:𝑛∈ℤ}

with

⟨ 𝑒 𝑛 | 𝑒 𝑚 ⟩ = 1 2 𝑎 \int 𝑎 - 𝑎 𝑒 - 𝑖 𝑛 𝜋 𝑥 / 𝑎 𝑒 𝑖 𝑚 𝜋 𝑥 / 𝑎 𝑑 𝑥 = 𝛿 𝑛 𝑚