直交関数列
{1, cosx, sinx, cos2x, sin2x,...} は [-π, π] で直交関数系を成す。
∫ − π π 1 d x = 2 π {\displaystyle \int _{-\pi }^{\pi }1\,dx=2\pi }
∫ − π π cos  m x ⋅ cos  n x d x = ∫ − π π sin  m x ⋅ sin  n x d x = π δ m n ( m , n = 1 , 2 , ... ) {\displaystyle \int _{-\pi }^{\pi }\cos {mx}\cdot \cos {nx}\,dx=\int _{-\pi }^{\pi }\sin {mx}\cdot \sin {nx}\,dx=\pi \delta _{mn}\quad (m,n=1,2,\dots )}
∫ − π π 1 ⋅ cos  n x d x = ∫ − π π 1 ⋅ sin  n x d x = 0 ( n = 1 , 2 , ... ) {\displaystyle \int _{-\pi }^{\pi }1\cdot \cos {nx}\,dx=\int _{-\pi }^{\pi }1\cdot \sin {nx}\,dx=0\quad (n=1,2,\dots )}
∫ − π π cos  m x ⋅ sin  n x d x = 0 ( m , n = 1 , 2 , ... ) {\displaystyle \int _{-\pi }^{\pi }\cos {mx}\cdot \sin {nx}\,dx=0\quad (m,n=1,2,\dots )}
関係式
H n ( x ) = ( − 1 ) n e x 2 / 2 d n d x n e − x 2 / 2 ( n = 0 , 1 , 2 , ... ) {\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}/2}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}/2}\quad (n=0,1,2,\dots )}
で定義されるエルミート多項式は区間 (−∞, ∞) 上の重み e の直交関数系であり、
∫ − ∞ ∞ e − x 2 / 2 H m ( x ) H n ( x ) d x = n ! 2 π δ m n ( m , n = 0 , 1 , 2 , ... ) {\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}/2}H_{m}(x)H_{n}(x)\,dx=n!{\sqrt {2\pi }}\delta _{mn}\quad (m,n=0,1,2,\dots )}
を満たす。