Note 1 to entry: Examples:

• Legendre polynomials P constitute a system of orthogonal functions on the interval $[-1,+1]$ because ${\displaystyle {\int }_{-1}^{+1}{P}_k(x)P_l(x)\text{d}x=0}$ for any integers $k\ne l$.
• Laguerre polynomials L constitute a system of orthogonal functions on the interval $[0,+\infty ]$ with the weight $\text{e}\text{x}\text{p}(-x)$ because ${\displaystyle {\int }_0^{+\infty }{L}_k(x)L_l(x)\text{exp}(-x)\text{d}x=0}$ for any integers $k\ne l$.
• Trigonometric functions sine and cosine constitute a system of orthogonal functions on the interval $[0,2\pi ]$ because ${\displaystyle {\int }_0^{2\pi }\sin (kx)\text{sin}(lx)\text{d}x=0}$ and ${\displaystyle {\int }_0^{2\pi }\cos (kx)\cos (lx)\text{d}x=0}$ for any integers $k\ne l$, and ${\displaystyle {\int }_0^{2\pi }\text{sin}(kx)\cos (lx)\text{d}x=0}$ for any integer k and l.