      IEVref: 103-01-11 ID: Language: en Status: Standard    Term: system of orthogonal functions Synonym1: orthogonal system  Synonym2:  Synonym3:  Symbol: Definition: set of functions, such that each of them is orthogonal to any otherNote 1 to entry: Examples: Legendre polynomials P constitute a system of orthogonal functions on the interval $\left[-1,\text{\hspace{0.17em}}+1\right]$ because ${\int }_{\text{ }-1}^{\text{ }+1}{P}_{k}\left(x\right){P}_{l}\left(x\right)\text{d}x=0$ for any integers $k\ne l$. Laguerre polynomials L constitute a system of orthogonal functions on the interval $\left[0,\text{\hspace{0.17em}}+\infty \right]$ with the weight $\text{e}\text{x}\text{p}\left(-x\right)$ because ${\int }_{\text{ }0}^{\text{ }+\infty }{L}_{k}\left(x\right){L}_{l}\left(x\right)\text{exp}\left(-x\right)\text{d}x=0$ for any integers $k\ne l$. Trigonometric functions sine and cosine constitute a system of orthogonal functions on the interval $\left[0,\text{\hspace{0.17em}}2\pi \right]$ because ${\int }_{\text{ }0}^{\text{ }2\pi }\mathrm{sin}\left(kx\right)\text{sin}\left(lx\right)\text{d}x=0$ and ${\int }_{\text{ }0}^{\text{ }2\pi }\mathrm{cos}\left(kx\right)\mathrm{cos}\left(lx\right)\text{d}x=0$ for any integers $k\ne l$, and ${\int }_{\text{ }0}^{\text{ }2\pi }\text{sin}\left(kx\right)\mathrm{cos}\left(lx\right)\text{d}x=0$ for any integer k and l. Publication date: 2009-12 Source: Replaces: Internal notes: 2017-02-20: Editorial revisions in accordance with the information provided in C00020 (IEV 103) - evaluation. JGO CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: