eng:
id: 1030111
terms:
- type: expression
normativeStatus: preferred
designation: system of orthogonal functions
- type: expression
designation: orthogonal system
definition: set of functions, such that each of them is orthogonal to any other
language_code: eng
notes:
- "Examples: \n\n<ul> \n\n<li>Legendre polynomials P constitute a system of orthogonal
functions on the interval $$- 1 , + 1$$ because $$∫_( - 1)^( + 1) P_k
\ ( x ) P_l ( x ) dx = 0$$ for any integers $$k != l$$.</li> \n\n<li>Laguerre
polynomials L constitute a system of orthogonal functions on the interval $$0
, + oo$$ with the weight $$exp ( - x )$$ because $$∫_( 0)^( + oo ) L_k
\ ( x ) L_l ( x ) exp ( - x ) dx = 0$$ for any integers $$k != l$$.</li> \n\n<li>Trigonometric
functions sine and cosine constitute a system of orthogonal functions on the
interval $$0 , 2 \\pi$$ because $$∫_( 0)^( 2 \\pi ) sin ( kx ) sin ( lx
) dx = 0$$ and $$∫_( 0)^( 2 \\pi ) cos ( kx ) cos ( lx ) dx = 0$$ for any
integers $$k != l$$, and $$∫_( 0)^( 2 \\pi ) sin ( kx ) cos ( lx ) dx = 0$$
for any integer $$\x01$$ and $$\x01$$. </li></ul>"
e.g.