Queries, comments, suggestions? Please contact us.



Area Mathematics - Functions / General concepts

IEV ref103-01-11

en
system of orthogonal functions
orthogonal system
set of functions, such that each of them is orthogonal to any other

Note 1 to entry: Examples:

  • Legendre polynomials P constitute a system of orthogonal functions on the interval [1,+1] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaacUfacqGHsislju gabiaaigdakiaacYcacaaMe8Uaey4kaSscLbqacaaIXaGccaGGDbaa aa@3D84@ because 1 +1 P k (x) P l (x)dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaamaapedabaqcLbsaca GGqbGcdaWgaaWcbaGaam4AaaqabaaabaGaaGjcVlabgkHiTiaaigda aeaacaaMi8Uaey4kaSIaaGymaaqdcqGHRiI8aOGaaiikaiaadIhaca GGPaqcLbsacaGGqbGcdaWgaaWcbaGaamiBaaqabaGccaGGOaGaamiE aiaacMcajugabiaacsgakiaadIhacqGH9aqpjugabiaaicdaaaa@4C35@ for any integers kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadUgacqGHGjsUca WGSbaaaa@38F8@ .
  • Laguerre polynomials L constitute a system of orthogonal functions on the interval [0,+] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaacUfajugabiaaic dakiaacYcacaaMe8Uaey4kaSIaeyOhIuQaaiyxaaaa@3CD3@ with the weight exp(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaKqzaeGaaiyzaiaacI hacaGGWbGccaGGOaGaeyOeI0IaamiEaiaacMcaaaa@3BE5@ because 0 + L k (x) L l (x)exp(x)dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaapedabaqcLbsaca GGmbGcdaWgaaWcbaGaam4AaaqabaaabaGaaGjcVNqzGdGaaGimaaWc baGaaGjcVlabgUcaRiabg6HiLcqdcqGHRiI8aOGaaiikaiaadIhaca GGPaqcLbsacaGGmbGcdaWgaaWcbaGaamiBaaqabaGccaGGOaGaamiE aiaacMcajugabiGacwgacaGG4bGaaiiCaOGaciikaiabgkHiTiaadI hacaGGPaqcLbqacaGGKbGccaWG4bGaeyypa0tcLbqacaaIWaaaaa@53E5@ for any integers kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadUgacqGHGjsUca WGSbaaaa@38F8@ .
  • Trigonometric functions sine and cosine constitute a system of orthogonal functions on the interval [0,2π] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaacUfajugabiaaic dakiaacYcacaaMe8EcLbqacaaIYaacdaGccqWFapaCcaGGDbaaaa@3D78@ because 0 2π sin(kx)sin(lx) dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaapedabaqcLbqaci GGZbGaaiyAaiaac6gakiGacIcacaWGRbGaamiEaiaacMcajugabiGa cohacaGGPbGaaiOBaOGaciikaiaadYgacaWG4bGaaiykaaWcbaGaaG jcVNqzGdGaaGimaaWcbaGaaGjcVNqzGdGaaGOmaGWaaSGae8hWdaha niabgUIiYdqcLbqacaGGKbGccaWG4bGaeyypa0tcLbqacaaIWaaaaa@5244@ and 0 2π cos(kx)cos(lx) dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaamaapedabaqcLbqaci GGJbGaai4BaiaacohakiaacIcacaWGRbGaamiEaiaacMcajugabiGa cogacaGGVbGaai4CaOGaaiikaiaadYgacaWG4bGaaiykaaWcbaGaaG jcVNqzGdGaaGimaaWcbaGaaGjcVNqzGdGaaGOmaGWaaSGae8hWdaha niabgUIiYdqcLbqacaGGKbGccaWG4bGaeyypa0tcLbqacaaIWaaaaa@5238@ for any integers kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadUgacqGHGjsUca WGSbaaaa@38F8@ , and 0 2π sin(kx)cos(lx) dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaapedabaqcLbqaci GGZbGaaiyAaiaac6gakiGacIcacaWGRbGaamiEaiaacMcajugabiGa cogacaGGVbGaai4CaOGaaiikaiaadYgacaWG4bGaaiykaaWcbaGaaG jcVNqzGdGaaGimaaWcbaGaaGjcVNqzGdGaaGOmaGWaaSGae8hWdaha niabgUIiYdqcLbqacaGGKbGccaWG4bGaeyypa0tcLbqacaaIWaaaaa@523D@ for any integer k and l.

fr
système de fonctions orthogonales, m
ensemble de fonctions dont chacune est orthogonale à toute autre

Note 1 à l'article: Exemples:

  • Les polynômes de Legendre P constituent un système de fonctions orthogonales sur l'intervalle [1,+1] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaacUfacqGHsislju gabiaaigdakiaacYcacaaMe8Uaey4kaSscLbqacaaIXaGccaGGDbaa aa@3D84@ parce que 1 +1 P k (x) P l (x)dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaamaapedabaqcLbsaca GGqbGcdaWgaaWcbaGaam4AaaqabaaabaGaaGjcVlabgkHiTiaaigda aeaacaaMi8Uaey4kaSIaaGymaaqdcqGHRiI8aOGaaiikaiaadIhaca GGPaqcLbsacaGGqbGcdaWgaaWcbaGaamiBaaqabaGccaGGOaGaamiE aiaacMcajugabiaacsgakiaadIhacqGH9aqpjugabiaaicdaaaa@4C35@ pour tous entiers kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadUgacqGHGjsUca WGSbaaaa@38F8@ .
  • Les polynômes de Laguerre L constituent un système de fonctions orthogonales sur l'intervalle [0,+] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaacUfajugabiaaic dakiaacYcacaaMe8Uaey4kaSIaeyOhIuQaaiyxaaaa@3CD3@ avec le poids exp(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaKqzaeGaaiyzaiaacI hacaGGWbGccaGGOaGaeyOeI0IaamiEaiaacMcaaaa@3BE5@ parce que 0 + L k (x) L l (x)exp(x)dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaapedabaqcLbsaca GGmbGcdaWgaaWcbaGaam4AaaqabaaabaGaaGjcVNqzGdGaaGimaaWc baGaaGjcVlabgUcaRiabg6HiLcqdcqGHRiI8aOGaaiikaiaadIhaca GGPaqcLbsacaGGmbGcdaWgaaWcbaGaamiBaaqabaGccaGGOaGaamiE aiaacMcajugabiGacwgacaGG4bGaaiiCaOGaciikaiabgkHiTiaadI hacaGGPaqcLbqacaGGKbGccaWG4bGaeyypa0tcLbqacaaIWaaaaa@53E5@ pour tous entiers kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadUgacqGHGjsUca WGSbaaaa@38F8@ .
  • Les fonctions trigonométriques sinus et cosinus constituent un système de fonctions orthogonales sur l'intervalle [0,2π] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaacUfajugabiaaic dakiaacYcacaaMe8EcLbqacaaIYaacdaGccqWFapaCcaGGDbaaaa@3D78@ parce que 0 2π sin(kx)sin(lx) dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaapedabaqcLbqaci GGZbGaaiyAaiaac6gakiGacIcacaWGRbGaamiEaiaacMcajugabiGa cohacaGGPbGaaiOBaOGaciikaiaadYgacaWG4bGaaiykaaWcbaGaaG jcVNqzGdGaaGimaaWcbaGaaGjcVNqzGdGaaGOmaGWaaSGae8hWdaha niabgUIiYdqcLbqacaGGKbGccaWG4bGaeyypa0tcLbqacaaIWaaaaa@5244@ et 0 2π cos(kx)cos(lx) dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaamaapedabaqcLbqaci GGJbGaai4BaiaacohakiaacIcacaWGRbGaamiEaiaacMcajugabiGa cogacaGGVbGaai4CaOGaaiikaiaadYgacaWG4bGaaiykaaWcbaGaaG jcVNqzGdGaaGimaaWcbaGaaGjcVNqzGdGaaGOmaGWaaSGae8hWdaha niabgUIiYdqcLbqacaGGKbGccaWG4bGaeyypa0tcLbqacaaIWaaaaa@5238@ pour tous entiers kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadUgacqGHGjsUca WGSbaaaa@38F8@ , et 0 2π sin(kx)cos(lx) dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaapedabaqcLbqaci GGZbGaaiyAaiaac6gakiGacIcacaWGRbGaamiEaiaacMcajugabiGa cogacaGGVbGaai4CaOGaaiikaiaadYgacaWG4bGaaiykaaWcbaGaaG jcVNqzGdGaaGimaaWcbaGaaGjcVNqzGdGaaGOmaGWaaSGae8hWdaha niabgUIiYdqcLbqacaGGKbGccaWG4bGaeyypa0tcLbqacaaIWaaaaa@523D@ pour tous entiers k et l.

ar
مجموعة الدوال المتعامدة
نظام متعامد

de
System orthogonaler Funktionen, n
Orthogonalsystem, n

es
sistema de funciones ortogonales

it
sistema di funzioni ortogonali
sistema ortogonale

ja
直交関数系
直交系

pl
układ funkcji ortogonalnych
układ ortogonalny

pt
sistema de funcionais ortogonais
sistema ortogonal

sr
систем ортогоналних функција, м јд
ортогонални систем, м јд

sv
system av ortogonala funktioner
ortogonalsystem

zh
正交函数系
正交系

Publication date: 2009-12
Copyright © IEC 2017. All Rights Reserved.