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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

ko
직교 함수 계

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
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