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IEVref:103-01-11ID:
Language:enStatus: 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 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.



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
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