IEVref:
103-01-09
ID:
Language:
en
Status:
Standard
Term:
weighted inner product
Synonym1:
Synonym2:
Synonym3:
Symbol:
Definition:
for two complex-valued functions,
f
_
MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaqcLbwaca WGMbaaaaaa@3718@
and
g
_
MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaqcLbwaca WGNbaaaaaa@3719@
, defined on the interval
[
a
,
b
]
MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaadmaabaGaamyyai aacYcacaWGIbaacaGLBbGaayzxaaaaaa@39BD@
of ℝ, complex number
〈
f
_
,
g
_
〉
=
∫
a
b
f
_
(
x
)
g
_
∗
(
x
)
w
(
x
)
d
x
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaamaabaWaaWaaaeaacaWGMbaaaiaacYcadaadaaqaaiaadEgaaaaacaGLPmIaayPkJaGaeyypa0Zaa8qmaeaadaadaaqaaiaadAgaaaGaaiikaiaadIhaaSqaaiaayIW7caWGHbaabaGaaGjcVlaadkgaa0Gaey4kIipakiaacMcacaaMc8+aaWaaaeaacaWGNbaaamaaCaaaleqabaGaey4fIOcaaOGaaGzaVlaacIcacaWG4bGaaiykaiaaykW7caWG3bGaaiikaiaadIhacaGGPaqcLbuacaGGKbGaamiEaaaa@543B@
, where
w
(
x
) is a real function of
x
called weight
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:
for two complex-valued functions,
f
_
MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaqcLbwaca WGMbaaaaaa@3718@
and
g
_
MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaqcLbwaca WGNbaaaaaa@3719@
, defined on the interval
[
a
,
b
]
MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaadmaabaGaamyyai aacYcacaWGIbaacaGLBbGaayzxaaaaaa@39BD@
of ℝ, complex number
〈
f
_
,
g
_
〉
=
∫
a
b
f
_
(
x
)
g
_
∗
(
x
)
w
(
x
)
d
x
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaamaabaWaaWaaaeaacaWGMbaaaiaacYcadaadaaqaaiaadEgaaaaacaGLPmIaayPkJaGaeyypa0Zaa8qmaeaadaadaaqaaiaadAgaaaGaaiikaiaadIhaaSqaaiaayIW7caWGHbaabaGaaGjcVlaadkgaa0Gaey4kIipakiaacMcacaaMc8+aaWaaaeaacaWGNbaaamaaCaaaleqabaGaey4fIOcaaOGaaGzaVlaacIcacaWG4bGaaiykaiaaykW7caWG3bGaaiikaiaadIhacaGGPaqcLbuacaGGKbGaamiEaaaa@543B@
, where
w
(
x
) is a real function of
x
called weight