Solve integration containing exponential inner product function
$begingroup$
Let $ f:Re^{n} hookrightarrow Re^{n} $ be a contionuous function such that $ int_{Re^{n}} {|f(x)|} dx < infty $ .
Let $ A $ be a real $ ntimes n $ invertible matrix and for $ x.y in Re^{n} $ , let $ <x,y> $ denote the standard inner product in $ Re^{n} $ . Then $ int_{Re^{n}} {f(Ax) ; e^{i <y,x>}} dx = $
$ int_{Re^{n}} {f(x) ; e^{i <( A^{-1})^{T} y , x >}} ; frac{dx}{|det A|} $
$ int_{Re^{n}} {f(x) ; e^{i < A^{T} y , x >}} ; frac{dx}{|det A|} $
$ int_{Re^{n}} {f(x) ; e^{i <( A^{T})^{-1} y , x >}} ; dx $
$ int_{Re^{n}} {f(x) ; e^{i <A^{-1} y , x >}} ; frac{dx}{|det A|} $
linear-algebra
asked Dec 11 '18 at 12:29
Nirvik MasantaNirvik Masanta
114
$endgroup$
add a comment |
$begingroup$
Let $ f:Re^{n} hookrightarrow Re^{n} $ be a contionuous function such that $ int_{Re^{n}} {|f(x)|} dx < infty $ .
Let $ A $ be a real $ ntimes n $ invertible matrix and for $ x.y in Re^{n} $ , let $ <x,y> $ denote the standard inner product in $ Re^{n} $ . Then $ int_{Re^{n}} {f(Ax) ; e^{i <y,x>}} dx = $
$ int_{Re^{n}} {f(x) ; e^{i <( A^{-1})^{T} y , x >}} ; frac{dx}{|det A|} $
$ int_{Re^{n}} {f(x) ; e^{i < A^{T} y , x >}} ; frac{dx}{|det A|} $
$ int_{Re^{n}} {f(x) ; e^{i <( A^{T})^{-1} y , x >}} ; dx $
$ int_{Re^{n}} {f(x) ; e^{i <A^{-1} y , x >}} ; frac{dx}{|det A|} $
linear-algebra
asked Dec 11 '18 at 12:29
Nirvik MasantaNirvik Masanta
114
$endgroup$
add a comment |
$begingroup$
Let $ f:Re^{n} hookrightarrow Re^{n} $ be a contionuous function such that $ int_{Re^{n}} {|f(x)|} dx < infty $ .
Let $ A $ be a real $ ntimes n $ invertible matrix and for $ x.y in Re^{n} $ , let $ <x,y> $ denote the standard inner product in $ Re^{n} $ . Then $ int_{Re^{n}} {f(Ax) ; e^{i <y,x>}} dx = $
$ int_{Re^{n}} {f(x) ; e^{i <( A^{-1})^{T} y , x >}} ; frac{dx}{|det A|} $
$ int_{Re^{n}} {f(x) ; e^{i < A^{T} y , x >}} ; frac{dx}{|det A|} $
$ int_{Re^{n}} {f(x) ; e^{i <( A^{T})^{-1} y , x >}} ; dx $
$ int_{Re^{n}} {f(x) ; e^{i <A^{-1} y , x >}} ; frac{dx}{|det A|} $
linear-algebra
asked Dec 11 '18 at 12:29
Nirvik MasantaNirvik Masanta
114
$endgroup$
Let $ f:Re^{n} hookrightarrow Re^{n} $ be a contionuous function such that $ int_{Re^{n}} {|f(x)|} dx < infty $ .
Let $ A $ be a real $ ntimes n $ invertible matrix and for $ x.y in Re^{n} $ , let $ <x,y> $ denote the standard inner product in $ Re^{n} $ . Then $ int_{Re^{n}} {f(Ax) ; e^{i <y,x>}} dx = $
$ int_{Re^{n}} {f(x) ; e^{i <( A^{-1})^{T} y , x >}} ; frac{dx}{|det A|} $
$ int_{Re^{n}} {f(x) ; e^{i < A^{T} y , x >}} ; frac{dx}{|det A|} $
$ int_{Re^{n}} {f(x) ; e^{i <( A^{T})^{-1} y , x >}} ; dx $
$ int_{Re^{n}} {f(x) ; e^{i <A^{-1} y , x >}} ; frac{dx}{|det A|} $
linear-algebra
linear-algebra
asked Dec 11 '18 at 12:29
Nirvik MasantaNirvik Masanta
114
asked Dec 11 '18 at 12:29
Nirvik MasantaNirvik Masanta
114
asked Dec 11 '18 at 12:29
Nirvik MasantaNirvik Masanta
114
asked Dec 11 '18 at 12:29
Nirvik MasantaNirvik Masanta
114
asked Dec 11 '18 at 12:29
Nirvik MasantaNirvik Masanta
114
114
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add a comment |
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