Docomposition/transformation other than fourier transformation.
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It has been known that Fourier transformation is a very useful tool since it transform a variable from its form $x$ to a correlated basis.
However, is there any other decomposition or transformation method that can transform variables and functions into other basis, and what's their advantage and disadvantage compared to Fourier transformation? (for example, it seems to me that Laplace transformation can be used to solve partial differential equations, but why it is so rarely used in daily occasions?)
functional-analysis analysis transformation fourier-transform
edited Nov 18 at 6:39
Daniele Tampieri
1,5651619
asked Nov 18 at 6:10
user9976437
558
add a comment |
up vote
0
down vote
favorite
It has been known that Fourier transformation is a very useful tool since it transform a variable from its form $x$ to a correlated basis.
However, is there any other decomposition or transformation method that can transform variables and functions into other basis, and what's their advantage and disadvantage compared to Fourier transformation? (for example, it seems to me that Laplace transformation can be used to solve partial differential equations, but why it is so rarely used in daily occasions?)
functional-analysis analysis transformation fourier-transform
edited Nov 18 at 6:39
Daniele Tampieri
1,5651619
asked Nov 18 at 6:10
user9976437
558
1
$int_0^infty f(t) e^{-st}dt$ is the Fourier transform of $f(t) e^{-sigma t} 1_{t > 0}$. The main reason why we look at it for every $sigma$ (instead of only one) is the analytic continuation and the tools from complex analysis. Do you know the orthogonal polynomials ? Some generalizations : the spectral theorem for compact operators in Hilbert spaces, providing an orthogonal basis where the operator acts diagonally, the diagonalization of the Laplacian in smooth manifolds, the representation theory in topological groups.
– reuns
Nov 18 at 6:45
1
en.wikipedia.org/wiki/Integral_transform#Table_of_transforms
– Rahul
Nov 18 at 7:53
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
It has been known that Fourier transformation is a very useful tool since it transform a variable from its form $x$ to a correlated basis.
However, is there any other decomposition or transformation method that can transform variables and functions into other basis, and what's their advantage and disadvantage compared to Fourier transformation? (for example, it seems to me that Laplace transformation can be used to solve partial differential equations, but why it is so rarely used in daily occasions?)
functional-analysis analysis transformation fourier-transform
edited Nov 18 at 6:39
Daniele Tampieri
1,5651619
asked Nov 18 at 6:10
user9976437
558
It has been known that Fourier transformation is a very useful tool since it transform a variable from its form $x$ to a correlated basis.
However, is there any other decomposition or transformation method that can transform variables and functions into other basis, and what's their advantage and disadvantage compared to Fourier transformation? (for example, it seems to me that Laplace transformation can be used to solve partial differential equations, but why it is so rarely used in daily occasions?)
functional-analysis analysis transformation fourier-transform
functional-analysis analysis transformation fourier-transform
edited Nov 18 at 6:39
Daniele Tampieri
1,5651619
asked Nov 18 at 6:10
user9976437
558
edited Nov 18 at 6:39
Daniele Tampieri
1,5651619
asked Nov 18 at 6:10
user9976437
558
edited Nov 18 at 6:39
Daniele Tampieri
1,5651619
edited Nov 18 at 6:39
Daniele Tampieri
1,5651619
edited Nov 18 at 6:39
Daniele Tampieri
1,5651619
1,5651619
asked Nov 18 at 6:10
user9976437
558
asked Nov 18 at 6:10
user9976437
558
asked Nov 18 at 6:10
user9976437
558
558
1
$int_0^infty f(t) e^{-st}dt$ is the Fourier transform of $f(t) e^{-sigma t} 1_{t > 0}$. The main reason why we look at it for every $sigma$ (instead of only one) is the analytic continuation and the tools from complex analysis. Do you know the orthogonal polynomials ? Some generalizations : the spectral theorem for compact operators in Hilbert spaces, providing an orthogonal basis where the operator acts diagonally, the diagonalization of the Laplacian in smooth manifolds, the representation theory in topological groups.
– reuns
Nov 18 at 6:45
1
en.wikipedia.org/wiki/Integral_transform#Table_of_transforms
– Rahul
Nov 18 at 7:53
add a comment |
1
$int_0^infty f(t) e^{-st}dt$ is the Fourier transform of $f(t) e^{-sigma t} 1_{t > 0}$. The main reason why we look at it for every $sigma$ (instead of only one) is the analytic continuation and the tools from complex analysis. Do you know the orthogonal polynomials ? Some generalizations : the spectral theorem for compact operators in Hilbert spaces, providing an orthogonal basis where the operator acts diagonally, the diagonalization of the Laplacian in smooth manifolds, the representation theory in topological groups.
– reuns
Nov 18 at 6:45
1
en.wikipedia.org/wiki/Integral_transform#Table_of_transforms
– Rahul
Nov 18 at 7:53
1
1
$int_0^infty f(t) e^{-st}dt$ is the Fourier transform of $f(t) e^{-sigma t} 1_{t > 0}$. The main reason why we look at it for every $sigma$ (instead of only one) is the analytic continuation and the tools from complex analysis. Do you know the orthogonal polynomials ? Some generalizations : the spectral theorem for compact operators in Hilbert spaces, providing an orthogonal basis where the operator acts diagonally, the diagonalization of the Laplacian in smooth manifolds, the representation theory in topological groups.
– reuns
Nov 18 at 6:45
$int_0^infty f(t) e^{-st}dt$ is the Fourier transform of $f(t) e^{-sigma t} 1_{t > 0}$. The main reason why we look at it for every $sigma$ (instead of only one) is the analytic continuation and the tools from complex analysis. Do you know the orthogonal polynomials ? Some generalizations : the spectral theorem for compact operators in Hilbert spaces, providing an orthogonal basis where the operator acts diagonally, the diagonalization of the Laplacian in smooth manifolds, the representation theory in topological groups.
– reuns
Nov 18 at 6:45
1
1
en.wikipedia.org/wiki/Integral_transform#Table_of_transforms
– Rahul
Nov 18 at 7:53
en.wikipedia.org/wiki/Integral_transform#Table_of_transforms
– Rahul
Nov 18 at 7:53
add a comment |
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$int_0^infty f(t) e^{-st}dt$ is the Fourier transform of $f(t) e^{-sigma t} 1_{t > 0}$. The main reason why we look at it for every $sigma$ (instead of only one) is the analytic continuation and the tools from complex analysis. Do you know the orthogonal polynomials ? Some generalizations : the spectral theorem for compact operators in Hilbert spaces, providing an orthogonal basis where the operator acts diagonally, the diagonalization of the Laplacian in smooth manifolds, the representation theory in topological groups.
– reuns
Nov 18 at 6:45
1
en.wikipedia.org/wiki/Integral_transform#Table_of_transforms
– Rahul
Nov 18 at 7:53