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Parseval equality with $int |f|^1$
Clash Royale CLAN TAG#URR8PPP
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The famous Parseval equality states that
[
int|f|^2 = int |F|^2,
]
where $f$ and $F$ are related to one another by
[
int f e^-2 pi i r y dr = F(y).
]
My question is, whether there is any result for
[
int |f|
]
given its Fourier transform.
fourier-transform parsevals-identity
edited Jul 26 at 4:50
Parcly Taxel
33.5k136588
asked Jul 26 at 4:48
Grown pains
9510
 |Â
show 2 more comments
up vote
0
down vote
favorite
The famous Parseval equality states that
[
int|f|^2 = int |F|^2,
]
where $f$ and $F$ are related to one another by
[
int f e^-2 pi i r y dr = F(y).
]
My question is, whether there is any result for
[
int |f|
]
given its Fourier transform.
fourier-transform parsevals-identity
edited Jul 26 at 4:50
Parcly Taxel
33.5k136588
asked Jul 26 at 4:48
Grown pains
9510
If I remember correctly, if $f in L^2$, then the $L^2$ norm of $f$ is equal to its Fourier transform norm: math.mit.edu/~jerison/103/handouts/fourierint1.13.pdf. For your case of $e^2 pi I r y$, the constant is 1
– Raymond Chu
Jul 26 at 4:51
If $f in L^2$, then can we say anything about $int |f|$? (not the mean square)
– Grown pains
Jul 26 at 5:01
Not necessarily, there are many $f in L^2$ that are not in $L^1$. Let $f = frac1x chi_[1,infty)$ for one example
– Raymond Chu
Jul 26 at 5:13
How about for those belonging to both spaces?
– Grown pains
Jul 26 at 5:22
You don't even have inequalities of the type $int |f| leq C (int |F|^2 )^1/2$ or the reverse of this inequality.
– Kavi Rama Murthy
Jul 26 at 5:39
 |Â
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The famous Parseval equality states that
[
int|f|^2 = int |F|^2,
]
where $f$ and $F$ are related to one another by
[
int f e^-2 pi i r y dr = F(y).
]
My question is, whether there is any result for
[
int |f|
]
given its Fourier transform.
fourier-transform parsevals-identity
edited Jul 26 at 4:50
Parcly Taxel
33.5k136588
asked Jul 26 at 4:48
Grown pains
9510
The famous Parseval equality states that
[
int|f|^2 = int |F|^2,
]
where $f$ and $F$ are related to one another by
[
int f e^-2 pi i r y dr = F(y).
]
My question is, whether there is any result for
[
int |f|
]
given its Fourier transform.
fourier-transform parsevals-identity
edited Jul 26 at 4:50
Parcly Taxel
33.5k136588
asked Jul 26 at 4:48
Grown pains
9510
edited Jul 26 at 4:50
Parcly Taxel
33.5k136588
edited Jul 26 at 4:50
Parcly Taxel
33.5k136588
edited Jul 26 at 4:50
Parcly Taxel
33.5k136588
33.5k136588
asked Jul 26 at 4:48
Grown pains
9510
asked Jul 26 at 4:48
Grown pains
9510
asked Jul 26 at 4:48
Grown pains
9510
9510
If I remember correctly, if $f in L^2$, then the $L^2$ norm of $f$ is equal to its Fourier transform norm: math.mit.edu/~jerison/103/handouts/fourierint1.13.pdf. For your case of $e^2 pi I r y$, the constant is 1
– Raymond Chu
Jul 26 at 4:51
If $f in L^2$, then can we say anything about $int |f|$? (not the mean square)
– Grown pains
Jul 26 at 5:01
Not necessarily, there are many $f in L^2$ that are not in $L^1$. Let $f = frac1x chi_[1,infty)$ for one example
– Raymond Chu
Jul 26 at 5:13
How about for those belonging to both spaces?
– Grown pains
Jul 26 at 5:22
You don't even have inequalities of the type $int |f| leq C (int |F|^2 )^1/2$ or the reverse of this inequality.
– Kavi Rama Murthy
Jul 26 at 5:39
 |Â
show 2 more comments
If I remember correctly, if $f in L^2$, then the $L^2$ norm of $f$ is equal to its Fourier transform norm: math.mit.edu/~jerison/103/handouts/fourierint1.13.pdf. For your case of $e^2 pi I r y$, the constant is 1
– Raymond Chu
Jul 26 at 4:51
If $f in L^2$, then can we say anything about $int |f|$? (not the mean square)
– Grown pains
Jul 26 at 5:01
Not necessarily, there are many $f in L^2$ that are not in $L^1$. Let $f = frac1x chi_[1,infty)$ for one example
– Raymond Chu
Jul 26 at 5:13
How about for those belonging to both spaces?
– Grown pains
Jul 26 at 5:22
You don't even have inequalities of the type $int |f| leq C (int |F|^2 )^1/2$ or the reverse of this inequality.
– Kavi Rama Murthy
Jul 26 at 5:39
If I remember correctly, if $f in L^2$, then the $L^2$ norm of $f$ is equal to its Fourier transform norm: math.mit.edu/~jerison/103/handouts/fourierint1.13.pdf. For your case of $e^2 pi I r y$, the constant is 1
– Raymond Chu
Jul 26 at 4:51
If I remember correctly, if $f in L^2$, then the $L^2$ norm of $f$ is equal to its Fourier transform norm: math.mit.edu/~jerison/103/handouts/fourierint1.13.pdf. For your case of $e^2 pi I r y$, the constant is 1
– Raymond Chu
Jul 26 at 4:51
If $f in L^2$, then can we say anything about $int |f|$? (not the mean square)
– Grown pains
Jul 26 at 5:01
If $f in L^2$, then can we say anything about $int |f|$? (not the mean square)
– Grown pains
Jul 26 at 5:01
Not necessarily, there are many $f in L^2$ that are not in $L^1$. Let $f = frac1x chi_[1,infty)$ for one example
– Raymond Chu
Jul 26 at 5:13
Not necessarily, there are many $f in L^2$ that are not in $L^1$. Let $f = frac1x chi_[1,infty)$ for one example
– Raymond Chu
Jul 26 at 5:13
How about for those belonging to both spaces?
– Grown pains
Jul 26 at 5:22
How about for those belonging to both spaces?
– Grown pains
Jul 26 at 5:22
You don't even have inequalities of the type $int |f| leq C (int |F|^2 )^1/2$ or the reverse of this inequality.
– Kavi Rama Murthy
Jul 26 at 5:39
You don't even have inequalities of the type $int |f| leq C (int |F|^2 )^1/2$ or the reverse of this inequality.
– Kavi Rama Murthy
Jul 26 at 5:39
 |Â
show 2 more comments
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If I remember correctly, if $f in L^2$, then the $L^2$ norm of $f$ is equal to its Fourier transform norm: math.mit.edu/~jerison/103/handouts/fourierint1.13.pdf. For your case of $e^2 pi I r y$, the constant is 1
– Raymond Chu
Jul 26 at 4:51
If $f in L^2$, then can we say anything about $int |f|$? (not the mean square)
– Grown pains
Jul 26 at 5:01
Not necessarily, there are many $f in L^2$ that are not in $L^1$. Let $f = frac1x chi_[1,infty)$ for one example
– Raymond Chu
Jul 26 at 5:13
How about for those belonging to both spaces?
– Grown pains
Jul 26 at 5:22
You don't even have inequalities of the type $int |f| leq C (int |F|^2 )^1/2$ or the reverse of this inequality.
– Kavi Rama Murthy
Jul 26 at 5:39