Let $fin L_2(mathbb R)$ be a function. Is $displaystylesum_kinmathbbZ|f(k)|^2<infty $?
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Let $fin L_2(mathbb R)$ be a function. Is $displaystylesum_kinmathbbZ|f(k)|^2<infty $ ?
I am trying to prove the relation $displaystylesum_kinmathbbZ|f(k)|^2<int_mathbb R|f(x)|^2dx$.
I am not able to produce examples to say above relation is false, so I guess that the above relation is true. But, I am not able to prove that.
Please help me!
analysis fourier-analysis harmonic-analysis
asked Aug 19 at 3:36
S. Pitchai Murugan
287111
add a comment |Â
up vote
0
down vote
favorite
Let $fin L_2(mathbb R)$ be a function. Is $displaystylesum_kinmathbbZ|f(k)|^2<infty $ ?
I am trying to prove the relation $displaystylesum_kinmathbbZ|f(k)|^2<int_mathbb R|f(x)|^2dx$.
I am not able to produce examples to say above relation is false, so I guess that the above relation is true. But, I am not able to prove that.
Please help me!
analysis fourier-analysis harmonic-analysis
asked Aug 19 at 3:36
S. Pitchai Murugan
287111
For a counter-example, take a function that is almost everywhere $0$...
– peter a g
Aug 19 at 3:38
You cannot expect it to be true, since the LHS only makes reference to $f$ on a null set.
– Taisuke Yasuda
Aug 19 at 3:39
nope nada null zip
– timur
Aug 19 at 3:44
If we assume $f$ is continuous on $mathbb R$, then Is my result true?
– S. Pitchai Murugan
Aug 19 at 4:30
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $fin L_2(mathbb R)$ be a function. Is $displaystylesum_kinmathbbZ|f(k)|^2<infty $ ?
I am trying to prove the relation $displaystylesum_kinmathbbZ|f(k)|^2<int_mathbb R|f(x)|^2dx$.
I am not able to produce examples to say above relation is false, so I guess that the above relation is true. But, I am not able to prove that.
Please help me!
analysis fourier-analysis harmonic-analysis
asked Aug 19 at 3:36
S. Pitchai Murugan
287111
Let $fin L_2(mathbb R)$ be a function. Is $displaystylesum_kinmathbbZ|f(k)|^2<infty $ ?
I am trying to prove the relation $displaystylesum_kinmathbbZ|f(k)|^2<int_mathbb R|f(x)|^2dx$.
I am not able to produce examples to say above relation is false, so I guess that the above relation is true. But, I am not able to prove that.
Please help me!
analysis fourier-analysis harmonic-analysis
asked Aug 19 at 3:36
S. Pitchai Murugan
287111
asked Aug 19 at 3:36
S. Pitchai Murugan
287111
asked Aug 19 at 3:36
S. Pitchai Murugan
287111
asked Aug 19 at 3:36
S. Pitchai Murugan
287111
287111
For a counter-example, take a function that is almost everywhere $0$...
– peter a g
Aug 19 at 3:38
You cannot expect it to be true, since the LHS only makes reference to $f$ on a null set.
– Taisuke Yasuda
Aug 19 at 3:39
nope nada null zip
– timur
Aug 19 at 3:44
If we assume $f$ is continuous on $mathbb R$, then Is my result true?
– S. Pitchai Murugan
Aug 19 at 4:30
add a comment |Â
For a counter-example, take a function that is almost everywhere $0$...
– peter a g
Aug 19 at 3:38
You cannot expect it to be true, since the LHS only makes reference to $f$ on a null set.
– Taisuke Yasuda
Aug 19 at 3:39
nope nada null zip
– timur
Aug 19 at 3:44
If we assume $f$ is continuous on $mathbb R$, then Is my result true?
– S. Pitchai Murugan
Aug 19 at 4:30
For a counter-example, take a function that is almost everywhere $0$...
– peter a g
Aug 19 at 3:38
For a counter-example, take a function that is almost everywhere $0$...
– peter a g
Aug 19 at 3:38
You cannot expect it to be true, since the LHS only makes reference to $f$ on a null set.
– Taisuke Yasuda
Aug 19 at 3:39
You cannot expect it to be true, since the LHS only makes reference to $f$ on a null set.
– Taisuke Yasuda
Aug 19 at 3:39
nope nada null zip
– timur
Aug 19 at 3:44
nope nada null zip
– timur
Aug 19 at 3:44
If we assume $f$ is continuous on $mathbb R$, then Is my result true?
– S. Pitchai Murugan
Aug 19 at 4:30
If we assume $f$ is continuous on $mathbb R$, then Is my result true?
– S. Pitchai Murugan
Aug 19 at 4:30
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
This result is false. Let $ f: mathbbR to mathbbR $ via $ f(x) = x chi_mathbbZ(x) $, where $ chi_mathbbZ$ is the characteristic function of the integers. Then $ f $ is zero almost everywhere, so $ f in L^2(mathbbR) $. However, $ sum_k in mathbbZ |f(k)|^2 = sum_k in mathbbZ k^2 $, which is certainly not finite.
answered Aug 19 at 3:41
user571438
9817
ok. Will it be true if we assume $f$ is continuous on $mathbb R$?
– S. Pitchai Murugan
Aug 19 at 4:28
You can still find counterexamples. In fact, I'm confident there are smooth counterexamples, but I can't think of an explicit one. For a continuous counterexample, consider the triangle function which is 0 everywhere except an interval. On that interval it rises linearly to a peak then decreases back to 0. Then let f be the sum of triangle functions with intervals around the integers. If you pick the heights and widths of these triangles appropriately, you will come up with a counterexample. This sort of counterexample is nice when you want continuity. You can try and modify it to be smooth.
– user571438
Aug 19 at 4:42
Ok. Thank you very much.
– S. Pitchai Murugan
Aug 19 at 4:46
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
This result is false. Let $ f: mathbbR to mathbbR $ via $ f(x) = x chi_mathbbZ(x) $, where $ chi_mathbbZ$ is the characteristic function of the integers. Then $ f $ is zero almost everywhere, so $ f in L^2(mathbbR) $. However, $ sum_k in mathbbZ |f(k)|^2 = sum_k in mathbbZ k^2 $, which is certainly not finite.
answered Aug 19 at 3:41
user571438
9817
ok. Will it be true if we assume $f$ is continuous on $mathbb R$?
– S. Pitchai Murugan
Aug 19 at 4:28
You can still find counterexamples. In fact, I'm confident there are smooth counterexamples, but I can't think of an explicit one. For a continuous counterexample, consider the triangle function which is 0 everywhere except an interval. On that interval it rises linearly to a peak then decreases back to 0. Then let f be the sum of triangle functions with intervals around the integers. If you pick the heights and widths of these triangles appropriately, you will come up with a counterexample. This sort of counterexample is nice when you want continuity. You can try and modify it to be smooth.
– user571438
Aug 19 at 4:42
Ok. Thank you very much.
– S. Pitchai Murugan
Aug 19 at 4:46
add a comment |Â
up vote
1
down vote
accepted
This result is false. Let $ f: mathbbR to mathbbR $ via $ f(x) = x chi_mathbbZ(x) $, where $ chi_mathbbZ$ is the characteristic function of the integers. Then $ f $ is zero almost everywhere, so $ f in L^2(mathbbR) $. However, $ sum_k in mathbbZ |f(k)|^2 = sum_k in mathbbZ k^2 $, which is certainly not finite.
answered Aug 19 at 3:41
user571438
9817
ok. Will it be true if we assume $f$ is continuous on $mathbb R$?
– S. Pitchai Murugan
Aug 19 at 4:28
You can still find counterexamples. In fact, I'm confident there are smooth counterexamples, but I can't think of an explicit one. For a continuous counterexample, consider the triangle function which is 0 everywhere except an interval. On that interval it rises linearly to a peak then decreases back to 0. Then let f be the sum of triangle functions with intervals around the integers. If you pick the heights and widths of these triangles appropriately, you will come up with a counterexample. This sort of counterexample is nice when you want continuity. You can try and modify it to be smooth.
– user571438
Aug 19 at 4:42
Ok. Thank you very much.
– S. Pitchai Murugan
Aug 19 at 4:46
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
This result is false. Let $ f: mathbbR to mathbbR $ via $ f(x) = x chi_mathbbZ(x) $, where $ chi_mathbbZ$ is the characteristic function of the integers. Then $ f $ is zero almost everywhere, so $ f in L^2(mathbbR) $. However, $ sum_k in mathbbZ |f(k)|^2 = sum_k in mathbbZ k^2 $, which is certainly not finite.
answered Aug 19 at 3:41
user571438
9817
This result is false. Let $ f: mathbbR to mathbbR $ via $ f(x) = x chi_mathbbZ(x) $, where $ chi_mathbbZ$ is the characteristic function of the integers. Then $ f $ is zero almost everywhere, so $ f in L^2(mathbbR) $. However, $ sum_k in mathbbZ |f(k)|^2 = sum_k in mathbbZ k^2 $, which is certainly not finite.
answered Aug 19 at 3:41
user571438
9817
answered Aug 19 at 3:41
user571438
9817
answered Aug 19 at 3:41
user571438
9817
answered Aug 19 at 3:41
user571438
9817
9817
ok. Will it be true if we assume $f$ is continuous on $mathbb R$?
– S. Pitchai Murugan
Aug 19 at 4:28
You can still find counterexamples. In fact, I'm confident there are smooth counterexamples, but I can't think of an explicit one. For a continuous counterexample, consider the triangle function which is 0 everywhere except an interval. On that interval it rises linearly to a peak then decreases back to 0. Then let f be the sum of triangle functions with intervals around the integers. If you pick the heights and widths of these triangles appropriately, you will come up with a counterexample. This sort of counterexample is nice when you want continuity. You can try and modify it to be smooth.
– user571438
Aug 19 at 4:42
Ok. Thank you very much.
– S. Pitchai Murugan
Aug 19 at 4:46
add a comment |Â
ok. Will it be true if we assume $f$ is continuous on $mathbb R$?
– S. Pitchai Murugan
Aug 19 at 4:28
You can still find counterexamples. In fact, I'm confident there are smooth counterexamples, but I can't think of an explicit one. For a continuous counterexample, consider the triangle function which is 0 everywhere except an interval. On that interval it rises linearly to a peak then decreases back to 0. Then let f be the sum of triangle functions with intervals around the integers. If you pick the heights and widths of these triangles appropriately, you will come up with a counterexample. This sort of counterexample is nice when you want continuity. You can try and modify it to be smooth.
– user571438
Aug 19 at 4:42
Ok. Thank you very much.
– S. Pitchai Murugan
Aug 19 at 4:46
ok. Will it be true if we assume $f$ is continuous on $mathbb R$?
– S. Pitchai Murugan
Aug 19 at 4:28
ok. Will it be true if we assume $f$ is continuous on $mathbb R$?
– S. Pitchai Murugan
Aug 19 at 4:28
You can still find counterexamples. In fact, I'm confident there are smooth counterexamples, but I can't think of an explicit one. For a continuous counterexample, consider the triangle function which is 0 everywhere except an interval. On that interval it rises linearly to a peak then decreases back to 0. Then let f be the sum of triangle functions with intervals around the integers. If you pick the heights and widths of these triangles appropriately, you will come up with a counterexample. This sort of counterexample is nice when you want continuity. You can try and modify it to be smooth.
– user571438
Aug 19 at 4:42
You can still find counterexamples. In fact, I'm confident there are smooth counterexamples, but I can't think of an explicit one. For a continuous counterexample, consider the triangle function which is 0 everywhere except an interval. On that interval it rises linearly to a peak then decreases back to 0. Then let f be the sum of triangle functions with intervals around the integers. If you pick the heights and widths of these triangles appropriately, you will come up with a counterexample. This sort of counterexample is nice when you want continuity. You can try and modify it to be smooth.
– user571438
Aug 19 at 4:42
Ok. Thank you very much.
– S. Pitchai Murugan
Aug 19 at 4:46
Ok. Thank you very much.
– S. Pitchai Murugan
Aug 19 at 4:46
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2887331%2flet-f-in-l-2-mathbb-r-be-a-function-is-displaystyle-sum-k-in-mathbbz%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
For a counter-example, take a function that is almost everywhere $0$...
– peter a g
Aug 19 at 3:38
You cannot expect it to be true, since the LHS only makes reference to $f$ on a null set.
– Taisuke Yasuda
Aug 19 at 3:39
nope nada null zip
– timur
Aug 19 at 3:44
If we assume $f$ is continuous on $mathbb R$, then Is my result true?
– S. Pitchai Murugan
Aug 19 at 4:30