Fourier coefficients of a Holder continous function
up vote
1
down vote
favorite
I'm trying to do an exercise in Pinsky's "Introduction to Fourier analysis and wavelets":
Suppose that $f$ satisfies $L^{2}$ Holder condition with $alpha=1$. Prove that $sum_{nin mathbb{Z}} |n|^{2}|hat{f}(n)|^{2}<infty$.
The author suggests applying Fatou's lemma to the fomula:
$$||f_{h}-f||_{2}^{2}=sum_{nin mathbb{Z}}|e^{inh}-1|^{2}|hat{f}(n)|^{2},$$
here $||f_{h}-f||_{2}^{2}=int_{0}^{2pi}|f(x+h)-f(x)|^{2}dx$ and the $L^{2}$ Holder condition means $||f_{h}-f||_{2}^{2}le Kh^{alpha}$. However I don't know how to use this hint. Can anyone help me? Thanks a lot.
fourier-analysis fourier-series harmonic-analysis
asked Nov 25 at 11:49
vutuanhien
518
add a comment |
up vote
1
down vote
favorite
I'm trying to do an exercise in Pinsky's "Introduction to Fourier analysis and wavelets":
Suppose that $f$ satisfies $L^{2}$ Holder condition with $alpha=1$. Prove that $sum_{nin mathbb{Z}} |n|^{2}|hat{f}(n)|^{2}<infty$.
The author suggests applying Fatou's lemma to the fomula:
$$||f_{h}-f||_{2}^{2}=sum_{nin mathbb{Z}}|e^{inh}-1|^{2}|hat{f}(n)|^{2},$$
here $||f_{h}-f||_{2}^{2}=int_{0}^{2pi}|f(x+h)-f(x)|^{2}dx$ and the $L^{2}$ Holder condition means $||f_{h}-f||_{2}^{2}le Kh^{alpha}$. However I don't know how to use this hint. Can anyone help me? Thanks a lot.
fourier-analysis fourier-series harmonic-analysis
asked Nov 25 at 11:49
vutuanhien
518
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm trying to do an exercise in Pinsky's "Introduction to Fourier analysis and wavelets":
Suppose that $f$ satisfies $L^{2}$ Holder condition with $alpha=1$. Prove that $sum_{nin mathbb{Z}} |n|^{2}|hat{f}(n)|^{2}<infty$.
The author suggests applying Fatou's lemma to the fomula:
$$||f_{h}-f||_{2}^{2}=sum_{nin mathbb{Z}}|e^{inh}-1|^{2}|hat{f}(n)|^{2},$$
here $||f_{h}-f||_{2}^{2}=int_{0}^{2pi}|f(x+h)-f(x)|^{2}dx$ and the $L^{2}$ Holder condition means $||f_{h}-f||_{2}^{2}le Kh^{alpha}$. However I don't know how to use this hint. Can anyone help me? Thanks a lot.
fourier-analysis fourier-series harmonic-analysis
asked Nov 25 at 11:49
vutuanhien
518
I'm trying to do an exercise in Pinsky's "Introduction to Fourier analysis and wavelets":
Suppose that $f$ satisfies $L^{2}$ Holder condition with $alpha=1$. Prove that $sum_{nin mathbb{Z}} |n|^{2}|hat{f}(n)|^{2}<infty$.
The author suggests applying Fatou's lemma to the fomula:
$$||f_{h}-f||_{2}^{2}=sum_{nin mathbb{Z}}|e^{inh}-1|^{2}|hat{f}(n)|^{2},$$
here $||f_{h}-f||_{2}^{2}=int_{0}^{2pi}|f(x+h)-f(x)|^{2}dx$ and the $L^{2}$ Holder condition means $||f_{h}-f||_{2}^{2}le Kh^{alpha}$. However I don't know how to use this hint. Can anyone help me? Thanks a lot.
fourier-analysis fourier-series harmonic-analysis
fourier-analysis fourier-series harmonic-analysis
asked Nov 25 at 11:49
vutuanhien
518
asked Nov 25 at 11:49
vutuanhien
518
asked Nov 25 at 11:49
vutuanhien
518
asked Nov 25 at 11:49
vutuanhien
518
asked Nov 25 at 11:49
vutuanhien
518
518
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
I think you have stated Holder continuity wrongly. In the definition you have $|f_h-f|_2 leq Kh^{alpha}$ . (No square on the left). Hence We have $sum |frac {e^{inh}-1} h|^{2} |hat {f} (n)|^{2} leq K$. Now apply Fatou's Lemma. [$frac {e^{inh}-1} h to in$ as $h to 0$].
answered Nov 25 at 12:15
Kavi Rama Murthy
45.8k31853
Oh yes, it should be $||f_{h}-f||_{2}leq Kh^{alpha}$. But I only know Fatou's lemma for integral, can you tell me what is Fatou's lemma for series?
– vutuanhien
Nov 25 at 13:01
1
@vutuanhien Any infinite sum is an integral w.r.t. counting measure, so Fatou's Lemma applies to series. ($sum a_n = int f dmu$ where $f:mathbb N to mathbb R$ is defined by $f(n)=a_n$ and $mu (E)$ is the number of points of $E$ for any $E subset mathbb N$).
– Kavi Rama Murthy
Nov 25 at 23:14
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
I think you have stated Holder continuity wrongly. In the definition you have $|f_h-f|_2 leq Kh^{alpha}$ . (No square on the left). Hence We have $sum |frac {e^{inh}-1} h|^{2} |hat {f} (n)|^{2} leq K$. Now apply Fatou's Lemma. [$frac {e^{inh}-1} h to in$ as $h to 0$].
answered Nov 25 at 12:15
Kavi Rama Murthy
45.8k31853
Oh yes, it should be $||f_{h}-f||_{2}leq Kh^{alpha}$. But I only know Fatou's lemma for integral, can you tell me what is Fatou's lemma for series?
– vutuanhien
Nov 25 at 13:01
1
@vutuanhien Any infinite sum is an integral w.r.t. counting measure, so Fatou's Lemma applies to series. ($sum a_n = int f dmu$ where $f:mathbb N to mathbb R$ is defined by $f(n)=a_n$ and $mu (E)$ is the number of points of $E$ for any $E subset mathbb N$).
– Kavi Rama Murthy
Nov 25 at 23:14
add a comment |
up vote
1
down vote
I think you have stated Holder continuity wrongly. In the definition you have $|f_h-f|_2 leq Kh^{alpha}$ . (No square on the left). Hence We have $sum |frac {e^{inh}-1} h|^{2} |hat {f} (n)|^{2} leq K$. Now apply Fatou's Lemma. [$frac {e^{inh}-1} h to in$ as $h to 0$].
answered Nov 25 at 12:15
Kavi Rama Murthy
45.8k31853
Oh yes, it should be $||f_{h}-f||_{2}leq Kh^{alpha}$. But I only know Fatou's lemma for integral, can you tell me what is Fatou's lemma for series?
– vutuanhien
Nov 25 at 13:01
1
@vutuanhien Any infinite sum is an integral w.r.t. counting measure, so Fatou's Lemma applies to series. ($sum a_n = int f dmu$ where $f:mathbb N to mathbb R$ is defined by $f(n)=a_n$ and $mu (E)$ is the number of points of $E$ for any $E subset mathbb N$).
– Kavi Rama Murthy
Nov 25 at 23:14
add a comment |
up vote
1
down vote
up vote
1
down vote
I think you have stated Holder continuity wrongly. In the definition you have $|f_h-f|_2 leq Kh^{alpha}$ . (No square on the left). Hence We have $sum |frac {e^{inh}-1} h|^{2} |hat {f} (n)|^{2} leq K$. Now apply Fatou's Lemma. [$frac {e^{inh}-1} h to in$ as $h to 0$].
answered Nov 25 at 12:15
Kavi Rama Murthy
45.8k31853
I think you have stated Holder continuity wrongly. In the definition you have $|f_h-f|_2 leq Kh^{alpha}$ . (No square on the left). Hence We have $sum |frac {e^{inh}-1} h|^{2} |hat {f} (n)|^{2} leq K$. Now apply Fatou's Lemma. [$frac {e^{inh}-1} h to in$ as $h to 0$].
answered Nov 25 at 12:15
Kavi Rama Murthy
45.8k31853
answered Nov 25 at 12:15
Kavi Rama Murthy
45.8k31853
answered Nov 25 at 12:15
Kavi Rama Murthy
45.8k31853
answered Nov 25 at 12:15
Kavi Rama Murthy
45.8k31853
45.8k31853
Oh yes, it should be $||f_{h}-f||_{2}leq Kh^{alpha}$. But I only know Fatou's lemma for integral, can you tell me what is Fatou's lemma for series?
– vutuanhien
Nov 25 at 13:01
1
@vutuanhien Any infinite sum is an integral w.r.t. counting measure, so Fatou's Lemma applies to series. ($sum a_n = int f dmu$ where $f:mathbb N to mathbb R$ is defined by $f(n)=a_n$ and $mu (E)$ is the number of points of $E$ for any $E subset mathbb N$).
– Kavi Rama Murthy
Nov 25 at 23:14
add a comment |
Oh yes, it should be $||f_{h}-f||_{2}leq Kh^{alpha}$. But I only know Fatou's lemma for integral, can you tell me what is Fatou's lemma for series?
– vutuanhien
Nov 25 at 13:01
1
@vutuanhien Any infinite sum is an integral w.r.t. counting measure, so Fatou's Lemma applies to series. ($sum a_n = int f dmu$ where $f:mathbb N to mathbb R$ is defined by $f(n)=a_n$ and $mu (E)$ is the number of points of $E$ for any $E subset mathbb N$).
– Kavi Rama Murthy
Nov 25 at 23:14
Oh yes, it should be $||f_{h}-f||_{2}leq Kh^{alpha}$. But I only know Fatou's lemma for integral, can you tell me what is Fatou's lemma for series?
– vutuanhien
Nov 25 at 13:01
Oh yes, it should be $||f_{h}-f||_{2}leq Kh^{alpha}$. But I only know Fatou's lemma for integral, can you tell me what is Fatou's lemma for series?
– vutuanhien
Nov 25 at 13:01
1
1
@vutuanhien Any infinite sum is an integral w.r.t. counting measure, so Fatou's Lemma applies to series. ($sum a_n = int f dmu$ where $f:mathbb N to mathbb R$ is defined by $f(n)=a_n$ and $mu (E)$ is the number of points of $E$ for any $E subset mathbb N$).
– Kavi Rama Murthy
Nov 25 at 23:14
@vutuanhien Any infinite sum is an integral w.r.t. counting measure, so Fatou's Lemma applies to series. ($sum a_n = int f dmu$ where $f:mathbb N to mathbb R$ is defined by $f(n)=a_n$ and $mu (E)$ is the number of points of $E$ for any $E subset mathbb N$).
– Kavi Rama Murthy
Nov 25 at 23:14
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012736%2ffourier-coefficients-of-a-holder-continous-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
