Parseval Relation for Dilated Function on Integers
$begingroup$
Let $f:mathbb{Z}rightarrow mathbb{C}.$ If necessary assume that the support of $f$ is finite, and that $mid fmid$ is bounded. Define the fourier transform $$hat{f}:[0,1)rightarrow mathbb{C}$$ by
$$hat{f}=sum_{nin mathbb{Z}} f(n)~e^{2i pi n t}.$$
Parseval states that
$$
sum_{n in mathbb{Z}} mid f(n) mid^2 = int_0^1 midhat{f}(t) mid^2 ,dt
$$
holds. Now let $v$ be a positive integer $geq 2,$ and let
$$
f_v(n)=left{
begin{array}{ccc}
f(n/v), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
What is the Parseval relationship for this function?
Also define, with $v$ as before, the "sampled" function
$$
g_v(n)=left{
begin{array}{ccc}
f(n), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
What is the Parseval relationship for this function?
analysis fourier-analysis
asked May 8 '18 at 1:12
kodlukodlu
3,415816
$endgroup$
add a comment |
$begingroup$
Let $f:mathbb{Z}rightarrow mathbb{C}.$ If necessary assume that the support of $f$ is finite, and that $mid fmid$ is bounded. Define the fourier transform $$hat{f}:[0,1)rightarrow mathbb{C}$$ by
$$hat{f}=sum_{nin mathbb{Z}} f(n)~e^{2i pi n t}.$$
Parseval states that
$$
sum_{n in mathbb{Z}} mid f(n) mid^2 = int_0^1 midhat{f}(t) mid^2 ,dt
$$
holds. Now let $v$ be a positive integer $geq 2,$ and let
$$
f_v(n)=left{
begin{array}{ccc}
f(n/v), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
What is the Parseval relationship for this function?
Also define, with $v$ as before, the "sampled" function
$$
g_v(n)=left{
begin{array}{ccc}
f(n), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
What is the Parseval relationship for this function?
analysis fourier-analysis
asked May 8 '18 at 1:12
kodlukodlu
3,415816
$endgroup$
add a comment |
$begingroup$
Let $f:mathbb{Z}rightarrow mathbb{C}.$ If necessary assume that the support of $f$ is finite, and that $mid fmid$ is bounded. Define the fourier transform $$hat{f}:[0,1)rightarrow mathbb{C}$$ by
$$hat{f}=sum_{nin mathbb{Z}} f(n)~e^{2i pi n t}.$$
Parseval states that
$$
sum_{n in mathbb{Z}} mid f(n) mid^2 = int_0^1 midhat{f}(t) mid^2 ,dt
$$
holds. Now let $v$ be a positive integer $geq 2,$ and let
$$
f_v(n)=left{
begin{array}{ccc}
f(n/v), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
What is the Parseval relationship for this function?
Also define, with $v$ as before, the "sampled" function
$$
g_v(n)=left{
begin{array}{ccc}
f(n), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
What is the Parseval relationship for this function?
analysis fourier-analysis
asked May 8 '18 at 1:12
kodlukodlu
3,415816
$endgroup$
Let $f:mathbb{Z}rightarrow mathbb{C}.$ If necessary assume that the support of $f$ is finite, and that $mid fmid$ is bounded. Define the fourier transform $$hat{f}:[0,1)rightarrow mathbb{C}$$ by
$$hat{f}=sum_{nin mathbb{Z}} f(n)~e^{2i pi n t}.$$
Parseval states that
$$
sum_{n in mathbb{Z}} mid f(n) mid^2 = int_0^1 midhat{f}(t) mid^2 ,dt
$$
holds. Now let $v$ be a positive integer $geq 2,$ and let
$$
f_v(n)=left{
begin{array}{ccc}
f(n/v), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
What is the Parseval relationship for this function?
Also define, with $v$ as before, the "sampled" function
$$
g_v(n)=left{
begin{array}{ccc}
f(n), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
What is the Parseval relationship for this function?
analysis fourier-analysis
analysis fourier-analysis
asked May 8 '18 at 1:12
kodlukodlu
3,415816
asked May 8 '18 at 1:12
kodlukodlu
3,415816
asked May 8 '18 at 1:12
kodlukodlu
3,415816
asked May 8 '18 at 1:12
kodlukodlu
3,415816
asked May 8 '18 at 1:12
kodlukodlu
3,415816
3,415816
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This question has been answered in mathoverflow see here where you can go for the details.
Let
$$
f_v(n)=left{
begin{array}{ccc}
f(n), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
Let $f=f_1$ for simplicity's sake. Then
$$
sum_{v=1}^m sum_{n in mathbb{Z}} mid f_v(n) mid^2
geq left( ln m-frac{m}{N}right) int_0^1 midwidehat{f(t)} mid^2 ,dt.
$$
answered Jan 26 at 0:00
kodlukodlu
3,415816
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2f2771511%2fparseval-relation-for-dilated-function-on-integers%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This question has been answered in mathoverflow see here where you can go for the details.
Let
$$
f_v(n)=left{
begin{array}{ccc}
f(n), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
Let $f=f_1$ for simplicity's sake. Then
$$
sum_{v=1}^m sum_{n in mathbb{Z}} mid f_v(n) mid^2
geq left( ln m-frac{m}{N}right) int_0^1 midwidehat{f(t)} mid^2 ,dt.
$$
answered Jan 26 at 0:00
kodlukodlu
3,415816
$endgroup$
add a comment |
$begingroup$
This question has been answered in mathoverflow see here where you can go for the details.
Let
$$
f_v(n)=left{
begin{array}{ccc}
f(n), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
Let $f=f_1$ for simplicity's sake. Then
$$
sum_{v=1}^m sum_{n in mathbb{Z}} mid f_v(n) mid^2
geq left( ln m-frac{m}{N}right) int_0^1 midwidehat{f(t)} mid^2 ,dt.
$$
answered Jan 26 at 0:00
kodlukodlu
3,415816
$endgroup$
add a comment |
$begingroup$
This question has been answered in mathoverflow see here where you can go for the details.
Let
$$
f_v(n)=left{
begin{array}{ccc}
f(n), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
Let $f=f_1$ for simplicity's sake. Then
$$
sum_{v=1}^m sum_{n in mathbb{Z}} mid f_v(n) mid^2
geq left( ln m-frac{m}{N}right) int_0^1 midwidehat{f(t)} mid^2 ,dt.
$$
answered Jan 26 at 0:00
kodlukodlu
3,415816
$endgroup$
This question has been answered in mathoverflow see here where you can go for the details.
Let
$$
f_v(n)=left{
begin{array}{ccc}
f(n), & quadmathrm{if}quad & v|n,\
& & \
0 & & mathrm{otherwise}.
end{array}
right.
$$
Let $f=f_1$ for simplicity's sake. Then
$$
sum_{v=1}^m sum_{n in mathbb{Z}} mid f_v(n) mid^2
geq left( ln m-frac{m}{N}right) int_0^1 midwidehat{f(t)} mid^2 ,dt.
$$
answered Jan 26 at 0:00
kodlukodlu
3,415816
answered Jan 26 at 0:00
kodlukodlu
3,415816
answered Jan 26 at 0:00
kodlukodlu
3,415816
answered Jan 26 at 0:00
kodlukodlu
3,415816
3,415816
add a comment |
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.
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%2f2771511%2fparseval-relation-for-dilated-function-on-integers%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
