fourier transformation and differential equation
up vote
1
down vote
favorite
I already asked a very similar question but I did not get any satisfying answer. Now I will ask the question a little different and explain my problem a little bit.
We have given $f:mathbb{R}^d rightarrow mathbb{R}$ with $f(x) = e^{-frac{1}{2}mid xmid^2}$.
Show that the fourier trafsform $hat f $ is given with $(sqrt{2 pi})^d f$.
So we actually dont know anything about the fourier transformation. We had a task last week which introduces the fourier tranform of a function but we wont talk about this topic in class. Its just an excursion.
We defined the fourier transform of $f in L^1(mathbb{R^d,mathbb{C}})$ as
$hat f(xi) := displaystyleint_{mathbb{R}^d}f(x)e^{-ixi cdot x} mu(dx)$
also we know we following:
$partial_jhat f(xi) = -idisplaystyleint_mathbb{R^d} x_j f(x)e^{-ixi cdot x} mu(dx)$ if $(x_1,...,x_d) mapsto x_j f(x)$ is integrable for every $j in lbrace 1,...,d rbrace$.
Thats everything we know about the fourier transform. There is no other thing we know. We even didnt know whats the purpose of this transformation. Its just a training task for multidimensional integration.
So back to the task. What I shell do is to take a look at $d = 1$ first by constructing a ordinary differential equation for $hat f$ which provides the solution $hat f = sqrt{2 pi} f$.
I literally got stuck there. The last time I asked this question with less detail one of the answers solved the integral for $hat f$ but using a method from complex analysis which Im not allowed to use.
calculus differential-equations lebesgue-integral fourier-transform
asked Nov 22 at 20:26
Arjihad
378111
add a comment |
up vote
1
down vote
favorite
I already asked a very similar question but I did not get any satisfying answer. Now I will ask the question a little different and explain my problem a little bit.
We have given $f:mathbb{R}^d rightarrow mathbb{R}$ with $f(x) = e^{-frac{1}{2}mid xmid^2}$.
Show that the fourier trafsform $hat f $ is given with $(sqrt{2 pi})^d f$.
So we actually dont know anything about the fourier transformation. We had a task last week which introduces the fourier tranform of a function but we wont talk about this topic in class. Its just an excursion.
We defined the fourier transform of $f in L^1(mathbb{R^d,mathbb{C}})$ as
$hat f(xi) := displaystyleint_{mathbb{R}^d}f(x)e^{-ixi cdot x} mu(dx)$
also we know we following:
$partial_jhat f(xi) = -idisplaystyleint_mathbb{R^d} x_j f(x)e^{-ixi cdot x} mu(dx)$ if $(x_1,...,x_d) mapsto x_j f(x)$ is integrable for every $j in lbrace 1,...,d rbrace$.
Thats everything we know about the fourier transform. There is no other thing we know. We even didnt know whats the purpose of this transformation. Its just a training task for multidimensional integration.
So back to the task. What I shell do is to take a look at $d = 1$ first by constructing a ordinary differential equation for $hat f$ which provides the solution $hat f = sqrt{2 pi} f$.
I literally got stuck there. The last time I asked this question with less detail one of the answers solved the integral for $hat f$ but using a method from complex analysis which Im not allowed to use.
calculus differential-equations lebesgue-integral fourier-transform
asked Nov 22 at 20:26
Arjihad
378111
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I already asked a very similar question but I did not get any satisfying answer. Now I will ask the question a little different and explain my problem a little bit.
We have given $f:mathbb{R}^d rightarrow mathbb{R}$ with $f(x) = e^{-frac{1}{2}mid xmid^2}$.
Show that the fourier trafsform $hat f $ is given with $(sqrt{2 pi})^d f$.
So we actually dont know anything about the fourier transformation. We had a task last week which introduces the fourier tranform of a function but we wont talk about this topic in class. Its just an excursion.
We defined the fourier transform of $f in L^1(mathbb{R^d,mathbb{C}})$ as
$hat f(xi) := displaystyleint_{mathbb{R}^d}f(x)e^{-ixi cdot x} mu(dx)$
also we know we following:
$partial_jhat f(xi) = -idisplaystyleint_mathbb{R^d} x_j f(x)e^{-ixi cdot x} mu(dx)$ if $(x_1,...,x_d) mapsto x_j f(x)$ is integrable for every $j in lbrace 1,...,d rbrace$.
Thats everything we know about the fourier transform. There is no other thing we know. We even didnt know whats the purpose of this transformation. Its just a training task for multidimensional integration.
So back to the task. What I shell do is to take a look at $d = 1$ first by constructing a ordinary differential equation for $hat f$ which provides the solution $hat f = sqrt{2 pi} f$.
I literally got stuck there. The last time I asked this question with less detail one of the answers solved the integral for $hat f$ but using a method from complex analysis which Im not allowed to use.
calculus differential-equations lebesgue-integral fourier-transform
asked Nov 22 at 20:26
Arjihad
378111
I already asked a very similar question but I did not get any satisfying answer. Now I will ask the question a little different and explain my problem a little bit.
We have given $f:mathbb{R}^d rightarrow mathbb{R}$ with $f(x) = e^{-frac{1}{2}mid xmid^2}$.
Show that the fourier trafsform $hat f $ is given with $(sqrt{2 pi})^d f$.
So we actually dont know anything about the fourier transformation. We had a task last week which introduces the fourier tranform of a function but we wont talk about this topic in class. Its just an excursion.
We defined the fourier transform of $f in L^1(mathbb{R^d,mathbb{C}})$ as
$hat f(xi) := displaystyleint_{mathbb{R}^d}f(x)e^{-ixi cdot x} mu(dx)$
also we know we following:
$partial_jhat f(xi) = -idisplaystyleint_mathbb{R^d} x_j f(x)e^{-ixi cdot x} mu(dx)$ if $(x_1,...,x_d) mapsto x_j f(x)$ is integrable for every $j in lbrace 1,...,d rbrace$.
Thats everything we know about the fourier transform. There is no other thing we know. We even didnt know whats the purpose of this transformation. Its just a training task for multidimensional integration.
So back to the task. What I shell do is to take a look at $d = 1$ first by constructing a ordinary differential equation for $hat f$ which provides the solution $hat f = sqrt{2 pi} f$.
I literally got stuck there. The last time I asked this question with less detail one of the answers solved the integral for $hat f$ but using a method from complex analysis which Im not allowed to use.
calculus differential-equations lebesgue-integral fourier-transform
calculus differential-equations lebesgue-integral fourier-transform
asked Nov 22 at 20:26
Arjihad
378111
asked Nov 22 at 20:26
Arjihad
378111
asked Nov 22 at 20:26
Arjihad
378111
asked Nov 22 at 20:26
Arjihad
378111
asked Nov 22 at 20:26
Arjihad
378111
378111
add a comment |
add a comment |
active
oldest
votes
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',
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%2f3009612%2ffourier-transformation-and-differential-equation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3009612%2ffourier-transformation-and-differential-equation%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
