Multiplication formula for Fourier transform
Citing Stein and Weiss' "Introduction to Fourier Analysis", if $f,gin L^1$ then we have the so-called Multiplication formula:
$$ int hat f g,dx = int f,hat g, dx $$
The proof is just a couple of lines:
$$ int hat f(x) g(x),dx = intleft( int f(t)e^{-2pi i, tx},dtright) g(x), dx = \ intleft( int g(x)e^{-2pi i, tx},dxright) f(t), dt = int f(t),hat g(t), dt$$
My question is: in order to apply Fubini's theorem, shouldn't we require also that $~fgin L^1$? I don't see why the hypotheses of Fubini's theorem are met otherwise.
integration fourier-analysis lebesgue-integral
asked Nov 27 '18 at 23:07
Manlio
902719
add a comment |
Citing Stein and Weiss' "Introduction to Fourier Analysis", if $f,gin L^1$ then we have the so-called Multiplication formula:
$$ int hat f g,dx = int f,hat g, dx $$
The proof is just a couple of lines:
$$ int hat f(x) g(x),dx = intleft( int f(t)e^{-2pi i, tx},dtright) g(x), dx = \ intleft( int g(x)e^{-2pi i, tx},dxright) f(t), dt = int f(t),hat g(t), dt$$
My question is: in order to apply Fubini's theorem, shouldn't we require also that $~fgin L^1$? I don't see why the hypotheses of Fubini's theorem are met otherwise.
integration fourier-analysis lebesgue-integral
asked Nov 27 '18 at 23:07
Manlio
902719
add a comment |
Citing Stein and Weiss' "Introduction to Fourier Analysis", if $f,gin L^1$ then we have the so-called Multiplication formula:
$$ int hat f g,dx = int f,hat g, dx $$
The proof is just a couple of lines:
$$ int hat f(x) g(x),dx = intleft( int f(t)e^{-2pi i, tx},dtright) g(x), dx = \ intleft( int g(x)e^{-2pi i, tx},dxright) f(t), dt = int f(t),hat g(t), dt$$
My question is: in order to apply Fubini's theorem, shouldn't we require also that $~fgin L^1$? I don't see why the hypotheses of Fubini's theorem are met otherwise.
integration fourier-analysis lebesgue-integral
asked Nov 27 '18 at 23:07
Manlio
902719
Citing Stein and Weiss' "Introduction to Fourier Analysis", if $f,gin L^1$ then we have the so-called Multiplication formula:
$$ int hat f g,dx = int f,hat g, dx $$
The proof is just a couple of lines:
$$ int hat f(x) g(x),dx = intleft( int f(t)e^{-2pi i, tx},dtright) g(x), dx = \ intleft( int g(x)e^{-2pi i, tx},dxright) f(t), dt = int f(t),hat g(t), dt$$
My question is: in order to apply Fubini's theorem, shouldn't we require also that $~fgin L^1$? I don't see why the hypotheses of Fubini's theorem are met otherwise.
integration fourier-analysis lebesgue-integral
integration fourier-analysis lebesgue-integral
asked Nov 27 '18 at 23:07
Manlio
902719
asked Nov 27 '18 at 23:07
Manlio
902719
asked Nov 27 '18 at 23:07
Manlio
902719
asked Nov 27 '18 at 23:07
Manlio
902719
asked Nov 27 '18 at 23:07
Manlio
902719
902719
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$int int |f(t)g(x)|, dt, dx=(int|f(t)|, dt )(int |g(x)|, dx) <infty$ so Fubini's Theorem is applicable.
answered Nov 27 '18 at 23:12
Kavi Rama Murthy
50.4k31854
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%2f3016452%2fmultiplication-formula-for-fourier-transform%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
$int int |f(t)g(x)|, dt, dx=(int|f(t)|, dt )(int |g(x)|, dx) <infty$ so Fubini's Theorem is applicable.
answered Nov 27 '18 at 23:12
Kavi Rama Murthy
50.4k31854
add a comment |
$int int |f(t)g(x)|, dt, dx=(int|f(t)|, dt )(int |g(x)|, dx) <infty$ so Fubini's Theorem is applicable.
answered Nov 27 '18 at 23:12
Kavi Rama Murthy
50.4k31854
add a comment |
$int int |f(t)g(x)|, dt, dx=(int|f(t)|, dt )(int |g(x)|, dx) <infty$ so Fubini's Theorem is applicable.
answered Nov 27 '18 at 23:12
Kavi Rama Murthy
50.4k31854
$int int |f(t)g(x)|, dt, dx=(int|f(t)|, dt )(int |g(x)|, dx) <infty$ so Fubini's Theorem is applicable.
answered Nov 27 '18 at 23:12
Kavi Rama Murthy
50.4k31854
answered Nov 27 '18 at 23:12
Kavi Rama Murthy
50.4k31854
answered Nov 27 '18 at 23:12
Kavi Rama Murthy
50.4k31854
answered Nov 27 '18 at 23:12
Kavi Rama Murthy
50.4k31854
50.4k31854
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.
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%2f3016452%2fmultiplication-formula-for-fourier-transform%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
