Convergence in Schwartz Space of Fourier transforms?
$begingroup$
Let $f_n$ a sequence in Schwartz space. Suppose that $lim _{ nrightarrow infty }{ { left| { f }_{ n }-f right| }_{ alpha ,0 }=0 } $, where ${ left| f right| }_{ alpha ,0 }=underset { xin R }{ sup } { left| x right| }^{ alpha }left| fleft( x right) right|$. Is it true that its Fourier transforms converge in the same sense, in others words $lim _{ nrightarrow infty }{ { left| widehat { { f }_{ n }-f } right| }_{ alpha ,0 }= } 0$ or not?
convergence fourier-transform schwartz-space
edited yesterday
Davide Giraudo
126k16150261
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
$endgroup$
add a comment |
$begingroup$
Let $f_n$ a sequence in Schwartz space. Suppose that $lim _{ nrightarrow infty }{ { left| { f }_{ n }-f right| }_{ alpha ,0 }=0 } $, where ${ left| f right| }_{ alpha ,0 }=underset { xin R }{ sup } { left| x right| }^{ alpha }left| fleft( x right) right|$. Is it true that its Fourier transforms converge in the same sense, in others words $lim _{ nrightarrow infty }{ { left| widehat { { f }_{ n }-f } right| }_{ alpha ,0 }= } 0$ or not?
convergence fourier-transform schwartz-space
edited yesterday
Davide Giraudo
126k16150261
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
$endgroup$
1
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51
add a comment |
$begingroup$
Let $f_n$ a sequence in Schwartz space. Suppose that $lim _{ nrightarrow infty }{ { left| { f }_{ n }-f right| }_{ alpha ,0 }=0 } $, where ${ left| f right| }_{ alpha ,0 }=underset { xin R }{ sup } { left| x right| }^{ alpha }left| fleft( x right) right|$. Is it true that its Fourier transforms converge in the same sense, in others words $lim _{ nrightarrow infty }{ { left| widehat { { f }_{ n }-f } right| }_{ alpha ,0 }= } 0$ or not?
convergence fourier-transform schwartz-space
edited yesterday
Davide Giraudo
126k16150261
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
$endgroup$
Let $f_n$ a sequence in Schwartz space. Suppose that $lim _{ nrightarrow infty }{ { left| { f }_{ n }-f right| }_{ alpha ,0 }=0 } $, where ${ left| f right| }_{ alpha ,0 }=underset { xin R }{ sup } { left| x right| }^{ alpha }left| fleft( x right) right|$. Is it true that its Fourier transforms converge in the same sense, in others words $lim _{ nrightarrow infty }{ { left| widehat { { f }_{ n }-f } right| }_{ alpha ,0 }= } 0$ or not?
convergence fourier-transform schwartz-space
convergence fourier-transform schwartz-space
edited yesterday
Davide Giraudo
126k16150261
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
edited yesterday
Davide Giraudo
126k16150261
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
edited yesterday
Davide Giraudo
126k16150261
edited yesterday
Davide Giraudo
126k16150261
edited yesterday
Davide Giraudo
126k16150261
126k16150261
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
254
1
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51
add a comment |
1
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51
1
1
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51
add a comment |
0
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',
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%2f3065757%2fconvergence-in-schwartz-space-of-fourier-transforms%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
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.
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%2f3065757%2fconvergence-in-schwartz-space-of-fourier-transforms%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
1
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51