How do I show this equality using Plancherel's theorem in this paper?
$begingroup$
I refer to the paper written by T. Ozawa, K. M. Rogers, https://link.springer.com/article/10.1007/s11854-013-0031-0
I have 2 questions
In the proof of lemma 2.1,
first,
I really don't know how Plancherel's theorem is worked.
And second,
Why are blue line necessary?
I have spent lots of days.
The most plausible idea(only my opinion) is, the $D^{-s}e^{-itD^a}f(x)$ is the Fourier transform of $int_{mathbb{S}^{d-1}} r^{frac{d-a-s}{a} hat{f}(r^{1/a}omega})e^{ir^{1/a}omegacdot x}domega$ with respect to $t$ because of $e^{-itr}$.
But I think that is not true since $t$ and $r$ are defined $mathbb{R}$ and $mathbb{R}^+$, respectively.
Please, I am begging you. And I apologize about my poor English skill.
fourier-transform harmonic-analysis
asked Jan 12 at 19:00
IdkwhatIdkwhat
236
$endgroup$
add a comment |
$begingroup$
I refer to the paper written by T. Ozawa, K. M. Rogers, https://link.springer.com/article/10.1007/s11854-013-0031-0
I have 2 questions
In the proof of lemma 2.1,
first,
I really don't know how Plancherel's theorem is worked.
And second,
Why are blue line necessary?
I have spent lots of days.
The most plausible idea(only my opinion) is, the $D^{-s}e^{-itD^a}f(x)$ is the Fourier transform of $int_{mathbb{S}^{d-1}} r^{frac{d-a-s}{a} hat{f}(r^{1/a}omega})e^{ir^{1/a}omegacdot x}domega$ with respect to $t$ because of $e^{-itr}$.
But I think that is not true since $t$ and $r$ are defined $mathbb{R}$ and $mathbb{R}^+$, respectively.
Please, I am begging you. And I apologize about my poor English skill.
fourier-transform harmonic-analysis
asked Jan 12 at 19:00
IdkwhatIdkwhat
236
$endgroup$
$begingroup$
Just write to the authors. They may remember what they wrote.
$endgroup$
– ablmf
Jan 12 at 19:08
add a comment |
$begingroup$
I refer to the paper written by T. Ozawa, K. M. Rogers, https://link.springer.com/article/10.1007/s11854-013-0031-0
I have 2 questions
In the proof of lemma 2.1,
first,
I really don't know how Plancherel's theorem is worked.
And second,
Why are blue line necessary?
I have spent lots of days.
The most plausible idea(only my opinion) is, the $D^{-s}e^{-itD^a}f(x)$ is the Fourier transform of $int_{mathbb{S}^{d-1}} r^{frac{d-a-s}{a} hat{f}(r^{1/a}omega})e^{ir^{1/a}omegacdot x}domega$ with respect to $t$ because of $e^{-itr}$.
But I think that is not true since $t$ and $r$ are defined $mathbb{R}$ and $mathbb{R}^+$, respectively.
Please, I am begging you. And I apologize about my poor English skill.
fourier-transform harmonic-analysis
asked Jan 12 at 19:00
IdkwhatIdkwhat
236
$endgroup$
I refer to the paper written by T. Ozawa, K. M. Rogers, https://link.springer.com/article/10.1007/s11854-013-0031-0
I have 2 questions
In the proof of lemma 2.1,
first,
I really don't know how Plancherel's theorem is worked.
And second,
Why are blue line necessary?
I have spent lots of days.
The most plausible idea(only my opinion) is, the $D^{-s}e^{-itD^a}f(x)$ is the Fourier transform of $int_{mathbb{S}^{d-1}} r^{frac{d-a-s}{a} hat{f}(r^{1/a}omega})e^{ir^{1/a}omegacdot x}domega$ with respect to $t$ because of $e^{-itr}$.
But I think that is not true since $t$ and $r$ are defined $mathbb{R}$ and $mathbb{R}^+$, respectively.
Please, I am begging you. And I apologize about my poor English skill.
fourier-transform harmonic-analysis
fourier-transform harmonic-analysis
asked Jan 12 at 19:00
IdkwhatIdkwhat
236
asked Jan 12 at 19:00
IdkwhatIdkwhat
236
edited Jan 12 at 19:18
Idkwhat
asked Jan 12 at 19:00
IdkwhatIdkwhat
236
asked Jan 12 at 19:00
IdkwhatIdkwhat
236
asked Jan 12 at 19:00
IdkwhatIdkwhat
236
236
$begingroup$
Just write to the authors. They may remember what they wrote.
$endgroup$
– ablmf
Jan 12 at 19:08
add a comment |
$begingroup$
Just write to the authors. They may remember what they wrote.
$endgroup$
– ablmf
Jan 12 at 19:08
$begingroup$
Just write to the authors. They may remember what they wrote.
$endgroup$
– ablmf
Jan 12 at 19:08
$begingroup$
Just write to the authors. They may remember what they wrote.
$endgroup$
– ablmf
Jan 12 at 19:08
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%2f3071255%2fhow-do-i-show-this-equality-using-plancherels-theorem-in-this-paper%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%2f3071255%2fhow-do-i-show-this-equality-using-plancherels-theorem-in-this-paper%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
$begingroup$
Just write to the authors. They may remember what they wrote.
$endgroup$
– ablmf
Jan 12 at 19:08