Some sort of Laplace transform
Let $K$ be some compact set and $mu$ be a probability measure on $K$.
Let $f:Ktomathbb S^{d-1}$ be a continuous function, where $mathbb S^{d-1}$ is the unit sphere in $mathbb R^d$ And $g:Ktomathbb R^d$ be a measurable function. Assume that $f$ is injective.
Is it true that if, for all $yinmathbb R^d$,
$$int_K e^{y^{top} f(x)}y^top g(x) dmu(x)=0,$$
Then $g(x)=0$ $mu$-almost surely?
When $mu$ is finitely supported, it is true and quite easy to show, but I am not sure about the general case...
Thank you!
real-analysis laplace-transform
asked 2 days ago
TrivialPursuitTrivialPursuit
63
add a comment |
Let $K$ be some compact set and $mu$ be a probability measure on $K$.
Let $f:Ktomathbb S^{d-1}$ be a continuous function, where $mathbb S^{d-1}$ is the unit sphere in $mathbb R^d$ And $g:Ktomathbb R^d$ be a measurable function. Assume that $f$ is injective.
Is it true that if, for all $yinmathbb R^d$,
$$int_K e^{y^{top} f(x)}y^top g(x) dmu(x)=0,$$
Then $g(x)=0$ $mu$-almost surely?
When $mu$ is finitely supported, it is true and quite easy to show, but I am not sure about the general case...
Thank you!
real-analysis laplace-transform
asked 2 days ago
TrivialPursuitTrivialPursuit
63
add a comment |
Let $K$ be some compact set and $mu$ be a probability measure on $K$.
Let $f:Ktomathbb S^{d-1}$ be a continuous function, where $mathbb S^{d-1}$ is the unit sphere in $mathbb R^d$ And $g:Ktomathbb R^d$ be a measurable function. Assume that $f$ is injective.
Is it true that if, for all $yinmathbb R^d$,
$$int_K e^{y^{top} f(x)}y^top g(x) dmu(x)=0,$$
Then $g(x)=0$ $mu$-almost surely?
When $mu$ is finitely supported, it is true and quite easy to show, but I am not sure about the general case...
Thank you!
real-analysis laplace-transform
asked 2 days ago
TrivialPursuitTrivialPursuit
63
Let $K$ be some compact set and $mu$ be a probability measure on $K$.
Let $f:Ktomathbb S^{d-1}$ be a continuous function, where $mathbb S^{d-1}$ is the unit sphere in $mathbb R^d$ And $g:Ktomathbb R^d$ be a measurable function. Assume that $f$ is injective.
Is it true that if, for all $yinmathbb R^d$,
$$int_K e^{y^{top} f(x)}y^top g(x) dmu(x)=0,$$
Then $g(x)=0$ $mu$-almost surely?
When $mu$ is finitely supported, it is true and quite easy to show, but I am not sure about the general case...
Thank you!
real-analysis laplace-transform
real-analysis laplace-transform
asked 2 days ago
TrivialPursuitTrivialPursuit
63
asked 2 days ago
TrivialPursuitTrivialPursuit
63
asked 2 days ago
TrivialPursuitTrivialPursuit
63
asked 2 days ago
TrivialPursuitTrivialPursuit
63
asked 2 days ago
TrivialPursuitTrivialPursuit
63
63
add a comment |
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%2f3063058%2fsome-sort-of-laplace-transform%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.
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%2f3063058%2fsome-sort-of-laplace-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
