Value of an integral depending on a function and its cosine transform
$begingroup$
Let's consider $g(x)$ a function in $C^{infty}(mathbb{R}^+ to mathbb{C})$ such that $g(x)$ is asymptotic to $x^{alpha}$ for $x$ near $+infty$ with $g(0)=0$, and $alpha$ a complex number such that $Re(alpha)<-frac{1}{2}$.
If $g$ satisfy following condition:
$$intlimits_{0}^{infty} g(x) overline{x^{alpha} } dx=0 $$
then following integral is well defined:
$$ I=intlimits_{0}^infty frac{1}{x} intlimits_{x}^{infty} g(y) ;overline{y^{alpha} }+4 ; hat{g(y)} ;overline{hat{y^{alpha} }} ; dy$$
where $hat{f}(x)$ is the cosine transform ($hat{f}(x)=intlimits_{0}^infty f(t) cos(2pi xt) dt)$
It is quite intuitive that $I$ is not zero for all $g$ functions satisfying given conditions above, but how to prove it ?
How to prove that there exists $g$ functions such that $I ne 0$ ? Is there an elegant / simple way ? Which functions can be used for a simple proof ? I took linear combinations of functions of the form $x^b e^{-x^2}$ but calculation of $I$ is not so simple ! What type of functions can be good candidates for simple demonstration ? Any trick to show there is a contradiction if we assume all $g$ funcitons give $I=0$ ?
real-analysis integration complex-analysis fourier-analysis
asked Jan 3 at 11:41
BertrandBertrand
1776
$endgroup$
add a comment |
$begingroup$
Let's consider $g(x)$ a function in $C^{infty}(mathbb{R}^+ to mathbb{C})$ such that $g(x)$ is asymptotic to $x^{alpha}$ for $x$ near $+infty$ with $g(0)=0$, and $alpha$ a complex number such that $Re(alpha)<-frac{1}{2}$.
If $g$ satisfy following condition:
$$intlimits_{0}^{infty} g(x) overline{x^{alpha} } dx=0 $$
then following integral is well defined:
$$ I=intlimits_{0}^infty frac{1}{x} intlimits_{x}^{infty} g(y) ;overline{y^{alpha} }+4 ; hat{g(y)} ;overline{hat{y^{alpha} }} ; dy$$
where $hat{f}(x)$ is the cosine transform ($hat{f}(x)=intlimits_{0}^infty f(t) cos(2pi xt) dt)$
It is quite intuitive that $I$ is not zero for all $g$ functions satisfying given conditions above, but how to prove it ?
How to prove that there exists $g$ functions such that $I ne 0$ ? Is there an elegant / simple way ? Which functions can be used for a simple proof ? I took linear combinations of functions of the form $x^b e^{-x^2}$ but calculation of $I$ is not so simple ! What type of functions can be good candidates for simple demonstration ? Any trick to show there is a contradiction if we assume all $g$ funcitons give $I=0$ ?
real-analysis integration complex-analysis fourier-analysis
asked Jan 3 at 11:41
BertrandBertrand
1776
$endgroup$
add a comment |
$begingroup$
Let's consider $g(x)$ a function in $C^{infty}(mathbb{R}^+ to mathbb{C})$ such that $g(x)$ is asymptotic to $x^{alpha}$ for $x$ near $+infty$ with $g(0)=0$, and $alpha$ a complex number such that $Re(alpha)<-frac{1}{2}$.
If $g$ satisfy following condition:
$$intlimits_{0}^{infty} g(x) overline{x^{alpha} } dx=0 $$
then following integral is well defined:
$$ I=intlimits_{0}^infty frac{1}{x} intlimits_{x}^{infty} g(y) ;overline{y^{alpha} }+4 ; hat{g(y)} ;overline{hat{y^{alpha} }} ; dy$$
where $hat{f}(x)$ is the cosine transform ($hat{f}(x)=intlimits_{0}^infty f(t) cos(2pi xt) dt)$
It is quite intuitive that $I$ is not zero for all $g$ functions satisfying given conditions above, but how to prove it ?
How to prove that there exists $g$ functions such that $I ne 0$ ? Is there an elegant / simple way ? Which functions can be used for a simple proof ? I took linear combinations of functions of the form $x^b e^{-x^2}$ but calculation of $I$ is not so simple ! What type of functions can be good candidates for simple demonstration ? Any trick to show there is a contradiction if we assume all $g$ funcitons give $I=0$ ?
real-analysis integration complex-analysis fourier-analysis
asked Jan 3 at 11:41
BertrandBertrand
1776
$endgroup$
Let's consider $g(x)$ a function in $C^{infty}(mathbb{R}^+ to mathbb{C})$ such that $g(x)$ is asymptotic to $x^{alpha}$ for $x$ near $+infty$ with $g(0)=0$, and $alpha$ a complex number such that $Re(alpha)<-frac{1}{2}$.
If $g$ satisfy following condition:
$$intlimits_{0}^{infty} g(x) overline{x^{alpha} } dx=0 $$
then following integral is well defined:
$$ I=intlimits_{0}^infty frac{1}{x} intlimits_{x}^{infty} g(y) ;overline{y^{alpha} }+4 ; hat{g(y)} ;overline{hat{y^{alpha} }} ; dy$$
where $hat{f}(x)$ is the cosine transform ($hat{f}(x)=intlimits_{0}^infty f(t) cos(2pi xt) dt)$
It is quite intuitive that $I$ is not zero for all $g$ functions satisfying given conditions above, but how to prove it ?
How to prove that there exists $g$ functions such that $I ne 0$ ? Is there an elegant / simple way ? Which functions can be used for a simple proof ? I took linear combinations of functions of the form $x^b e^{-x^2}$ but calculation of $I$ is not so simple ! What type of functions can be good candidates for simple demonstration ? Any trick to show there is a contradiction if we assume all $g$ funcitons give $I=0$ ?
real-analysis integration complex-analysis fourier-analysis
real-analysis integration complex-analysis fourier-analysis
asked Jan 3 at 11:41
BertrandBertrand
1776
asked Jan 3 at 11:41
BertrandBertrand
1776
edited Jan 3 at 12:18
Bertrand
asked Jan 3 at 11:41
BertrandBertrand
1776
asked Jan 3 at 11:41
BertrandBertrand
1776
asked Jan 3 at 11:41
BertrandBertrand
1776
1776
add a comment |
add a comment |
0
active
oldest
votes
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%2f3060475%2fvalue-of-an-integral-depending-on-a-function-and-its-cosine-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.
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%2f3060475%2fvalue-of-an-integral-depending-on-a-function-and-its-cosine-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
