Sturm - Liouville Parseval proof
$begingroup$
Consider the expansion $𝑓(𝑥)=sum_{n=0}^{infty}A_𝑛phi_𝑛(𝑥)$
where $phi_𝑛(𝑥)$ is the set of eigenfunctions of the Sturm-Liouville problem, and $𝑓(𝑥)$ is any piecewise differentiable function on the interval $(𝑎,𝑏)$. Show that $int_{a}^{b}[𝑓(𝑥)]^2𝑤(𝑥)𝑑𝑥=sum_{n=0}^{infty}𝐴_𝑛^2𝑁_𝑛$
where $w(x)$ is weight function and $𝑁_𝑛$ is the normalization integrals of the eigenfunctions $phi_𝑛(𝑥)$. $(𝑁𝑛=〈phi_𝑛(𝑥),phi_𝑛(𝑥)〉)$
sturm-liouville parsevals-identity
asked Jan 8 at 23:15
SquanchSquanch
405
$endgroup$
add a comment |
$begingroup$
Consider the expansion $𝑓(𝑥)=sum_{n=0}^{infty}A_𝑛phi_𝑛(𝑥)$
where $phi_𝑛(𝑥)$ is the set of eigenfunctions of the Sturm-Liouville problem, and $𝑓(𝑥)$ is any piecewise differentiable function on the interval $(𝑎,𝑏)$. Show that $int_{a}^{b}[𝑓(𝑥)]^2𝑤(𝑥)𝑑𝑥=sum_{n=0}^{infty}𝐴_𝑛^2𝑁_𝑛$
where $w(x)$ is weight function and $𝑁_𝑛$ is the normalization integrals of the eigenfunctions $phi_𝑛(𝑥)$. $(𝑁𝑛=〈phi_𝑛(𝑥),phi_𝑛(𝑥)〉)$
sturm-liouville parsevals-identity
asked Jan 8 at 23:15
SquanchSquanch
405
$endgroup$
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44
add a comment |
$begingroup$
Consider the expansion $𝑓(𝑥)=sum_{n=0}^{infty}A_𝑛phi_𝑛(𝑥)$
where $phi_𝑛(𝑥)$ is the set of eigenfunctions of the Sturm-Liouville problem, and $𝑓(𝑥)$ is any piecewise differentiable function on the interval $(𝑎,𝑏)$. Show that $int_{a}^{b}[𝑓(𝑥)]^2𝑤(𝑥)𝑑𝑥=sum_{n=0}^{infty}𝐴_𝑛^2𝑁_𝑛$
where $w(x)$ is weight function and $𝑁_𝑛$ is the normalization integrals of the eigenfunctions $phi_𝑛(𝑥)$. $(𝑁𝑛=〈phi_𝑛(𝑥),phi_𝑛(𝑥)〉)$
sturm-liouville parsevals-identity
asked Jan 8 at 23:15
SquanchSquanch
405
$endgroup$
Consider the expansion $𝑓(𝑥)=sum_{n=0}^{infty}A_𝑛phi_𝑛(𝑥)$
where $phi_𝑛(𝑥)$ is the set of eigenfunctions of the Sturm-Liouville problem, and $𝑓(𝑥)$ is any piecewise differentiable function on the interval $(𝑎,𝑏)$. Show that $int_{a}^{b}[𝑓(𝑥)]^2𝑤(𝑥)𝑑𝑥=sum_{n=0}^{infty}𝐴_𝑛^2𝑁_𝑛$
where $w(x)$ is weight function and $𝑁_𝑛$ is the normalization integrals of the eigenfunctions $phi_𝑛(𝑥)$. $(𝑁𝑛=〈phi_𝑛(𝑥),phi_𝑛(𝑥)〉)$
sturm-liouville parsevals-identity
sturm-liouville parsevals-identity
asked Jan 8 at 23:15
SquanchSquanch
405
asked Jan 8 at 23:15
SquanchSquanch
405
asked Jan 8 at 23:15
SquanchSquanch
405
asked Jan 8 at 23:15
SquanchSquanch
405
asked Jan 8 at 23:15
SquanchSquanch
405
405
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44
add a comment |
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44
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%2f3066851%2fsturm-liouville-parseval-proof%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%2f3066851%2fsturm-liouville-parseval-proof%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
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44