Hilbert space for integral
Find numbers $alpha,beta, gamma in mathbb{C}$ so that the integral $$int_{-1}^{1}|x^3-alpha-beta x-gamma x^2|^2dx$$ is minimal.
Here I want to apply Hilbert space and projection lemma by considering the integral as the distance square between a point and a closed linear subspace. So how can I start my work?
functional-analysis
add a comment |
Find numbers $alpha,beta, gamma in mathbb{C}$ so that the integral $$int_{-1}^{1}|x^3-alpha-beta x-gamma x^2|^2dx$$ is minimal.
Here I want to apply Hilbert space and projection lemma by considering the integral as the distance square between a point and a closed linear subspace. So how can I start my work?
functional-analysis
add a comment |
Find numbers $alpha,beta, gamma in mathbb{C}$ so that the integral $$int_{-1}^{1}|x^3-alpha-beta x-gamma x^2|^2dx$$ is minimal.
Here I want to apply Hilbert space and projection lemma by considering the integral as the distance square between a point and a closed linear subspace. So how can I start my work?
functional-analysis
Find numbers $alpha,beta, gamma in mathbb{C}$ so that the integral $$int_{-1}^{1}|x^3-alpha-beta x-gamma x^2|^2dx$$ is minimal.
Here I want to apply Hilbert space and projection lemma by considering the integral as the distance square between a point and a closed linear subspace. So how can I start my work?
functional-analysis
functional-analysis
asked Mar 2 '16 at 2:22
UserAbUserAb
342110
342110
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
If $mathcal{M}$ is the subspace of polynomials $p(x)=alpha + beta x + gamma x^2$, then you are looking for $pin mathcal{M}$ closest to $x^3$. Equivalently, $(x^3-p)perpmathcal{M}$, which gives equations
$$
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)dx =0 \
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)xdx = 0 \
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)x^2dx = 0.
$$
That's 3 equations in the 3 unknowns $alpha,beta,gamma$.
Another way consists of constructing an ONB ${u,v,w}$ via Gram-Schmidt from ${1,x,x^2}$ for $M$ (which is easy since $1$ and $x$ are already orthogonal) and then simply projecting $x^3$ onto $M$: $P(x^3) = langle x^3,urangle u + langle x^3,vrangle v + langle x^3,wrangle w$.
– Friedrich Philipp
Mar 2 '16 at 5:06
@FriedrichPhilipp : Not much is gained in this case. The second equation gives $beta$ directly. The other two equations involve only $alpha$, $gamma$. If you want the Gram-Schmidt version, then can also look up the normalized Legendre polynomials and use them.
– DisintegratingByParts
Mar 2 '16 at 5:18
Yes. Also, when you have the projection, you have to go back to the coefficients which means to solve a 3x3 linear system anyway.
– Friedrich Philipp
Mar 2 '16 at 5:25
add a comment |
As the Hilbert space, choose the span of the functions $x^0,x^1,x^2,x^3$ (on $(-1,1)$), equipped with the scalar product
$$
langle f,grangle = int_{-1}^1 f(x)overline{g(x)},dx.
$$
The subspace $M$ that you want to project onto should be the span of $x^0,x^1,x^2$. In this setting you are looking for that function $f_0$ in $M$ which has the smallest distance to $x^3$.
add a comment |
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%2f1679383%2fhilbert-space-for-integral%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
If $mathcal{M}$ is the subspace of polynomials $p(x)=alpha + beta x + gamma x^2$, then you are looking for $pin mathcal{M}$ closest to $x^3$. Equivalently, $(x^3-p)perpmathcal{M}$, which gives equations
$$
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)dx =0 \
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)xdx = 0 \
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)x^2dx = 0.
$$
That's 3 equations in the 3 unknowns $alpha,beta,gamma$.
Another way consists of constructing an ONB ${u,v,w}$ via Gram-Schmidt from ${1,x,x^2}$ for $M$ (which is easy since $1$ and $x$ are already orthogonal) and then simply projecting $x^3$ onto $M$: $P(x^3) = langle x^3,urangle u + langle x^3,vrangle v + langle x^3,wrangle w$.
– Friedrich Philipp
Mar 2 '16 at 5:06
@FriedrichPhilipp : Not much is gained in this case. The second equation gives $beta$ directly. The other two equations involve only $alpha$, $gamma$. If you want the Gram-Schmidt version, then can also look up the normalized Legendre polynomials and use them.
– DisintegratingByParts
Mar 2 '16 at 5:18
Yes. Also, when you have the projection, you have to go back to the coefficients which means to solve a 3x3 linear system anyway.
– Friedrich Philipp
Mar 2 '16 at 5:25
add a comment |
If $mathcal{M}$ is the subspace of polynomials $p(x)=alpha + beta x + gamma x^2$, then you are looking for $pin mathcal{M}$ closest to $x^3$. Equivalently, $(x^3-p)perpmathcal{M}$, which gives equations
$$
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)dx =0 \
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)xdx = 0 \
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)x^2dx = 0.
$$
That's 3 equations in the 3 unknowns $alpha,beta,gamma$.
Another way consists of constructing an ONB ${u,v,w}$ via Gram-Schmidt from ${1,x,x^2}$ for $M$ (which is easy since $1$ and $x$ are already orthogonal) and then simply projecting $x^3$ onto $M$: $P(x^3) = langle x^3,urangle u + langle x^3,vrangle v + langle x^3,wrangle w$.
– Friedrich Philipp
Mar 2 '16 at 5:06
@FriedrichPhilipp : Not much is gained in this case. The second equation gives $beta$ directly. The other two equations involve only $alpha$, $gamma$. If you want the Gram-Schmidt version, then can also look up the normalized Legendre polynomials and use them.
– DisintegratingByParts
Mar 2 '16 at 5:18
Yes. Also, when you have the projection, you have to go back to the coefficients which means to solve a 3x3 linear system anyway.
– Friedrich Philipp
Mar 2 '16 at 5:25
add a comment |
If $mathcal{M}$ is the subspace of polynomials $p(x)=alpha + beta x + gamma x^2$, then you are looking for $pin mathcal{M}$ closest to $x^3$. Equivalently, $(x^3-p)perpmathcal{M}$, which gives equations
$$
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)dx =0 \
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)xdx = 0 \
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)x^2dx = 0.
$$
That's 3 equations in the 3 unknowns $alpha,beta,gamma$.
If $mathcal{M}$ is the subspace of polynomials $p(x)=alpha + beta x + gamma x^2$, then you are looking for $pin mathcal{M}$ closest to $x^3$. Equivalently, $(x^3-p)perpmathcal{M}$, which gives equations
$$
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)dx =0 \
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)xdx = 0 \
int_{-1}^{1}(x^3-alpha-beta x-gamma x^2)x^2dx = 0.
$$
That's 3 equations in the 3 unknowns $alpha,beta,gamma$.
answered Mar 2 '16 at 4:47
DisintegratingByPartsDisintegratingByParts
58.7k42579
58.7k42579
Another way consists of constructing an ONB ${u,v,w}$ via Gram-Schmidt from ${1,x,x^2}$ for $M$ (which is easy since $1$ and $x$ are already orthogonal) and then simply projecting $x^3$ onto $M$: $P(x^3) = langle x^3,urangle u + langle x^3,vrangle v + langle x^3,wrangle w$.
– Friedrich Philipp
Mar 2 '16 at 5:06
@FriedrichPhilipp : Not much is gained in this case. The second equation gives $beta$ directly. The other two equations involve only $alpha$, $gamma$. If you want the Gram-Schmidt version, then can also look up the normalized Legendre polynomials and use them.
– DisintegratingByParts
Mar 2 '16 at 5:18
Yes. Also, when you have the projection, you have to go back to the coefficients which means to solve a 3x3 linear system anyway.
– Friedrich Philipp
Mar 2 '16 at 5:25
add a comment |
Another way consists of constructing an ONB ${u,v,w}$ via Gram-Schmidt from ${1,x,x^2}$ for $M$ (which is easy since $1$ and $x$ are already orthogonal) and then simply projecting $x^3$ onto $M$: $P(x^3) = langle x^3,urangle u + langle x^3,vrangle v + langle x^3,wrangle w$.
– Friedrich Philipp
Mar 2 '16 at 5:06
@FriedrichPhilipp : Not much is gained in this case. The second equation gives $beta$ directly. The other two equations involve only $alpha$, $gamma$. If you want the Gram-Schmidt version, then can also look up the normalized Legendre polynomials and use them.
– DisintegratingByParts
Mar 2 '16 at 5:18
Yes. Also, when you have the projection, you have to go back to the coefficients which means to solve a 3x3 linear system anyway.
– Friedrich Philipp
Mar 2 '16 at 5:25
Another way consists of constructing an ONB ${u,v,w}$ via Gram-Schmidt from ${1,x,x^2}$ for $M$ (which is easy since $1$ and $x$ are already orthogonal) and then simply projecting $x^3$ onto $M$: $P(x^3) = langle x^3,urangle u + langle x^3,vrangle v + langle x^3,wrangle w$.
– Friedrich Philipp
Mar 2 '16 at 5:06
Another way consists of constructing an ONB ${u,v,w}$ via Gram-Schmidt from ${1,x,x^2}$ for $M$ (which is easy since $1$ and $x$ are already orthogonal) and then simply projecting $x^3$ onto $M$: $P(x^3) = langle x^3,urangle u + langle x^3,vrangle v + langle x^3,wrangle w$.
– Friedrich Philipp
Mar 2 '16 at 5:06
@FriedrichPhilipp : Not much is gained in this case. The second equation gives $beta$ directly. The other two equations involve only $alpha$, $gamma$. If you want the Gram-Schmidt version, then can also look up the normalized Legendre polynomials and use them.
– DisintegratingByParts
Mar 2 '16 at 5:18
@FriedrichPhilipp : Not much is gained in this case. The second equation gives $beta$ directly. The other two equations involve only $alpha$, $gamma$. If you want the Gram-Schmidt version, then can also look up the normalized Legendre polynomials and use them.
– DisintegratingByParts
Mar 2 '16 at 5:18
Yes. Also, when you have the projection, you have to go back to the coefficients which means to solve a 3x3 linear system anyway.
– Friedrich Philipp
Mar 2 '16 at 5:25
Yes. Also, when you have the projection, you have to go back to the coefficients which means to solve a 3x3 linear system anyway.
– Friedrich Philipp
Mar 2 '16 at 5:25
add a comment |
As the Hilbert space, choose the span of the functions $x^0,x^1,x^2,x^3$ (on $(-1,1)$), equipped with the scalar product
$$
langle f,grangle = int_{-1}^1 f(x)overline{g(x)},dx.
$$
The subspace $M$ that you want to project onto should be the span of $x^0,x^1,x^2$. In this setting you are looking for that function $f_0$ in $M$ which has the smallest distance to $x^3$.
add a comment |
As the Hilbert space, choose the span of the functions $x^0,x^1,x^2,x^3$ (on $(-1,1)$), equipped with the scalar product
$$
langle f,grangle = int_{-1}^1 f(x)overline{g(x)},dx.
$$
The subspace $M$ that you want to project onto should be the span of $x^0,x^1,x^2$. In this setting you are looking for that function $f_0$ in $M$ which has the smallest distance to $x^3$.
add a comment |
As the Hilbert space, choose the span of the functions $x^0,x^1,x^2,x^3$ (on $(-1,1)$), equipped with the scalar product
$$
langle f,grangle = int_{-1}^1 f(x)overline{g(x)},dx.
$$
The subspace $M$ that you want to project onto should be the span of $x^0,x^1,x^2$. In this setting you are looking for that function $f_0$ in $M$ which has the smallest distance to $x^3$.
As the Hilbert space, choose the span of the functions $x^0,x^1,x^2,x^3$ (on $(-1,1)$), equipped with the scalar product
$$
langle f,grangle = int_{-1}^1 f(x)overline{g(x)},dx.
$$
The subspace $M$ that you want to project onto should be the span of $x^0,x^1,x^2$. In this setting you are looking for that function $f_0$ in $M$ which has the smallest distance to $x^3$.
answered Mar 2 '16 at 2:27
Friedrich PhilippFriedrich Philipp
3,323314
3,323314
add a comment |
add a comment |
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%2f1679383%2fhilbert-space-for-integral%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