Approximation of arbitrary convex function
$begingroup$
I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations". During the proof of the first lemma, he claims that an arbitrary convex function $f$ can be approximated by a sequence of twice continously differentiable convex functions $f_m$, such that $f_m''(u)=0$ for large $|u|$, $f_m rightarrow f$ and $f_m'(u) rightarrow f'(u)$, where $f'(u)$ exists.
Having looked in several books about convex analysis (Rockafellar), I found out that a convex function should be twice differentiable almost everywhere. I further found in one of Rockafellar's books (convex analysis, thm. 25.7) a Theorem that shows the claim, except for the restriction on $f''$.
Now the question is how can we come up with that restriction on $f''$ and maybe someone has a reference for a book as well, since I would be interested in seeing the proof.
ordinary-differential-equations pde convex-analysis
asked Dec 5 '18 at 11:21
MaxMax
586
$endgroup$
add a comment |
$begingroup$
I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations". During the proof of the first lemma, he claims that an arbitrary convex function $f$ can be approximated by a sequence of twice continously differentiable convex functions $f_m$, such that $f_m''(u)=0$ for large $|u|$, $f_m rightarrow f$ and $f_m'(u) rightarrow f'(u)$, where $f'(u)$ exists.
Having looked in several books about convex analysis (Rockafellar), I found out that a convex function should be twice differentiable almost everywhere. I further found in one of Rockafellar's books (convex analysis, thm. 25.7) a Theorem that shows the claim, except for the restriction on $f''$.
Now the question is how can we come up with that restriction on $f''$ and maybe someone has a reference for a book as well, since I would be interested in seeing the proof.
ordinary-differential-equations pde convex-analysis
asked Dec 5 '18 at 11:21
MaxMax
586
$endgroup$
1
$begingroup$
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 11:52
$begingroup$
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
$endgroup$
– Max
Dec 5 '18 at 15:28
$begingroup$
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
$endgroup$
– Max
Dec 5 '18 at 15:41
add a comment |
$begingroup$
I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations". During the proof of the first lemma, he claims that an arbitrary convex function $f$ can be approximated by a sequence of twice continously differentiable convex functions $f_m$, such that $f_m''(u)=0$ for large $|u|$, $f_m rightarrow f$ and $f_m'(u) rightarrow f'(u)$, where $f'(u)$ exists.
Having looked in several books about convex analysis (Rockafellar), I found out that a convex function should be twice differentiable almost everywhere. I further found in one of Rockafellar's books (convex analysis, thm. 25.7) a Theorem that shows the claim, except for the restriction on $f''$.
Now the question is how can we come up with that restriction on $f''$ and maybe someone has a reference for a book as well, since I would be interested in seeing the proof.
ordinary-differential-equations pde convex-analysis
asked Dec 5 '18 at 11:21
MaxMax
586
$endgroup$
I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations". During the proof of the first lemma, he claims that an arbitrary convex function $f$ can be approximated by a sequence of twice continously differentiable convex functions $f_m$, such that $f_m''(u)=0$ for large $|u|$, $f_m rightarrow f$ and $f_m'(u) rightarrow f'(u)$, where $f'(u)$ exists.
Having looked in several books about convex analysis (Rockafellar), I found out that a convex function should be twice differentiable almost everywhere. I further found in one of Rockafellar's books (convex analysis, thm. 25.7) a Theorem that shows the claim, except for the restriction on $f''$.
Now the question is how can we come up with that restriction on $f''$ and maybe someone has a reference for a book as well, since I would be interested in seeing the proof.
ordinary-differential-equations pde convex-analysis
ordinary-differential-equations pde convex-analysis
asked Dec 5 '18 at 11:21
MaxMax
586
asked Dec 5 '18 at 11:21
MaxMax
586
edited Dec 5 '18 at 11:29
Max
asked Dec 5 '18 at 11:21
MaxMax
586
asked Dec 5 '18 at 11:21
MaxMax
586
asked Dec 5 '18 at 11:21
MaxMax
586
586
1
$begingroup$
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 11:52
$begingroup$
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
$endgroup$
– Max
Dec 5 '18 at 15:28
$begingroup$
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
$endgroup$
– Max
Dec 5 '18 at 15:41
add a comment |
1
$begingroup$
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 11:52
$begingroup$
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
$endgroup$
– Max
Dec 5 '18 at 15:28
$begingroup$
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
$endgroup$
– Max
Dec 5 '18 at 15:41
1
1
$begingroup$
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 11:52
$begingroup$
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 11:52
$begingroup$
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
$endgroup$
– Max
Dec 5 '18 at 15:28
$begingroup$
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
$endgroup$
– Max
Dec 5 '18 at 15:28
$begingroup$
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
$endgroup$
– Max
Dec 5 '18 at 15:41
$begingroup$
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
$endgroup$
– Max
Dec 5 '18 at 15:41
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
So I figured it out a few days ago and wanted to share it with you.
Most importantly it is a well known result that an arbitrary convex function can be uniformly approximated by a convex $C^{infty}$-function on any closed bounded subinterval of the domain. I personally am not interested in the construction, but if you are, you might find some answers in the later mentioned references. Since this Approximation is applied on the Lemma of Fatou and the Lemma of Lebesgue, pointwise convergence of the Approximation is actually enough.
Assume $g$ is the function which approximates our convex function $f$ uniformly on the set $[-n,n]$. Then define
$tilde{g}_n:= mathbb{1}_{(-infty,-n)} (g_n(-n)+g_n'(-n)(x+n)) + mathbb{1}_{[-n,n]} g_n +mathbb{1}_{(n,infty)}(g_n(n)+g_n'(n)(x-n))$.
This function then approximates the convex function $f$ pointwise, and the second derivative vanishes outside the bounded set $[-n,n]$.
Some good references on approximation of convex functions might be:
D. Azagra, “Global and fine approximation of convex functions,” Proc. London Math. Soc., vol. 107, p. 799–824, 2013
A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State University Annals
J. J. Koliha, "APPROXIMATION OF CONVEX FUNCTIONS", Real Analysis Exchange, Vol. 29(1), 2003/2004, pp. 465–471
R. Tyrrell Rockafellar, "SECOND-ORDER CONVEX ANALYSIS", Journal of Nonlinear and Convex Analysis 1 (1999), 1-16
answered Jan 13 at 21:12
MaxMax
586
$endgroup$
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%2f3026957%2fapproximation-of-arbitrary-convex-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
So I figured it out a few days ago and wanted to share it with you.
Most importantly it is a well known result that an arbitrary convex function can be uniformly approximated by a convex $C^{infty}$-function on any closed bounded subinterval of the domain. I personally am not interested in the construction, but if you are, you might find some answers in the later mentioned references. Since this Approximation is applied on the Lemma of Fatou and the Lemma of Lebesgue, pointwise convergence of the Approximation is actually enough.
Assume $g$ is the function which approximates our convex function $f$ uniformly on the set $[-n,n]$. Then define
$tilde{g}_n:= mathbb{1}_{(-infty,-n)} (g_n(-n)+g_n'(-n)(x+n)) + mathbb{1}_{[-n,n]} g_n +mathbb{1}_{(n,infty)}(g_n(n)+g_n'(n)(x-n))$.
This function then approximates the convex function $f$ pointwise, and the second derivative vanishes outside the bounded set $[-n,n]$.
Some good references on approximation of convex functions might be:
D. Azagra, “Global and fine approximation of convex functions,” Proc. London Math. Soc., vol. 107, p. 799–824, 2013
A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State University Annals
J. J. Koliha, "APPROXIMATION OF CONVEX FUNCTIONS", Real Analysis Exchange, Vol. 29(1), 2003/2004, pp. 465–471
R. Tyrrell Rockafellar, "SECOND-ORDER CONVEX ANALYSIS", Journal of Nonlinear and Convex Analysis 1 (1999), 1-16
answered Jan 13 at 21:12
MaxMax
586
$endgroup$
add a comment |
$begingroup$
So I figured it out a few days ago and wanted to share it with you.
Most importantly it is a well known result that an arbitrary convex function can be uniformly approximated by a convex $C^{infty}$-function on any closed bounded subinterval of the domain. I personally am not interested in the construction, but if you are, you might find some answers in the later mentioned references. Since this Approximation is applied on the Lemma of Fatou and the Lemma of Lebesgue, pointwise convergence of the Approximation is actually enough.
Assume $g$ is the function which approximates our convex function $f$ uniformly on the set $[-n,n]$. Then define
$tilde{g}_n:= mathbb{1}_{(-infty,-n)} (g_n(-n)+g_n'(-n)(x+n)) + mathbb{1}_{[-n,n]} g_n +mathbb{1}_{(n,infty)}(g_n(n)+g_n'(n)(x-n))$.
This function then approximates the convex function $f$ pointwise, and the second derivative vanishes outside the bounded set $[-n,n]$.
Some good references on approximation of convex functions might be:
D. Azagra, “Global and fine approximation of convex functions,” Proc. London Math. Soc., vol. 107, p. 799–824, 2013
A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State University Annals
J. J. Koliha, "APPROXIMATION OF CONVEX FUNCTIONS", Real Analysis Exchange, Vol. 29(1), 2003/2004, pp. 465–471
R. Tyrrell Rockafellar, "SECOND-ORDER CONVEX ANALYSIS", Journal of Nonlinear and Convex Analysis 1 (1999), 1-16
answered Jan 13 at 21:12
MaxMax
586
$endgroup$
add a comment |
$begingroup$
So I figured it out a few days ago and wanted to share it with you.
Most importantly it is a well known result that an arbitrary convex function can be uniformly approximated by a convex $C^{infty}$-function on any closed bounded subinterval of the domain. I personally am not interested in the construction, but if you are, you might find some answers in the later mentioned references. Since this Approximation is applied on the Lemma of Fatou and the Lemma of Lebesgue, pointwise convergence of the Approximation is actually enough.
Assume $g$ is the function which approximates our convex function $f$ uniformly on the set $[-n,n]$. Then define
$tilde{g}_n:= mathbb{1}_{(-infty,-n)} (g_n(-n)+g_n'(-n)(x+n)) + mathbb{1}_{[-n,n]} g_n +mathbb{1}_{(n,infty)}(g_n(n)+g_n'(n)(x-n))$.
This function then approximates the convex function $f$ pointwise, and the second derivative vanishes outside the bounded set $[-n,n]$.
Some good references on approximation of convex functions might be:
D. Azagra, “Global and fine approximation of convex functions,” Proc. London Math. Soc., vol. 107, p. 799–824, 2013
A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State University Annals
J. J. Koliha, "APPROXIMATION OF CONVEX FUNCTIONS", Real Analysis Exchange, Vol. 29(1), 2003/2004, pp. 465–471
R. Tyrrell Rockafellar, "SECOND-ORDER CONVEX ANALYSIS", Journal of Nonlinear and Convex Analysis 1 (1999), 1-16
answered Jan 13 at 21:12
MaxMax
586
$endgroup$
So I figured it out a few days ago and wanted to share it with you.
Most importantly it is a well known result that an arbitrary convex function can be uniformly approximated by a convex $C^{infty}$-function on any closed bounded subinterval of the domain. I personally am not interested in the construction, but if you are, you might find some answers in the later mentioned references. Since this Approximation is applied on the Lemma of Fatou and the Lemma of Lebesgue, pointwise convergence of the Approximation is actually enough.
Assume $g$ is the function which approximates our convex function $f$ uniformly on the set $[-n,n]$. Then define
$tilde{g}_n:= mathbb{1}_{(-infty,-n)} (g_n(-n)+g_n'(-n)(x+n)) + mathbb{1}_{[-n,n]} g_n +mathbb{1}_{(n,infty)}(g_n(n)+g_n'(n)(x-n))$.
This function then approximates the convex function $f$ pointwise, and the second derivative vanishes outside the bounded set $[-n,n]$.
Some good references on approximation of convex functions might be:
D. Azagra, “Global and fine approximation of convex functions,” Proc. London Math. Soc., vol. 107, p. 799–824, 2013
A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State University Annals
J. J. Koliha, "APPROXIMATION OF CONVEX FUNCTIONS", Real Analysis Exchange, Vol. 29(1), 2003/2004, pp. 465–471
R. Tyrrell Rockafellar, "SECOND-ORDER CONVEX ANALYSIS", Journal of Nonlinear and Convex Analysis 1 (1999), 1-16
answered Jan 13 at 21:12
MaxMax
586
answered Jan 13 at 21:12
MaxMax
586
answered Jan 13 at 21:12
MaxMax
586
answered Jan 13 at 21:12
MaxMax
586
586
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.
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%2f3026957%2fapproximation-of-arbitrary-convex-function%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
1
$begingroup$
Is this about pointwise convergence? Then you could just cut off at say $|u|>m$ and replace that bit by a linear function.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 11:52
$begingroup$
I then need to apply Fatou's Lemma so I think pointwise convergence almost everywhere should be enough. Then interpolation would actually do the job..
$endgroup$
– Max
Dec 5 '18 at 15:28
$begingroup$
I thought about it again and I am not certain that Interpolation would do the job. The function's derivate could only be defined on $mathbb{R}$ $ mathbb{Q}$. Then interpolation on the rational numbers wouldn't necessarily do the job, I guess. Any thoughts on that?
$endgroup$
– Max
Dec 5 '18 at 15:41