Functions in Sobolev space












0












$begingroup$


How to prove that $x_1=signt notin W{}_2^1[-1;1] $ ?



I know that signt=$frac{d}{dt}mid tmid$ and $ mid tmid '=frac{t}{mid tmid}$, derivative of module is not defined in t=0. Am I thinking right?



Thanks for any help










share|cite|improve this question











$endgroup$












  • $begingroup$
    By $W_2^1$ do you mean $W^{1,2}$?
    $endgroup$
    – Zachary Selk
    Dec 24 '18 at 22:04










  • $begingroup$
    @ZacharySelk yes
    $endgroup$
    – T. Elmira
    Dec 24 '18 at 22:09










  • $begingroup$
    What do you mean by $x_1$? Is that the name of your function depending on $t$? However, you need to prove that either your function does not live in $L^2[-1,1]$ or its weak derivative does not. Hint: Its the second case.
    $endgroup$
    – YoungMath
    Dec 24 '18 at 22:28












  • $begingroup$
    @YoungMath $x_1$ is the name of my function depending on t. Can you please explain how to prove that signt does not live in $L^2[-1;1]?
    $endgroup$
    – T. Elmira
    Dec 24 '18 at 22:33






  • 3




    $begingroup$
    Well, $text{sign}$ is indeed in $L^2[-1,1]$. But its weak derivative is not. Hint two: The weak derivative of $text{sign}$ only exists in a distributional manner. It is (twice) the famous Dirac-distribution.
    $endgroup$
    – YoungMath
    Dec 24 '18 at 22:40


















0












$begingroup$


How to prove that $x_1=signt notin W{}_2^1[-1;1] $ ?



I know that signt=$frac{d}{dt}mid tmid$ and $ mid tmid '=frac{t}{mid tmid}$, derivative of module is not defined in t=0. Am I thinking right?



Thanks for any help










share|cite|improve this question











$endgroup$












  • $begingroup$
    By $W_2^1$ do you mean $W^{1,2}$?
    $endgroup$
    – Zachary Selk
    Dec 24 '18 at 22:04










  • $begingroup$
    @ZacharySelk yes
    $endgroup$
    – T. Elmira
    Dec 24 '18 at 22:09










  • $begingroup$
    What do you mean by $x_1$? Is that the name of your function depending on $t$? However, you need to prove that either your function does not live in $L^2[-1,1]$ or its weak derivative does not. Hint: Its the second case.
    $endgroup$
    – YoungMath
    Dec 24 '18 at 22:28












  • $begingroup$
    @YoungMath $x_1$ is the name of my function depending on t. Can you please explain how to prove that signt does not live in $L^2[-1;1]?
    $endgroup$
    – T. Elmira
    Dec 24 '18 at 22:33






  • 3




    $begingroup$
    Well, $text{sign}$ is indeed in $L^2[-1,1]$. But its weak derivative is not. Hint two: The weak derivative of $text{sign}$ only exists in a distributional manner. It is (twice) the famous Dirac-distribution.
    $endgroup$
    – YoungMath
    Dec 24 '18 at 22:40
















0












0








0





$begingroup$


How to prove that $x_1=signt notin W{}_2^1[-1;1] $ ?



I know that signt=$frac{d}{dt}mid tmid$ and $ mid tmid '=frac{t}{mid tmid}$, derivative of module is not defined in t=0. Am I thinking right?



Thanks for any help










share|cite|improve this question











$endgroup$




How to prove that $x_1=signt notin W{}_2^1[-1;1] $ ?



I know that signt=$frac{d}{dt}mid tmid$ and $ mid tmid '=frac{t}{mid tmid}$, derivative of module is not defined in t=0. Am I thinking right?



Thanks for any help







functional-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 24 '18 at 21:45







T. Elmira

















asked Dec 24 '18 at 21:38









T. ElmiraT. Elmira

174




174












  • $begingroup$
    By $W_2^1$ do you mean $W^{1,2}$?
    $endgroup$
    – Zachary Selk
    Dec 24 '18 at 22:04










  • $begingroup$
    @ZacharySelk yes
    $endgroup$
    – T. Elmira
    Dec 24 '18 at 22:09










  • $begingroup$
    What do you mean by $x_1$? Is that the name of your function depending on $t$? However, you need to prove that either your function does not live in $L^2[-1,1]$ or its weak derivative does not. Hint: Its the second case.
    $endgroup$
    – YoungMath
    Dec 24 '18 at 22:28












  • $begingroup$
    @YoungMath $x_1$ is the name of my function depending on t. Can you please explain how to prove that signt does not live in $L^2[-1;1]?
    $endgroup$
    – T. Elmira
    Dec 24 '18 at 22:33






  • 3




    $begingroup$
    Well, $text{sign}$ is indeed in $L^2[-1,1]$. But its weak derivative is not. Hint two: The weak derivative of $text{sign}$ only exists in a distributional manner. It is (twice) the famous Dirac-distribution.
    $endgroup$
    – YoungMath
    Dec 24 '18 at 22:40




















  • $begingroup$
    By $W_2^1$ do you mean $W^{1,2}$?
    $endgroup$
    – Zachary Selk
    Dec 24 '18 at 22:04










  • $begingroup$
    @ZacharySelk yes
    $endgroup$
    – T. Elmira
    Dec 24 '18 at 22:09










  • $begingroup$
    What do you mean by $x_1$? Is that the name of your function depending on $t$? However, you need to prove that either your function does not live in $L^2[-1,1]$ or its weak derivative does not. Hint: Its the second case.
    $endgroup$
    – YoungMath
    Dec 24 '18 at 22:28












  • $begingroup$
    @YoungMath $x_1$ is the name of my function depending on t. Can you please explain how to prove that signt does not live in $L^2[-1;1]?
    $endgroup$
    – T. Elmira
    Dec 24 '18 at 22:33






  • 3




    $begingroup$
    Well, $text{sign}$ is indeed in $L^2[-1,1]$. But its weak derivative is not. Hint two: The weak derivative of $text{sign}$ only exists in a distributional manner. It is (twice) the famous Dirac-distribution.
    $endgroup$
    – YoungMath
    Dec 24 '18 at 22:40


















$begingroup$
By $W_2^1$ do you mean $W^{1,2}$?
$endgroup$
– Zachary Selk
Dec 24 '18 at 22:04




$begingroup$
By $W_2^1$ do you mean $W^{1,2}$?
$endgroup$
– Zachary Selk
Dec 24 '18 at 22:04












$begingroup$
@ZacharySelk yes
$endgroup$
– T. Elmira
Dec 24 '18 at 22:09




$begingroup$
@ZacharySelk yes
$endgroup$
– T. Elmira
Dec 24 '18 at 22:09












$begingroup$
What do you mean by $x_1$? Is that the name of your function depending on $t$? However, you need to prove that either your function does not live in $L^2[-1,1]$ or its weak derivative does not. Hint: Its the second case.
$endgroup$
– YoungMath
Dec 24 '18 at 22:28






$begingroup$
What do you mean by $x_1$? Is that the name of your function depending on $t$? However, you need to prove that either your function does not live in $L^2[-1,1]$ or its weak derivative does not. Hint: Its the second case.
$endgroup$
– YoungMath
Dec 24 '18 at 22:28














$begingroup$
@YoungMath $x_1$ is the name of my function depending on t. Can you please explain how to prove that signt does not live in $L^2[-1;1]?
$endgroup$
– T. Elmira
Dec 24 '18 at 22:33




$begingroup$
@YoungMath $x_1$ is the name of my function depending on t. Can you please explain how to prove that signt does not live in $L^2[-1;1]?
$endgroup$
– T. Elmira
Dec 24 '18 at 22:33




3




3




$begingroup$
Well, $text{sign}$ is indeed in $L^2[-1,1]$. But its weak derivative is not. Hint two: The weak derivative of $text{sign}$ only exists in a distributional manner. It is (twice) the famous Dirac-distribution.
$endgroup$
– YoungMath
Dec 24 '18 at 22:40






$begingroup$
Well, $text{sign}$ is indeed in $L^2[-1,1]$. But its weak derivative is not. Hint two: The weak derivative of $text{sign}$ only exists in a distributional manner. It is (twice) the famous Dirac-distribution.
$endgroup$
– YoungMath
Dec 24 '18 at 22:40












1 Answer
1






active

oldest

votes


















1












$begingroup$

Problem. A weak derivative $v(t)$ of $sign(t)$ will satisfy, for all smooth functions $phi$ such that $phi(-1) = phi(1) = 0$,
$$int_{-1}^1sign(t)phi'(t),dt = -int_{-1}^1v(t)phi(t),dt$$ so we want to see that it can't exist.





Reason. You can evaluate $$int_{-1}^1 sign(t)phi'(t)dt = int_0^1phi'(t)dt - int_{-1}^0phi'(t)dt = -2phi(0)$$
As mentioned in the comments above this is a $delta$ distribution, $v(t) = 2delta_0$ which is not a function.





Finishing the proof. If you haven't done so already, the most straightforward way to prove that $delta$ distributions are not representable is by applying the dominated convergence theorem to a sequence $varphi_n$ of bump functions $0 leq varphi_n(t) leq 1$ whose support shrinks to a point. Someone wrote this down over here: Why Dirac's Delta is not an ordinary function?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    That is not entirely true. As far as I know, $phi$ has to be a bump function, i.e. $phi$ has compact support in the interior of $[-1,1]$. Thus, $phi(-1)=phi(1)=0$.
    $endgroup$
    – YoungMath
    Dec 25 '18 at 11:25










  • $begingroup$
    @youngmath Sorry, I don’t understand your comment.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:21










  • $begingroup$
    Long story short: $phi$ has to vanish on the boundary. See for instance aforementioned link to Wikipedia about weak derivatives.
    $endgroup$
    – YoungMath
    Dec 25 '18 at 12:32












  • $begingroup$
    @YoungMath Oh I see what you are saying - thanks! I'll fix that.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:35










  • $begingroup$
    @YoungMath, if you shift the test function $varphi$ by a constant to $varphi + C$ it doesn't affect the left hand side but the right hand side changes by $C cdot int v,dt$, so yes what I wrote the first time was wrong.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:39











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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









1












$begingroup$

Problem. A weak derivative $v(t)$ of $sign(t)$ will satisfy, for all smooth functions $phi$ such that $phi(-1) = phi(1) = 0$,
$$int_{-1}^1sign(t)phi'(t),dt = -int_{-1}^1v(t)phi(t),dt$$ so we want to see that it can't exist.





Reason. You can evaluate $$int_{-1}^1 sign(t)phi'(t)dt = int_0^1phi'(t)dt - int_{-1}^0phi'(t)dt = -2phi(0)$$
As mentioned in the comments above this is a $delta$ distribution, $v(t) = 2delta_0$ which is not a function.





Finishing the proof. If you haven't done so already, the most straightforward way to prove that $delta$ distributions are not representable is by applying the dominated convergence theorem to a sequence $varphi_n$ of bump functions $0 leq varphi_n(t) leq 1$ whose support shrinks to a point. Someone wrote this down over here: Why Dirac's Delta is not an ordinary function?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    That is not entirely true. As far as I know, $phi$ has to be a bump function, i.e. $phi$ has compact support in the interior of $[-1,1]$. Thus, $phi(-1)=phi(1)=0$.
    $endgroup$
    – YoungMath
    Dec 25 '18 at 11:25










  • $begingroup$
    @youngmath Sorry, I don’t understand your comment.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:21










  • $begingroup$
    Long story short: $phi$ has to vanish on the boundary. See for instance aforementioned link to Wikipedia about weak derivatives.
    $endgroup$
    – YoungMath
    Dec 25 '18 at 12:32












  • $begingroup$
    @YoungMath Oh I see what you are saying - thanks! I'll fix that.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:35










  • $begingroup$
    @YoungMath, if you shift the test function $varphi$ by a constant to $varphi + C$ it doesn't affect the left hand side but the right hand side changes by $C cdot int v,dt$, so yes what I wrote the first time was wrong.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:39
















1












$begingroup$

Problem. A weak derivative $v(t)$ of $sign(t)$ will satisfy, for all smooth functions $phi$ such that $phi(-1) = phi(1) = 0$,
$$int_{-1}^1sign(t)phi'(t),dt = -int_{-1}^1v(t)phi(t),dt$$ so we want to see that it can't exist.





Reason. You can evaluate $$int_{-1}^1 sign(t)phi'(t)dt = int_0^1phi'(t)dt - int_{-1}^0phi'(t)dt = -2phi(0)$$
As mentioned in the comments above this is a $delta$ distribution, $v(t) = 2delta_0$ which is not a function.





Finishing the proof. If you haven't done so already, the most straightforward way to prove that $delta$ distributions are not representable is by applying the dominated convergence theorem to a sequence $varphi_n$ of bump functions $0 leq varphi_n(t) leq 1$ whose support shrinks to a point. Someone wrote this down over here: Why Dirac's Delta is not an ordinary function?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    That is not entirely true. As far as I know, $phi$ has to be a bump function, i.e. $phi$ has compact support in the interior of $[-1,1]$. Thus, $phi(-1)=phi(1)=0$.
    $endgroup$
    – YoungMath
    Dec 25 '18 at 11:25










  • $begingroup$
    @youngmath Sorry, I don’t understand your comment.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:21










  • $begingroup$
    Long story short: $phi$ has to vanish on the boundary. See for instance aforementioned link to Wikipedia about weak derivatives.
    $endgroup$
    – YoungMath
    Dec 25 '18 at 12:32












  • $begingroup$
    @YoungMath Oh I see what you are saying - thanks! I'll fix that.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:35










  • $begingroup$
    @YoungMath, if you shift the test function $varphi$ by a constant to $varphi + C$ it doesn't affect the left hand side but the right hand side changes by $C cdot int v,dt$, so yes what I wrote the first time was wrong.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:39














1












1








1





$begingroup$

Problem. A weak derivative $v(t)$ of $sign(t)$ will satisfy, for all smooth functions $phi$ such that $phi(-1) = phi(1) = 0$,
$$int_{-1}^1sign(t)phi'(t),dt = -int_{-1}^1v(t)phi(t),dt$$ so we want to see that it can't exist.





Reason. You can evaluate $$int_{-1}^1 sign(t)phi'(t)dt = int_0^1phi'(t)dt - int_{-1}^0phi'(t)dt = -2phi(0)$$
As mentioned in the comments above this is a $delta$ distribution, $v(t) = 2delta_0$ which is not a function.





Finishing the proof. If you haven't done so already, the most straightforward way to prove that $delta$ distributions are not representable is by applying the dominated convergence theorem to a sequence $varphi_n$ of bump functions $0 leq varphi_n(t) leq 1$ whose support shrinks to a point. Someone wrote this down over here: Why Dirac's Delta is not an ordinary function?






share|cite|improve this answer











$endgroup$



Problem. A weak derivative $v(t)$ of $sign(t)$ will satisfy, for all smooth functions $phi$ such that $phi(-1) = phi(1) = 0$,
$$int_{-1}^1sign(t)phi'(t),dt = -int_{-1}^1v(t)phi(t),dt$$ so we want to see that it can't exist.





Reason. You can evaluate $$int_{-1}^1 sign(t)phi'(t)dt = int_0^1phi'(t)dt - int_{-1}^0phi'(t)dt = -2phi(0)$$
As mentioned in the comments above this is a $delta$ distribution, $v(t) = 2delta_0$ which is not a function.





Finishing the proof. If you haven't done so already, the most straightforward way to prove that $delta$ distributions are not representable is by applying the dominated convergence theorem to a sequence $varphi_n$ of bump functions $0 leq varphi_n(t) leq 1$ whose support shrinks to a point. Someone wrote this down over here: Why Dirac's Delta is not an ordinary function?







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 25 '18 at 12:37

























answered Dec 25 '18 at 2:31









BenBen

4,048617




4,048617












  • $begingroup$
    That is not entirely true. As far as I know, $phi$ has to be a bump function, i.e. $phi$ has compact support in the interior of $[-1,1]$. Thus, $phi(-1)=phi(1)=0$.
    $endgroup$
    – YoungMath
    Dec 25 '18 at 11:25










  • $begingroup$
    @youngmath Sorry, I don’t understand your comment.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:21










  • $begingroup$
    Long story short: $phi$ has to vanish on the boundary. See for instance aforementioned link to Wikipedia about weak derivatives.
    $endgroup$
    – YoungMath
    Dec 25 '18 at 12:32












  • $begingroup$
    @YoungMath Oh I see what you are saying - thanks! I'll fix that.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:35










  • $begingroup$
    @YoungMath, if you shift the test function $varphi$ by a constant to $varphi + C$ it doesn't affect the left hand side but the right hand side changes by $C cdot int v,dt$, so yes what I wrote the first time was wrong.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:39


















  • $begingroup$
    That is not entirely true. As far as I know, $phi$ has to be a bump function, i.e. $phi$ has compact support in the interior of $[-1,1]$. Thus, $phi(-1)=phi(1)=0$.
    $endgroup$
    – YoungMath
    Dec 25 '18 at 11:25










  • $begingroup$
    @youngmath Sorry, I don’t understand your comment.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:21










  • $begingroup$
    Long story short: $phi$ has to vanish on the boundary. See for instance aforementioned link to Wikipedia about weak derivatives.
    $endgroup$
    – YoungMath
    Dec 25 '18 at 12:32












  • $begingroup$
    @YoungMath Oh I see what you are saying - thanks! I'll fix that.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:35










  • $begingroup$
    @YoungMath, if you shift the test function $varphi$ by a constant to $varphi + C$ it doesn't affect the left hand side but the right hand side changes by $C cdot int v,dt$, so yes what I wrote the first time was wrong.
    $endgroup$
    – Ben
    Dec 25 '18 at 12:39
















$begingroup$
That is not entirely true. As far as I know, $phi$ has to be a bump function, i.e. $phi$ has compact support in the interior of $[-1,1]$. Thus, $phi(-1)=phi(1)=0$.
$endgroup$
– YoungMath
Dec 25 '18 at 11:25




$begingroup$
That is not entirely true. As far as I know, $phi$ has to be a bump function, i.e. $phi$ has compact support in the interior of $[-1,1]$. Thus, $phi(-1)=phi(1)=0$.
$endgroup$
– YoungMath
Dec 25 '18 at 11:25












$begingroup$
@youngmath Sorry, I don’t understand your comment.
$endgroup$
– Ben
Dec 25 '18 at 12:21




$begingroup$
@youngmath Sorry, I don’t understand your comment.
$endgroup$
– Ben
Dec 25 '18 at 12:21












$begingroup$
Long story short: $phi$ has to vanish on the boundary. See for instance aforementioned link to Wikipedia about weak derivatives.
$endgroup$
– YoungMath
Dec 25 '18 at 12:32






$begingroup$
Long story short: $phi$ has to vanish on the boundary. See for instance aforementioned link to Wikipedia about weak derivatives.
$endgroup$
– YoungMath
Dec 25 '18 at 12:32














$begingroup$
@YoungMath Oh I see what you are saying - thanks! I'll fix that.
$endgroup$
– Ben
Dec 25 '18 at 12:35




$begingroup$
@YoungMath Oh I see what you are saying - thanks! I'll fix that.
$endgroup$
– Ben
Dec 25 '18 at 12:35












$begingroup$
@YoungMath, if you shift the test function $varphi$ by a constant to $varphi + C$ it doesn't affect the left hand side but the right hand side changes by $C cdot int v,dt$, so yes what I wrote the first time was wrong.
$endgroup$
– Ben
Dec 25 '18 at 12:39




$begingroup$
@YoungMath, if you shift the test function $varphi$ by a constant to $varphi + C$ it doesn't affect the left hand side but the right hand side changes by $C cdot int v,dt$, so yes what I wrote the first time was wrong.
$endgroup$
– Ben
Dec 25 '18 at 12:39


















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