About Sobolev Space definition.
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I would like to know if this two definitions for Sobolev Spaces are equivalent:
$H_2^m([a,b]):={f: f',...,f^{(m-1)} text{absolutely continuous}, int _a^b (f^{(m)})^2dx<infty }$
and $H_2^m([a,b])$ contains all functions $f$ such that $D^mu f in L_2([a,b])$ for $mu=0,1,...,m$
The first definition requires the $m$th derivative to be square integrable, while in the second one all derivatives (up to order m) must belong to $L_2([a,b])$ and no continuity is required.
I guess it has to do with the fact that absolute continuity is required, but I want to be sure that both are valid.
Any hint will be greatly appreciated.
functional-analysis vector-spaces sobolev-spaces
asked Dec 14 '18 at 16:24
Ramiro ScorolliRamiro Scorolli
644114
$endgroup$
add a comment |
$begingroup$
I would like to know if this two definitions for Sobolev Spaces are equivalent:
$H_2^m([a,b]):={f: f',...,f^{(m-1)} text{absolutely continuous}, int _a^b (f^{(m)})^2dx<infty }$
and $H_2^m([a,b])$ contains all functions $f$ such that $D^mu f in L_2([a,b])$ for $mu=0,1,...,m$
The first definition requires the $m$th derivative to be square integrable, while in the second one all derivatives (up to order m) must belong to $L_2([a,b])$ and no continuity is required.
I guess it has to do with the fact that absolute continuity is required, but I want to be sure that both are valid.
Any hint will be greatly appreciated.
functional-analysis vector-spaces sobolev-spaces
asked Dec 14 '18 at 16:24
Ramiro ScorolliRamiro Scorolli
644114
$endgroup$
$begingroup$
@orange en.wikipedia.org/wiki/Absolute_continuity
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– Ramiro Scorolli
Dec 14 '18 at 20:01
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Or could it be that the second is not actually a definition in the formal sense, but just a state? It came to my mind that maybe the author was not actually defining the space but just making a characterization
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– Ramiro Scorolli
Dec 14 '18 at 22:00
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@orange this is the article where the second statement is made, link.springer.com/article/10.1023/A:1019250217452
$endgroup$
– Ramiro Scorolli
Dec 15 '18 at 9:48
add a comment |
$begingroup$
I would like to know if this two definitions for Sobolev Spaces are equivalent:
$H_2^m([a,b]):={f: f',...,f^{(m-1)} text{absolutely continuous}, int _a^b (f^{(m)})^2dx<infty }$
and $H_2^m([a,b])$ contains all functions $f$ such that $D^mu f in L_2([a,b])$ for $mu=0,1,...,m$
The first definition requires the $m$th derivative to be square integrable, while in the second one all derivatives (up to order m) must belong to $L_2([a,b])$ and no continuity is required.
I guess it has to do with the fact that absolute continuity is required, but I want to be sure that both are valid.
Any hint will be greatly appreciated.
functional-analysis vector-spaces sobolev-spaces
asked Dec 14 '18 at 16:24
Ramiro ScorolliRamiro Scorolli
644114
$endgroup$
I would like to know if this two definitions for Sobolev Spaces are equivalent:
$H_2^m([a,b]):={f: f',...,f^{(m-1)} text{absolutely continuous}, int _a^b (f^{(m)})^2dx<infty }$
and $H_2^m([a,b])$ contains all functions $f$ such that $D^mu f in L_2([a,b])$ for $mu=0,1,...,m$
The first definition requires the $m$th derivative to be square integrable, while in the second one all derivatives (up to order m) must belong to $L_2([a,b])$ and no continuity is required.
I guess it has to do with the fact that absolute continuity is required, but I want to be sure that both are valid.
Any hint will be greatly appreciated.
functional-analysis vector-spaces sobolev-spaces
functional-analysis vector-spaces sobolev-spaces
asked Dec 14 '18 at 16:24
Ramiro ScorolliRamiro Scorolli
644114
asked Dec 14 '18 at 16:24
Ramiro ScorolliRamiro Scorolli
644114
asked Dec 14 '18 at 16:24
Ramiro ScorolliRamiro Scorolli
644114
asked Dec 14 '18 at 16:24
Ramiro ScorolliRamiro Scorolli
644114
asked Dec 14 '18 at 16:24
Ramiro ScorolliRamiro Scorolli
644114
644114
$begingroup$
@orange en.wikipedia.org/wiki/Absolute_continuity
$endgroup$
– Ramiro Scorolli
Dec 14 '18 at 20:01
$begingroup$
Or could it be that the second is not actually a definition in the formal sense, but just a state? It came to my mind that maybe the author was not actually defining the space but just making a characterization
$endgroup$
– Ramiro Scorolli
Dec 14 '18 at 22:00
$begingroup$
@orange this is the article where the second statement is made, link.springer.com/article/10.1023/A:1019250217452
$endgroup$
– Ramiro Scorolli
Dec 15 '18 at 9:48
add a comment |
$begingroup$
@orange en.wikipedia.org/wiki/Absolute_continuity
$endgroup$
– Ramiro Scorolli
Dec 14 '18 at 20:01
$begingroup$
Or could it be that the second is not actually a definition in the formal sense, but just a state? It came to my mind that maybe the author was not actually defining the space but just making a characterization
$endgroup$
– Ramiro Scorolli
Dec 14 '18 at 22:00
$begingroup$
@orange this is the article where the second statement is made, link.springer.com/article/10.1023/A:1019250217452
$endgroup$
– Ramiro Scorolli
Dec 15 '18 at 9:48
$begingroup$
@orange en.wikipedia.org/wiki/Absolute_continuity
$endgroup$
– Ramiro Scorolli
Dec 14 '18 at 20:01
$begingroup$
@orange en.wikipedia.org/wiki/Absolute_continuity
$endgroup$
– Ramiro Scorolli
Dec 14 '18 at 20:01
$begingroup$
Or could it be that the second is not actually a definition in the formal sense, but just a state? It came to my mind that maybe the author was not actually defining the space but just making a characterization
$endgroup$
– Ramiro Scorolli
Dec 14 '18 at 22:00
$begingroup$
Or could it be that the second is not actually a definition in the formal sense, but just a state? It came to my mind that maybe the author was not actually defining the space but just making a characterization
$endgroup$
– Ramiro Scorolli
Dec 14 '18 at 22:00
$begingroup$
@orange this is the article where the second statement is made, link.springer.com/article/10.1023/A:1019250217452
$endgroup$
– Ramiro Scorolli
Dec 15 '18 at 9:48
$begingroup$
@orange this is the article where the second statement is made, link.springer.com/article/10.1023/A:1019250217452
$endgroup$
– Ramiro Scorolli
Dec 15 '18 at 9:48
add a comment |
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$begingroup$
@orange en.wikipedia.org/wiki/Absolute_continuity
$endgroup$
– Ramiro Scorolli
Dec 14 '18 at 20:01
$begingroup$
Or could it be that the second is not actually a definition in the formal sense, but just a state? It came to my mind that maybe the author was not actually defining the space but just making a characterization
$endgroup$
– Ramiro Scorolli
Dec 14 '18 at 22:00
$begingroup$
@orange this is the article where the second statement is made, link.springer.com/article/10.1023/A:1019250217452
$endgroup$
– Ramiro Scorolli
Dec 15 '18 at 9:48