Approximation of Cantor function by piecewise constant function in $L^1$
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Let $c(x)$ be Cantor function.
How can we prove that constant $frac{1}{2}$ gives the best approximation in $L^1$ metric?
Let $ h(x)=
begin{cases}
frac14 quadtext{for}quad xin[0,frac13]\
frac12quadtext{for}quad xin[frac13,frac23]\
frac34quadtext{for}quad xin[frac23,1]
end{cases} $
Will this piecewise constant function $h(x)$ give the best approximation in $L^1$? If so how to prove that?
My thoughts on this are this piecewise constant function $h(x)$ will not give the best approximation. But then how can we get the best function with three constancy intervals. Or may be somehow express $c(x)$ to get explicitly extremal problem.
real-analysis approximation approximation-theory cantor-set
asked Nov 16 at 19:00
ModeGen
312
add a comment |
up vote
0
down vote
favorite
Let $c(x)$ be Cantor function.
How can we prove that constant $frac{1}{2}$ gives the best approximation in $L^1$ metric?
Let $ h(x)=
begin{cases}
frac14 quadtext{for}quad xin[0,frac13]\
frac12quadtext{for}quad xin[frac13,frac23]\
frac34quadtext{for}quad xin[frac23,1]
end{cases} $
Will this piecewise constant function $h(x)$ give the best approximation in $L^1$? If so how to prove that?
My thoughts on this are this piecewise constant function $h(x)$ will not give the best approximation. But then how can we get the best function with three constancy intervals. Or may be somehow express $c(x)$ to get explicitly extremal problem.
real-analysis approximation approximation-theory cantor-set
asked Nov 16 at 19:00
ModeGen
312
What do you mean by "best" approximation? How can both $frac{1}{2}$ and $h(x)$ give the "best" approximation? Is it not clear that one is much better?
– Xander Henderson
Nov 16 at 19:11
The question says "in $L_1$ metric", so the goodness of an approximation would be $lVert h(x)-c(x)rVert_1$ (where smaller is better)
– Carmeister
Nov 16 at 19:17
@XanderHenderson, best approximation means that $ inf_{r in mathbb{R}} lVert c(x) - r rVert_{L^1} = lVert c(x)-r_0rVert_{L^1}$, i.e. constant $r_0$ gives best approximation and it pretend to be $frac12$. $h(x)$ is the function that has three constanty intervals, not just one.
– ModeGen
Nov 16 at 19:33
@Carmeister My confusion is not with the metric, but with the function(s?) approximating the Cantor function. Is the question "Is $x mapsto frac{1}{2}$ the best approximation?" or "Is $x mapsto h(x)$ the best approximation?" "Best" in what sense? That is, what are the other candidate approximations against which these functions are being compared? Perhaps I am being dense, but I am having trouble parsing the question.
– Xander Henderson
Nov 16 at 20:09
@XanderHenderson, there are two questions actually: what is the best constant and what is the best 3 constanty interval function approximating $c(x)$. For example, best 3 constanty interval function $h(x)$ may has discontinuity points not in $frac13$ and $frac23$ or takes other values.
– ModeGen
Nov 16 at 20:32
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $c(x)$ be Cantor function.
How can we prove that constant $frac{1}{2}$ gives the best approximation in $L^1$ metric?
Let $ h(x)=
begin{cases}
frac14 quadtext{for}quad xin[0,frac13]\
frac12quadtext{for}quad xin[frac13,frac23]\
frac34quadtext{for}quad xin[frac23,1]
end{cases} $
Will this piecewise constant function $h(x)$ give the best approximation in $L^1$? If so how to prove that?
My thoughts on this are this piecewise constant function $h(x)$ will not give the best approximation. But then how can we get the best function with three constancy intervals. Or may be somehow express $c(x)$ to get explicitly extremal problem.
real-analysis approximation approximation-theory cantor-set
asked Nov 16 at 19:00
ModeGen
312
Let $c(x)$ be Cantor function.
How can we prove that constant $frac{1}{2}$ gives the best approximation in $L^1$ metric?
Let $ h(x)=
begin{cases}
frac14 quadtext{for}quad xin[0,frac13]\
frac12quadtext{for}quad xin[frac13,frac23]\
frac34quadtext{for}quad xin[frac23,1]
end{cases} $
Will this piecewise constant function $h(x)$ give the best approximation in $L^1$? If so how to prove that?
My thoughts on this are this piecewise constant function $h(x)$ will not give the best approximation. But then how can we get the best function with three constancy intervals. Or may be somehow express $c(x)$ to get explicitly extremal problem.
real-analysis approximation approximation-theory cantor-set
real-analysis approximation approximation-theory cantor-set
asked Nov 16 at 19:00
ModeGen
312
asked Nov 16 at 19:00
ModeGen
312
asked Nov 16 at 19:00
ModeGen
312
asked Nov 16 at 19:00
ModeGen
312
asked Nov 16 at 19:00
ModeGen
312
312
What do you mean by "best" approximation? How can both $frac{1}{2}$ and $h(x)$ give the "best" approximation? Is it not clear that one is much better?
– Xander Henderson
Nov 16 at 19:11
The question says "in $L_1$ metric", so the goodness of an approximation would be $lVert h(x)-c(x)rVert_1$ (where smaller is better)
– Carmeister
Nov 16 at 19:17
@XanderHenderson, best approximation means that $ inf_{r in mathbb{R}} lVert c(x) - r rVert_{L^1} = lVert c(x)-r_0rVert_{L^1}$, i.e. constant $r_0$ gives best approximation and it pretend to be $frac12$. $h(x)$ is the function that has three constanty intervals, not just one.
– ModeGen
Nov 16 at 19:33
@Carmeister My confusion is not with the metric, but with the function(s?) approximating the Cantor function. Is the question "Is $x mapsto frac{1}{2}$ the best approximation?" or "Is $x mapsto h(x)$ the best approximation?" "Best" in what sense? That is, what are the other candidate approximations against which these functions are being compared? Perhaps I am being dense, but I am having trouble parsing the question.
– Xander Henderson
Nov 16 at 20:09
@XanderHenderson, there are two questions actually: what is the best constant and what is the best 3 constanty interval function approximating $c(x)$. For example, best 3 constanty interval function $h(x)$ may has discontinuity points not in $frac13$ and $frac23$ or takes other values.
– ModeGen
Nov 16 at 20:32
add a comment |
What do you mean by "best" approximation? How can both $frac{1}{2}$ and $h(x)$ give the "best" approximation? Is it not clear that one is much better?
– Xander Henderson
Nov 16 at 19:11
The question says "in $L_1$ metric", so the goodness of an approximation would be $lVert h(x)-c(x)rVert_1$ (where smaller is better)
– Carmeister
Nov 16 at 19:17
@XanderHenderson, best approximation means that $ inf_{r in mathbb{R}} lVert c(x) - r rVert_{L^1} = lVert c(x)-r_0rVert_{L^1}$, i.e. constant $r_0$ gives best approximation and it pretend to be $frac12$. $h(x)$ is the function that has three constanty intervals, not just one.
– ModeGen
Nov 16 at 19:33
@Carmeister My confusion is not with the metric, but with the function(s?) approximating the Cantor function. Is the question "Is $x mapsto frac{1}{2}$ the best approximation?" or "Is $x mapsto h(x)$ the best approximation?" "Best" in what sense? That is, what are the other candidate approximations against which these functions are being compared? Perhaps I am being dense, but I am having trouble parsing the question.
– Xander Henderson
Nov 16 at 20:09
@XanderHenderson, there are two questions actually: what is the best constant and what is the best 3 constanty interval function approximating $c(x)$. For example, best 3 constanty interval function $h(x)$ may has discontinuity points not in $frac13$ and $frac23$ or takes other values.
– ModeGen
Nov 16 at 20:32
What do you mean by "best" approximation? How can both $frac{1}{2}$ and $h(x)$ give the "best" approximation? Is it not clear that one is much better?
– Xander Henderson
Nov 16 at 19:11
What do you mean by "best" approximation? How can both $frac{1}{2}$ and $h(x)$ give the "best" approximation? Is it not clear that one is much better?
– Xander Henderson
Nov 16 at 19:11
The question says "in $L_1$ metric", so the goodness of an approximation would be $lVert h(x)-c(x)rVert_1$ (where smaller is better)
– Carmeister
Nov 16 at 19:17
The question says "in $L_1$ metric", so the goodness of an approximation would be $lVert h(x)-c(x)rVert_1$ (where smaller is better)
– Carmeister
Nov 16 at 19:17
@XanderHenderson, best approximation means that $ inf_{r in mathbb{R}} lVert c(x) - r rVert_{L^1} = lVert c(x)-r_0rVert_{L^1}$, i.e. constant $r_0$ gives best approximation and it pretend to be $frac12$. $h(x)$ is the function that has three constanty intervals, not just one.
– ModeGen
Nov 16 at 19:33
@XanderHenderson, best approximation means that $ inf_{r in mathbb{R}} lVert c(x) - r rVert_{L^1} = lVert c(x)-r_0rVert_{L^1}$, i.e. constant $r_0$ gives best approximation and it pretend to be $frac12$. $h(x)$ is the function that has three constanty intervals, not just one.
– ModeGen
Nov 16 at 19:33
@Carmeister My confusion is not with the metric, but with the function(s?) approximating the Cantor function. Is the question "Is $x mapsto frac{1}{2}$ the best approximation?" or "Is $x mapsto h(x)$ the best approximation?" "Best" in what sense? That is, what are the other candidate approximations against which these functions are being compared? Perhaps I am being dense, but I am having trouble parsing the question.
– Xander Henderson
Nov 16 at 20:09
@Carmeister My confusion is not with the metric, but with the function(s?) approximating the Cantor function. Is the question "Is $x mapsto frac{1}{2}$ the best approximation?" or "Is $x mapsto h(x)$ the best approximation?" "Best" in what sense? That is, what are the other candidate approximations against which these functions are being compared? Perhaps I am being dense, but I am having trouble parsing the question.
– Xander Henderson
Nov 16 at 20:09
@XanderHenderson, there are two questions actually: what is the best constant and what is the best 3 constanty interval function approximating $c(x)$. For example, best 3 constanty interval function $h(x)$ may has discontinuity points not in $frac13$ and $frac23$ or takes other values.
– ModeGen
Nov 16 at 20:32
@XanderHenderson, there are two questions actually: what is the best constant and what is the best 3 constanty interval function approximating $c(x)$. For example, best 3 constanty interval function $h(x)$ may has discontinuity points not in $frac13$ and $frac23$ or takes other values.
– ModeGen
Nov 16 at 20:32
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What do you mean by "best" approximation? How can both $frac{1}{2}$ and $h(x)$ give the "best" approximation? Is it not clear that one is much better?
– Xander Henderson
Nov 16 at 19:11
The question says "in $L_1$ metric", so the goodness of an approximation would be $lVert h(x)-c(x)rVert_1$ (where smaller is better)
– Carmeister
Nov 16 at 19:17
@XanderHenderson, best approximation means that $ inf_{r in mathbb{R}} lVert c(x) - r rVert_{L^1} = lVert c(x)-r_0rVert_{L^1}$, i.e. constant $r_0$ gives best approximation and it pretend to be $frac12$. $h(x)$ is the function that has three constanty intervals, not just one.
– ModeGen
Nov 16 at 19:33
@Carmeister My confusion is not with the metric, but with the function(s?) approximating the Cantor function. Is the question "Is $x mapsto frac{1}{2}$ the best approximation?" or "Is $x mapsto h(x)$ the best approximation?" "Best" in what sense? That is, what are the other candidate approximations against which these functions are being compared? Perhaps I am being dense, but I am having trouble parsing the question.
– Xander Henderson
Nov 16 at 20:09
@XanderHenderson, there are two questions actually: what is the best constant and what is the best 3 constanty interval function approximating $c(x)$. For example, best 3 constanty interval function $h(x)$ may has discontinuity points not in $frac13$ and $frac23$ or takes other values.
– ModeGen
Nov 16 at 20:32