Write a series of piecewise linear functions that converges to $f(x) = x^2$ on the interval $[0,1]$.
$begingroup$
Write a series of piecewise linear functions that converges to $f(x) = x^2$ on the interval $[0,1]$.
A sequence of piecewise linear functions that converges to $f(x) = x^2$ is as follows:
For $k in [0,j-1]$:
begin{align*}
f_j(x) &= begin{cases}
frac{1}{j} cdot x & text{if $0 le x le frac{1}{j}$} \
frac{2k+1}{j} cdot left(x - frac{k}{j} right) + frac{k^2}{j^2} & text{if $frac{k}{j} le x le frac{k+1}{j}$} \
frac{2j-1}{j} cdot left(x - frac{j-1}{j} right) + frac{j^2-2j+1}{j^2} & text{if $frac{j-1}{j} le x le 1$} \
end{cases} \
end{align*}
begin{align*}
limlimits_{j to infty} f_j(x) &= f(x) = x^2 \
end{align*}
What would be a series of piecewise linear functions such that $sum_{j=0}^infty f_j(x) = f(x) = x^2$?
real-analysis sequences-and-series sequence-of-function
asked Dec 4 '18 at 6:50
clayclay
765414
$endgroup$
add a comment |
$begingroup$
Write a series of piecewise linear functions that converges to $f(x) = x^2$ on the interval $[0,1]$.
A sequence of piecewise linear functions that converges to $f(x) = x^2$ is as follows:
For $k in [0,j-1]$:
begin{align*}
f_j(x) &= begin{cases}
frac{1}{j} cdot x & text{if $0 le x le frac{1}{j}$} \
frac{2k+1}{j} cdot left(x - frac{k}{j} right) + frac{k^2}{j^2} & text{if $frac{k}{j} le x le frac{k+1}{j}$} \
frac{2j-1}{j} cdot left(x - frac{j-1}{j} right) + frac{j^2-2j+1}{j^2} & text{if $frac{j-1}{j} le x le 1$} \
end{cases} \
end{align*}
begin{align*}
limlimits_{j to infty} f_j(x) &= f(x) = x^2 \
end{align*}
What would be a series of piecewise linear functions such that $sum_{j=0}^infty f_j(x) = f(x) = x^2$?
real-analysis sequences-and-series sequence-of-function
asked Dec 4 '18 at 6:50
clayclay
765414
$endgroup$
1
$begingroup$
A thing to try is if to consider what is $f_{j+1}(x) - f_{j}(x)$. Then use that to define partial sum $g_n(x)$ (this will be quite messy since you have to take into account domains and the like) and show that this converges to $f(x)$. I am aware that this is very broad, but the object to consider is $f_{j+1}(x) - f_{j}(x)$.
$endgroup$
– twnly
Dec 4 '18 at 7:03
add a comment |
$begingroup$
Write a series of piecewise linear functions that converges to $f(x) = x^2$ on the interval $[0,1]$.
A sequence of piecewise linear functions that converges to $f(x) = x^2$ is as follows:
For $k in [0,j-1]$:
begin{align*}
f_j(x) &= begin{cases}
frac{1}{j} cdot x & text{if $0 le x le frac{1}{j}$} \
frac{2k+1}{j} cdot left(x - frac{k}{j} right) + frac{k^2}{j^2} & text{if $frac{k}{j} le x le frac{k+1}{j}$} \
frac{2j-1}{j} cdot left(x - frac{j-1}{j} right) + frac{j^2-2j+1}{j^2} & text{if $frac{j-1}{j} le x le 1$} \
end{cases} \
end{align*}
begin{align*}
limlimits_{j to infty} f_j(x) &= f(x) = x^2 \
end{align*}
What would be a series of piecewise linear functions such that $sum_{j=0}^infty f_j(x) = f(x) = x^2$?
real-analysis sequences-and-series sequence-of-function
asked Dec 4 '18 at 6:50
clayclay
765414
$endgroup$
Write a series of piecewise linear functions that converges to $f(x) = x^2$ on the interval $[0,1]$.
A sequence of piecewise linear functions that converges to $f(x) = x^2$ is as follows:
For $k in [0,j-1]$:
begin{align*}
f_j(x) &= begin{cases}
frac{1}{j} cdot x & text{if $0 le x le frac{1}{j}$} \
frac{2k+1}{j} cdot left(x - frac{k}{j} right) + frac{k^2}{j^2} & text{if $frac{k}{j} le x le frac{k+1}{j}$} \
frac{2j-1}{j} cdot left(x - frac{j-1}{j} right) + frac{j^2-2j+1}{j^2} & text{if $frac{j-1}{j} le x le 1$} \
end{cases} \
end{align*}
begin{align*}
limlimits_{j to infty} f_j(x) &= f(x) = x^2 \
end{align*}
What would be a series of piecewise linear functions such that $sum_{j=0}^infty f_j(x) = f(x) = x^2$?
real-analysis sequences-and-series sequence-of-function
real-analysis sequences-and-series sequence-of-function
asked Dec 4 '18 at 6:50
clayclay
765414
asked Dec 4 '18 at 6:50
clayclay
765414
asked Dec 4 '18 at 6:50
clayclay
765414
asked Dec 4 '18 at 6:50
clayclay
765414
asked Dec 4 '18 at 6:50
clayclay
765414
765414
1
$begingroup$
A thing to try is if to consider what is $f_{j+1}(x) - f_{j}(x)$. Then use that to define partial sum $g_n(x)$ (this will be quite messy since you have to take into account domains and the like) and show that this converges to $f(x)$. I am aware that this is very broad, but the object to consider is $f_{j+1}(x) - f_{j}(x)$.
$endgroup$
– twnly
Dec 4 '18 at 7:03
add a comment |
1
$begingroup$
A thing to try is if to consider what is $f_{j+1}(x) - f_{j}(x)$. Then use that to define partial sum $g_n(x)$ (this will be quite messy since you have to take into account domains and the like) and show that this converges to $f(x)$. I am aware that this is very broad, but the object to consider is $f_{j+1}(x) - f_{j}(x)$.
$endgroup$
– twnly
Dec 4 '18 at 7:03
1
1
$begingroup$
A thing to try is if to consider what is $f_{j+1}(x) - f_{j}(x)$. Then use that to define partial sum $g_n(x)$ (this will be quite messy since you have to take into account domains and the like) and show that this converges to $f(x)$. I am aware that this is very broad, but the object to consider is $f_{j+1}(x) - f_{j}(x)$.
$endgroup$
– twnly
Dec 4 '18 at 7:03
$begingroup$
A thing to try is if to consider what is $f_{j+1}(x) - f_{j}(x)$. Then use that to define partial sum $g_n(x)$ (this will be quite messy since you have to take into account domains and the like) and show that this converges to $f(x)$. I am aware that this is very broad, but the object to consider is $f_{j+1}(x) - f_{j}(x)$.
$endgroup$
– twnly
Dec 4 '18 at 7:03
add a comment |
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$begingroup$
A thing to try is if to consider what is $f_{j+1}(x) - f_{j}(x)$. Then use that to define partial sum $g_n(x)$ (this will be quite messy since you have to take into account domains and the like) and show that this converges to $f(x)$. I am aware that this is very broad, but the object to consider is $f_{j+1}(x) - f_{j}(x)$.
$endgroup$
– twnly
Dec 4 '18 at 7:03