Approximation of one function with smooth functions
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There is a continuous but non-smooth function at $x=0$:
$$
f(x)=left[frac{2}{1+e^{-2x}}-1 right]_{+},$$
where $[u]_+equiv
begin{cases}
u,quad u geqslant 0\
0, quad u<0
end{cases}
$
So, $f(x)$ is a sigmoid function on the right half-line $xgeqslant 0$ and equals to $0$ when $x<0$. I need to find a sequence of differentiable functions ${f_k(x)}$ such that it converges uniformly on $mathbb{R}$ to the function $f(x)$.
real-analysis approximation smooth-functions
asked Nov 28 at 22:13
Artem Zefirov
979
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up vote
0
down vote
favorite
There is a continuous but non-smooth function at $x=0$:
$$
f(x)=left[frac{2}{1+e^{-2x}}-1 right]_{+},$$
where $[u]_+equiv
begin{cases}
u,quad u geqslant 0\
0, quad u<0
end{cases}
$
So, $f(x)$ is a sigmoid function on the right half-line $xgeqslant 0$ and equals to $0$ when $x<0$. I need to find a sequence of differentiable functions ${f_k(x)}$ such that it converges uniformly on $mathbb{R}$ to the function $f(x)$.
real-analysis approximation smooth-functions
asked Nov 28 at 22:13
Artem Zefirov
979
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
There is a continuous but non-smooth function at $x=0$:
$$
f(x)=left[frac{2}{1+e^{-2x}}-1 right]_{+},$$
where $[u]_+equiv
begin{cases}
u,quad u geqslant 0\
0, quad u<0
end{cases}
$
So, $f(x)$ is a sigmoid function on the right half-line $xgeqslant 0$ and equals to $0$ when $x<0$. I need to find a sequence of differentiable functions ${f_k(x)}$ such that it converges uniformly on $mathbb{R}$ to the function $f(x)$.
real-analysis approximation smooth-functions
asked Nov 28 at 22:13
Artem Zefirov
979
There is a continuous but non-smooth function at $x=0$:
$$
f(x)=left[frac{2}{1+e^{-2x}}-1 right]_{+},$$
where $[u]_+equiv
begin{cases}
u,quad u geqslant 0\
0, quad u<0
end{cases}
$
So, $f(x)$ is a sigmoid function on the right half-line $xgeqslant 0$ and equals to $0$ when $x<0$. I need to find a sequence of differentiable functions ${f_k(x)}$ such that it converges uniformly on $mathbb{R}$ to the function $f(x)$.
real-analysis approximation smooth-functions
real-analysis approximation smooth-functions
asked Nov 28 at 22:13
Artem Zefirov
979
asked Nov 28 at 22:13
Artem Zefirov
979
edited Nov 29 at 8:31
asked Nov 28 at 22:13
Artem Zefirov
979
asked Nov 28 at 22:13
Artem Zefirov
979
asked Nov 28 at 22:13
Artem Zefirov
979
979
add a comment |
add a comment |
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