Covering Number of $C^mleft[ 0, 1 right]$ in $L^2 left[ 0, 1 right]$
$begingroup$
Let $m$ be a positive integer, for any $epsilon gt 0$, denote $N(epsilon)$ the covering number of $C^mleft[ 0, 1 right]$ in $L^2 left[ 0, 1 right]$, i.e., the smallest positive integer $N$ such that there exists $N$ functions $f_1, cdots, f_N$ in $C^mleft[ 0, 1 right]$, whose $m$-th derivatives are uniformly bounded by $M$, s.t. for any $f in C^mleft[ 0, 1 right], lVert f^{(m)} rVert leq M$,
begin{equation}
exists 1 leq i leq N, lVert f-f_i rVert_{L^2} lt epsilon.
end{equation}
I conjecture that $log N(epsilon)$ is approximately $C_0 epsilon^{-frac{2}{2m-1}}$ and I believe that there must be some previous results on it, but I didn't find any.
real-analysis functional-analysis analysis
asked Dec 23 '18 at 15:18
Martin ChowMartin Chow
1189
$endgroup$
add a comment |
$begingroup$
Let $m$ be a positive integer, for any $epsilon gt 0$, denote $N(epsilon)$ the covering number of $C^mleft[ 0, 1 right]$ in $L^2 left[ 0, 1 right]$, i.e., the smallest positive integer $N$ such that there exists $N$ functions $f_1, cdots, f_N$ in $C^mleft[ 0, 1 right]$, whose $m$-th derivatives are uniformly bounded by $M$, s.t. for any $f in C^mleft[ 0, 1 right], lVert f^{(m)} rVert leq M$,
begin{equation}
exists 1 leq i leq N, lVert f-f_i rVert_{L^2} lt epsilon.
end{equation}
I conjecture that $log N(epsilon)$ is approximately $C_0 epsilon^{-frac{2}{2m-1}}$ and I believe that there must be some previous results on it, but I didn't find any.
real-analysis functional-analysis analysis
asked Dec 23 '18 at 15:18
Martin ChowMartin Chow
1189
$endgroup$
add a comment |
$begingroup$
Let $m$ be a positive integer, for any $epsilon gt 0$, denote $N(epsilon)$ the covering number of $C^mleft[ 0, 1 right]$ in $L^2 left[ 0, 1 right]$, i.e., the smallest positive integer $N$ such that there exists $N$ functions $f_1, cdots, f_N$ in $C^mleft[ 0, 1 right]$, whose $m$-th derivatives are uniformly bounded by $M$, s.t. for any $f in C^mleft[ 0, 1 right], lVert f^{(m)} rVert leq M$,
begin{equation}
exists 1 leq i leq N, lVert f-f_i rVert_{L^2} lt epsilon.
end{equation}
I conjecture that $log N(epsilon)$ is approximately $C_0 epsilon^{-frac{2}{2m-1}}$ and I believe that there must be some previous results on it, but I didn't find any.
real-analysis functional-analysis analysis
asked Dec 23 '18 at 15:18
Martin ChowMartin Chow
1189
$endgroup$
Let $m$ be a positive integer, for any $epsilon gt 0$, denote $N(epsilon)$ the covering number of $C^mleft[ 0, 1 right]$ in $L^2 left[ 0, 1 right]$, i.e., the smallest positive integer $N$ such that there exists $N$ functions $f_1, cdots, f_N$ in $C^mleft[ 0, 1 right]$, whose $m$-th derivatives are uniformly bounded by $M$, s.t. for any $f in C^mleft[ 0, 1 right], lVert f^{(m)} rVert leq M$,
begin{equation}
exists 1 leq i leq N, lVert f-f_i rVert_{L^2} lt epsilon.
end{equation}
I conjecture that $log N(epsilon)$ is approximately $C_0 epsilon^{-frac{2}{2m-1}}$ and I believe that there must be some previous results on it, but I didn't find any.
real-analysis functional-analysis analysis
real-analysis functional-analysis analysis
asked Dec 23 '18 at 15:18
Martin ChowMartin Chow
1189
asked Dec 23 '18 at 15:18
Martin ChowMartin Chow
1189
edited Dec 23 '18 at 15:26
Martin Chow
asked Dec 23 '18 at 15:18
Martin ChowMartin Chow
1189
asked Dec 23 '18 at 15:18
Martin ChowMartin Chow
1189
asked Dec 23 '18 at 15:18
Martin ChowMartin Chow
1189
1189
add a comment |
add a comment |
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