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Simultaneous asymptotic expansion in multiple points
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Let $Omegasubsetmathbb R$ be open and connected. Assume I have some non-linear, smooth function $g:Omega tomathbb R$.
Given disjoint base points $a_1,ldots,a_nin Omega$ (and possible $pminfty)$, I want to find an asymptotic expansion $hat g $ of $g$ such that at each base point $a_k$, $hat g$ is an asymptotic expansion of $g$ up to order $c_k$.
For example if $g(x) = sqrt1+frac1x^2$, then $hat g = 1+frac 1x$ is asymptotic to $g$ both at $0$ and $infty$ up to first order.
(at $0$, $g(x)approx frac 1x + frac 12 x - frac 18 x^3 pmldots$ and at $infty$, $g(x)approx 1+frac12x^2- frac18x^4pmldots$)
One naive way to do it would be to take an appropriate bump function $phi$ and do $hat g = sum phi(x-a_k)cdot hat g_k(x)$ where $hat g_k$ is the local asymptotic expansion at $a_k$. However this is a bad way to do it since we would like to have that, for "nice" functions $g$, the expansion converges globally against the target function when we increase the orders.
Are there any common techniques to tackle these kinds of problems?
reference-request asymptotics approximation-theory
asked Aug 1 at 16:52
Hyperplane
3,1831624
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up vote
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Let $Omegasubsetmathbb R$ be open and connected. Assume I have some non-linear, smooth function $g:Omega tomathbb R$.
Given disjoint base points $a_1,ldots,a_nin Omega$ (and possible $pminfty)$, I want to find an asymptotic expansion $hat g $ of $g$ such that at each base point $a_k$, $hat g$ is an asymptotic expansion of $g$ up to order $c_k$.
For example if $g(x) = sqrt1+frac1x^2$, then $hat g = 1+frac 1x$ is asymptotic to $g$ both at $0$ and $infty$ up to first order.
(at $0$, $g(x)approx frac 1x + frac 12 x - frac 18 x^3 pmldots$ and at $infty$, $g(x)approx 1+frac12x^2- frac18x^4pmldots$)
One naive way to do it would be to take an appropriate bump function $phi$ and do $hat g = sum phi(x-a_k)cdot hat g_k(x)$ where $hat g_k$ is the local asymptotic expansion at $a_k$. However this is a bad way to do it since we would like to have that, for "nice" functions $g$, the expansion converges globally against the target function when we increase the orders.
Are there any common techniques to tackle these kinds of problems?
reference-request asymptotics approximation-theory
asked Aug 1 at 16:52
Hyperplane
3,1831624
Perhaps matched asymptotic expansions might be something similar to what you want?
– Antonio Vargas
Aug 1 at 23:50
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $Omegasubsetmathbb R$ be open and connected. Assume I have some non-linear, smooth function $g:Omega tomathbb R$.
Given disjoint base points $a_1,ldots,a_nin Omega$ (and possible $pminfty)$, I want to find an asymptotic expansion $hat g $ of $g$ such that at each base point $a_k$, $hat g$ is an asymptotic expansion of $g$ up to order $c_k$.
For example if $g(x) = sqrt1+frac1x^2$, then $hat g = 1+frac 1x$ is asymptotic to $g$ both at $0$ and $infty$ up to first order.
(at $0$, $g(x)approx frac 1x + frac 12 x - frac 18 x^3 pmldots$ and at $infty$, $g(x)approx 1+frac12x^2- frac18x^4pmldots$)
One naive way to do it would be to take an appropriate bump function $phi$ and do $hat g = sum phi(x-a_k)cdot hat g_k(x)$ where $hat g_k$ is the local asymptotic expansion at $a_k$. However this is a bad way to do it since we would like to have that, for "nice" functions $g$, the expansion converges globally against the target function when we increase the orders.
Are there any common techniques to tackle these kinds of problems?
reference-request asymptotics approximation-theory
asked Aug 1 at 16:52
Hyperplane
3,1831624
Let $Omegasubsetmathbb R$ be open and connected. Assume I have some non-linear, smooth function $g:Omega tomathbb R$.
Given disjoint base points $a_1,ldots,a_nin Omega$ (and possible $pminfty)$, I want to find an asymptotic expansion $hat g $ of $g$ such that at each base point $a_k$, $hat g$ is an asymptotic expansion of $g$ up to order $c_k$.
For example if $g(x) = sqrt1+frac1x^2$, then $hat g = 1+frac 1x$ is asymptotic to $g$ both at $0$ and $infty$ up to first order.
(at $0$, $g(x)approx frac 1x + frac 12 x - frac 18 x^3 pmldots$ and at $infty$, $g(x)approx 1+frac12x^2- frac18x^4pmldots$)
One naive way to do it would be to take an appropriate bump function $phi$ and do $hat g = sum phi(x-a_k)cdot hat g_k(x)$ where $hat g_k$ is the local asymptotic expansion at $a_k$. However this is a bad way to do it since we would like to have that, for "nice" functions $g$, the expansion converges globally against the target function when we increase the orders.
Are there any common techniques to tackle these kinds of problems?
reference-request asymptotics approximation-theory
asked Aug 1 at 16:52
Hyperplane
3,1831624
asked Aug 1 at 16:52
Hyperplane
3,1831624
asked Aug 1 at 16:52
Hyperplane
3,1831624
asked Aug 1 at 16:52
Hyperplane
3,1831624
3,1831624
Perhaps matched asymptotic expansions might be something similar to what you want?
– Antonio Vargas
Aug 1 at 23:50
add a comment |Â
Perhaps matched asymptotic expansions might be something similar to what you want?
– Antonio Vargas
Aug 1 at 23:50
Perhaps matched asymptotic expansions might be something similar to what you want?
– Antonio Vargas
Aug 1 at 23:50
Perhaps matched asymptotic expansions might be something similar to what you want?
– Antonio Vargas
Aug 1 at 23:50
add a comment |Â
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Perhaps matched asymptotic expansions might be something similar to what you want?
– Antonio Vargas
Aug 1 at 23:50