Constructing a connection between two functions
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Let $f(x)$ and $g(x)$ be two functions on the real numbers. At some point, say $x=0$ for convenience, the two functions intersect: $f(0) = g(0)$. The first and higher derivatives of $f$ and $g$ can be assumed to be unequal at $x=0$.
The aim is to construct a new function $h(x)$, which is equal to $f(x)$ for $x < -d$ and equal to $g(x)$ for $x > d$, with $d$ a small parameter. We want the function $h$ to be as smooth as possible. For that reason we introduced the small interval $(-d, d)$ where the transition between $f$ and $g$ takes place.
There are two approaches that come to mind:
One of the simplest methods is presumably to choose a sigmoidal function $s(x)$ which runs between $0$ and $1$, for example $s(x) = 0.5*(1+tanh(x/d))$ will do the job. We can then construct $h$ as follows: $h(x) = (1-s(x))*f(x) + s(x)*g(x)$. However, I am afraid that the result is often a bit more curved than anticipated.
Another method is to define a polynomial of degree $N$ on the interval $(-d, d)$. The coefficients are determined by matching the lowest Taylor coefficients of the polynomial on the one hand and $f$ or $g$ on the other hand, at $x=-d$ and $x=d$.
I wonder: is there a particular method that is recommended in the mathematical literature?
real-analysis functions
asked Nov 20 at 0:37
M. Wind
2,086715
add a comment |
up vote
0
down vote
favorite
Let $f(x)$ and $g(x)$ be two functions on the real numbers. At some point, say $x=0$ for convenience, the two functions intersect: $f(0) = g(0)$. The first and higher derivatives of $f$ and $g$ can be assumed to be unequal at $x=0$.
The aim is to construct a new function $h(x)$, which is equal to $f(x)$ for $x < -d$ and equal to $g(x)$ for $x > d$, with $d$ a small parameter. We want the function $h$ to be as smooth as possible. For that reason we introduced the small interval $(-d, d)$ where the transition between $f$ and $g$ takes place.
There are two approaches that come to mind:
One of the simplest methods is presumably to choose a sigmoidal function $s(x)$ which runs between $0$ and $1$, for example $s(x) = 0.5*(1+tanh(x/d))$ will do the job. We can then construct $h$ as follows: $h(x) = (1-s(x))*f(x) + s(x)*g(x)$. However, I am afraid that the result is often a bit more curved than anticipated.
Another method is to define a polynomial of degree $N$ on the interval $(-d, d)$. The coefficients are determined by matching the lowest Taylor coefficients of the polynomial on the one hand and $f$ or $g$ on the other hand, at $x=-d$ and $x=d$.
I wonder: is there a particular method that is recommended in the mathematical literature?
real-analysis functions
asked Nov 20 at 0:37
M. Wind
2,086715
If $f$ is $C^infty$ near $-d$ and $g$ is $C^infty$ near $d$, you can glue them together with a $C^infty$ function using en.wikipedia.org/wiki/…
– Federico
Nov 20 at 0:45
This is the best you can get from the point of view of regularity, but the transition function can be chosen quite arbitrarily, so you might prefer some more specific gluing methods
– Federico
Nov 20 at 0:48
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $f(x)$ and $g(x)$ be two functions on the real numbers. At some point, say $x=0$ for convenience, the two functions intersect: $f(0) = g(0)$. The first and higher derivatives of $f$ and $g$ can be assumed to be unequal at $x=0$.
The aim is to construct a new function $h(x)$, which is equal to $f(x)$ for $x < -d$ and equal to $g(x)$ for $x > d$, with $d$ a small parameter. We want the function $h$ to be as smooth as possible. For that reason we introduced the small interval $(-d, d)$ where the transition between $f$ and $g$ takes place.
There are two approaches that come to mind:
One of the simplest methods is presumably to choose a sigmoidal function $s(x)$ which runs between $0$ and $1$, for example $s(x) = 0.5*(1+tanh(x/d))$ will do the job. We can then construct $h$ as follows: $h(x) = (1-s(x))*f(x) + s(x)*g(x)$. However, I am afraid that the result is often a bit more curved than anticipated.
Another method is to define a polynomial of degree $N$ on the interval $(-d, d)$. The coefficients are determined by matching the lowest Taylor coefficients of the polynomial on the one hand and $f$ or $g$ on the other hand, at $x=-d$ and $x=d$.
I wonder: is there a particular method that is recommended in the mathematical literature?
real-analysis functions
asked Nov 20 at 0:37
M. Wind
2,086715
Let $f(x)$ and $g(x)$ be two functions on the real numbers. At some point, say $x=0$ for convenience, the two functions intersect: $f(0) = g(0)$. The first and higher derivatives of $f$ and $g$ can be assumed to be unequal at $x=0$.
The aim is to construct a new function $h(x)$, which is equal to $f(x)$ for $x < -d$ and equal to $g(x)$ for $x > d$, with $d$ a small parameter. We want the function $h$ to be as smooth as possible. For that reason we introduced the small interval $(-d, d)$ where the transition between $f$ and $g$ takes place.
There are two approaches that come to mind:
One of the simplest methods is presumably to choose a sigmoidal function $s(x)$ which runs between $0$ and $1$, for example $s(x) = 0.5*(1+tanh(x/d))$ will do the job. We can then construct $h$ as follows: $h(x) = (1-s(x))*f(x) + s(x)*g(x)$. However, I am afraid that the result is often a bit more curved than anticipated.
Another method is to define a polynomial of degree $N$ on the interval $(-d, d)$. The coefficients are determined by matching the lowest Taylor coefficients of the polynomial on the one hand and $f$ or $g$ on the other hand, at $x=-d$ and $x=d$.
I wonder: is there a particular method that is recommended in the mathematical literature?
real-analysis functions
real-analysis functions
asked Nov 20 at 0:37
M. Wind
2,086715
asked Nov 20 at 0:37
M. Wind
2,086715
asked Nov 20 at 0:37
M. Wind
2,086715
asked Nov 20 at 0:37
M. Wind
2,086715
asked Nov 20 at 0:37
M. Wind
2,086715
2,086715
If $f$ is $C^infty$ near $-d$ and $g$ is $C^infty$ near $d$, you can glue them together with a $C^infty$ function using en.wikipedia.org/wiki/…
– Federico
Nov 20 at 0:45
This is the best you can get from the point of view of regularity, but the transition function can be chosen quite arbitrarily, so you might prefer some more specific gluing methods
– Federico
Nov 20 at 0:48
add a comment |
If $f$ is $C^infty$ near $-d$ and $g$ is $C^infty$ near $d$, you can glue them together with a $C^infty$ function using en.wikipedia.org/wiki/…
– Federico
Nov 20 at 0:45
This is the best you can get from the point of view of regularity, but the transition function can be chosen quite arbitrarily, so you might prefer some more specific gluing methods
– Federico
Nov 20 at 0:48
If $f$ is $C^infty$ near $-d$ and $g$ is $C^infty$ near $d$, you can glue them together with a $C^infty$ function using en.wikipedia.org/wiki/…
– Federico
Nov 20 at 0:45
If $f$ is $C^infty$ near $-d$ and $g$ is $C^infty$ near $d$, you can glue them together with a $C^infty$ function using en.wikipedia.org/wiki/…
– Federico
Nov 20 at 0:45
This is the best you can get from the point of view of regularity, but the transition function can be chosen quite arbitrarily, so you might prefer some more specific gluing methods
– Federico
Nov 20 at 0:48
This is the best you can get from the point of view of regularity, but the transition function can be chosen quite arbitrarily, so you might prefer some more specific gluing methods
– Federico
Nov 20 at 0:48
add a comment |
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If $f$ is $C^infty$ near $-d$ and $g$ is $C^infty$ near $d$, you can glue them together with a $C^infty$ function using en.wikipedia.org/wiki/…
– Federico
Nov 20 at 0:45
This is the best you can get from the point of view of regularity, but the transition function can be chosen quite arbitrarily, so you might prefer some more specific gluing methods
– Federico
Nov 20 at 0:48