Particularize the Collage Theorem in $mathbb{R}$
$begingroup$
Consider the collage theorem stated as below (Interpolation and Approximation with Splines and Fractals by P. Massopust):
I want to get the particular case for $mathbb{R}$ with its usual distance, which is a complete metric space (the idea is to help me to remember the theorem). The only compact sets are intervals of the form $[a,b]$. So fix $A = [a,b]$ and some $epsilon > 0$.
The problem here, is to determine what is $F(A)$, so that we can get a clearer idea of how Hausdorff distance behaves. We have $F(A) = cup_{i = 1}^N overline{f_i}(A)$ where $f_i$ are contractions and $overline{f_i}(A)$ is defined as the image of $A$ under $f_i$.
Since $f_i$ are lipschitz, they are continuous and $overline{f_i}(A)$ will have to be a compact set (we could get no matter what compact set in $mathbb{R}$ right?). So we get a family of $N$ compact sets: $[a_i,b_i]$ for $i = 1, ldots , N$. Since $f_i$ are contractions we further know that these compact sets have less size than $A$.
Can I get any further intuition from here? In other words, what is the collage theorem saying about this finite collection of compact sets?
real-analysis functional-analysis metric-spaces fractals
asked Dec 3 '18 at 16:08
JavierJavier
2,01621133
$endgroup$
add a comment |
$begingroup$
Consider the collage theorem stated as below (Interpolation and Approximation with Splines and Fractals by P. Massopust):
I want to get the particular case for $mathbb{R}$ with its usual distance, which is a complete metric space (the idea is to help me to remember the theorem). The only compact sets are intervals of the form $[a,b]$. So fix $A = [a,b]$ and some $epsilon > 0$.
The problem here, is to determine what is $F(A)$, so that we can get a clearer idea of how Hausdorff distance behaves. We have $F(A) = cup_{i = 1}^N overline{f_i}(A)$ where $f_i$ are contractions and $overline{f_i}(A)$ is defined as the image of $A$ under $f_i$.
Since $f_i$ are lipschitz, they are continuous and $overline{f_i}(A)$ will have to be a compact set (we could get no matter what compact set in $mathbb{R}$ right?). So we get a family of $N$ compact sets: $[a_i,b_i]$ for $i = 1, ldots , N$. Since $f_i$ are contractions we further know that these compact sets have less size than $A$.
Can I get any further intuition from here? In other words, what is the collage theorem saying about this finite collection of compact sets?
real-analysis functional-analysis metric-spaces fractals
asked Dec 3 '18 at 16:08
JavierJavier
2,01621133
$endgroup$
add a comment |
$begingroup$
Consider the collage theorem stated as below (Interpolation and Approximation with Splines and Fractals by P. Massopust):
I want to get the particular case for $mathbb{R}$ with its usual distance, which is a complete metric space (the idea is to help me to remember the theorem). The only compact sets are intervals of the form $[a,b]$. So fix $A = [a,b]$ and some $epsilon > 0$.
The problem here, is to determine what is $F(A)$, so that we can get a clearer idea of how Hausdorff distance behaves. We have $F(A) = cup_{i = 1}^N overline{f_i}(A)$ where $f_i$ are contractions and $overline{f_i}(A)$ is defined as the image of $A$ under $f_i$.
Since $f_i$ are lipschitz, they are continuous and $overline{f_i}(A)$ will have to be a compact set (we could get no matter what compact set in $mathbb{R}$ right?). So we get a family of $N$ compact sets: $[a_i,b_i]$ for $i = 1, ldots , N$. Since $f_i$ are contractions we further know that these compact sets have less size than $A$.
Can I get any further intuition from here? In other words, what is the collage theorem saying about this finite collection of compact sets?
real-analysis functional-analysis metric-spaces fractals
asked Dec 3 '18 at 16:08
JavierJavier
2,01621133
$endgroup$
Consider the collage theorem stated as below (Interpolation and Approximation with Splines and Fractals by P. Massopust):
I want to get the particular case for $mathbb{R}$ with its usual distance, which is a complete metric space (the idea is to help me to remember the theorem). The only compact sets are intervals of the form $[a,b]$. So fix $A = [a,b]$ and some $epsilon > 0$.
The problem here, is to determine what is $F(A)$, so that we can get a clearer idea of how Hausdorff distance behaves. We have $F(A) = cup_{i = 1}^N overline{f_i}(A)$ where $f_i$ are contractions and $overline{f_i}(A)$ is defined as the image of $A$ under $f_i$.
Since $f_i$ are lipschitz, they are continuous and $overline{f_i}(A)$ will have to be a compact set (we could get no matter what compact set in $mathbb{R}$ right?). So we get a family of $N$ compact sets: $[a_i,b_i]$ for $i = 1, ldots , N$. Since $f_i$ are contractions we further know that these compact sets have less size than $A$.
Can I get any further intuition from here? In other words, what is the collage theorem saying about this finite collection of compact sets?
real-analysis functional-analysis metric-spaces fractals
real-analysis functional-analysis metric-spaces fractals
asked Dec 3 '18 at 16:08
JavierJavier
2,01621133
asked Dec 3 '18 at 16:08
JavierJavier
2,01621133
asked Dec 3 '18 at 16:08
JavierJavier
2,01621133
asked Dec 3 '18 at 16:08
JavierJavier
2,01621133
asked Dec 3 '18 at 16:08
JavierJavier
2,01621133
2,01621133
add a comment |
add a comment |
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