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The box dimension of the graph of a continuous function is $ge 1$
The 2019 Stack Overflow Developer Survey Results Are InIs the Hausdorff dimension less than the box counting dimension?How to correctly calculate the fractal dimension of a finite set of points?Relationship between the Hausdorff dimension and the Box-counting dimensionDefinition of Minkowski dimensionFractal dimension of the function $f(x)=sum_n=1^inftyfracmathrmsignleft(sin(nx)right)n$Lower Box Dimension InequalitySufficient condition for fractal dimension of continuous nowhere differentiable functionsBox dimension, graphs and sum of functionsA pointless characterization of the relation between a fractal and its code space.Computing the lipschitz constant of an affine IFS
$begingroup$
In the book Interpolation and Approximation with Splines and Fractals by Prof. Massopoust one studies the following theorem with regard two the box dimension of affine fractal interpolation functions:
Suppose that $f:[a,b] to mathbbR$ is an affine fractal interpolation funtion defined as above and $G$ its graph. If $sum_i = 1^N |lambda_i| > 1$ and if the interpolation set $Y$ is not collinear, then the box dimension of $G$ is given by the formula $dim_B ; G = 1 + log_N(sum_i = 1^N|lambda_i|)$
But more precisely, I'm interested in the following argument, which is used during the proof:
Given that f is a continuous function, then $dim_B G ge 1$
However, I cannot find a justification for this. Any hints?
You can find any missing piece of information in these lecture notes.
My try
To cover the graph of the function in the interval $[a,b]$ we need more than $lceil fracb-aepsilon rceil$ squares, so we have to compute $limlimits_epsilon to 0 fraclog lceil fracb-aepsilon rceillog frac1epsilon$.
We have $lceil fracb-aepsilon rceil ge fracb-aepsilon implies log lceil fracb-aepsilon rceil ge log fracb-aepsilon$. Thus, we get $fraclog lceil fracb-aepsilon rceillog frac1epsilon ge fraclog(b-a)-log(epsilon)-log(epsilon)$ and taking limits $dim_B ; G ge 1$.
But in fact, I didn't need the hypothesis that the function was continuous. So I guess my conclusion extends to any function $f:[a,b] to mathbbR$. Could you check my conclusions? (note the proof verification tag). I wonder if this can be generalized to higher dimensions.
analysis proof-verification fractals dimension-theory
asked Mar 11 at 17:05
JavierJavier
2,10521235
$endgroup$
add a comment |
$begingroup$
In the book Interpolation and Approximation with Splines and Fractals by Prof. Massopoust one studies the following theorem with regard two the box dimension of affine fractal interpolation functions:
Suppose that $f:[a,b] to mathbbR$ is an affine fractal interpolation funtion defined as above and $G$ its graph. If $sum_i = 1^N |lambda_i| > 1$ and if the interpolation set $Y$ is not collinear, then the box dimension of $G$ is given by the formula $dim_B ; G = 1 + log_N(sum_i = 1^N|lambda_i|)$
But more precisely, I'm interested in the following argument, which is used during the proof:
Given that f is a continuous function, then $dim_B G ge 1$
However, I cannot find a justification for this. Any hints?
You can find any missing piece of information in these lecture notes.
My try
To cover the graph of the function in the interval $[a,b]$ we need more than $lceil fracb-aepsilon rceil$ squares, so we have to compute $limlimits_epsilon to 0 fraclog lceil fracb-aepsilon rceillog frac1epsilon$.
We have $lceil fracb-aepsilon rceil ge fracb-aepsilon implies log lceil fracb-aepsilon rceil ge log fracb-aepsilon$. Thus, we get $fraclog lceil fracb-aepsilon rceillog frac1epsilon ge fraclog(b-a)-log(epsilon)-log(epsilon)$ and taking limits $dim_B ; G ge 1$.
But in fact, I didn't need the hypothesis that the function was continuous. So I guess my conclusion extends to any function $f:[a,b] to mathbbR$. Could you check my conclusions? (note the proof verification tag). I wonder if this can be generalized to higher dimensions.
analysis proof-verification fractals dimension-theory
asked Mar 11 at 17:05
JavierJavier
2,10521235
$endgroup$
$begingroup$
continuity could be needed to count boxes between the maximum and minimum of the function
$endgroup$
– Javier
Mar 17 at 15:59
add a comment |
$begingroup$
In the book Interpolation and Approximation with Splines and Fractals by Prof. Massopoust one studies the following theorem with regard two the box dimension of affine fractal interpolation functions:
Suppose that $f:[a,b] to mathbbR$ is an affine fractal interpolation funtion defined as above and $G$ its graph. If $sum_i = 1^N |lambda_i| > 1$ and if the interpolation set $Y$ is not collinear, then the box dimension of $G$ is given by the formula $dim_B ; G = 1 + log_N(sum_i = 1^N|lambda_i|)$
But more precisely, I'm interested in the following argument, which is used during the proof:
Given that f is a continuous function, then $dim_B G ge 1$
However, I cannot find a justification for this. Any hints?
You can find any missing piece of information in these lecture notes.
My try
To cover the graph of the function in the interval $[a,b]$ we need more than $lceil fracb-aepsilon rceil$ squares, so we have to compute $limlimits_epsilon to 0 fraclog lceil fracb-aepsilon rceillog frac1epsilon$.
We have $lceil fracb-aepsilon rceil ge fracb-aepsilon implies log lceil fracb-aepsilon rceil ge log fracb-aepsilon$. Thus, we get $fraclog lceil fracb-aepsilon rceillog frac1epsilon ge fraclog(b-a)-log(epsilon)-log(epsilon)$ and taking limits $dim_B ; G ge 1$.
But in fact, I didn't need the hypothesis that the function was continuous. So I guess my conclusion extends to any function $f:[a,b] to mathbbR$. Could you check my conclusions? (note the proof verification tag). I wonder if this can be generalized to higher dimensions.
analysis proof-verification fractals dimension-theory
asked Mar 11 at 17:05
JavierJavier
2,10521235
$endgroup$
In the book Interpolation and Approximation with Splines and Fractals by Prof. Massopoust one studies the following theorem with regard two the box dimension of affine fractal interpolation functions:
Suppose that $f:[a,b] to mathbbR$ is an affine fractal interpolation funtion defined as above and $G$ its graph. If $sum_i = 1^N |lambda_i| > 1$ and if the interpolation set $Y$ is not collinear, then the box dimension of $G$ is given by the formula $dim_B ; G = 1 + log_N(sum_i = 1^N|lambda_i|)$
But more precisely, I'm interested in the following argument, which is used during the proof:
Given that f is a continuous function, then $dim_B G ge 1$
However, I cannot find a justification for this. Any hints?
You can find any missing piece of information in these lecture notes.
My try
To cover the graph of the function in the interval $[a,b]$ we need more than $lceil fracb-aepsilon rceil$ squares, so we have to compute $limlimits_epsilon to 0 fraclog lceil fracb-aepsilon rceillog frac1epsilon$.
We have $lceil fracb-aepsilon rceil ge fracb-aepsilon implies log lceil fracb-aepsilon rceil ge log fracb-aepsilon$. Thus, we get $fraclog lceil fracb-aepsilon rceillog frac1epsilon ge fraclog(b-a)-log(epsilon)-log(epsilon)$ and taking limits $dim_B ; G ge 1$.
But in fact, I didn't need the hypothesis that the function was continuous. So I guess my conclusion extends to any function $f:[a,b] to mathbbR$. Could you check my conclusions? (note the proof verification tag). I wonder if this can be generalized to higher dimensions.
analysis proof-verification fractals dimension-theory
analysis proof-verification fractals dimension-theory
asked Mar 11 at 17:05
JavierJavier
2,10521235
asked Mar 11 at 17:05
JavierJavier
2,10521235
edited Mar 17 at 15:58
Javier
asked Mar 11 at 17:05
JavierJavier
2,10521235
asked Mar 11 at 17:05
JavierJavier
2,10521235
asked Mar 11 at 17:05
JavierJavier
2,10521235
2,10521235
$begingroup$
continuity could be needed to count boxes between the maximum and minimum of the function
$endgroup$
– Javier
Mar 17 at 15:59
add a comment |
$begingroup$
continuity could be needed to count boxes between the maximum and minimum of the function
$endgroup$
– Javier
Mar 17 at 15:59
$begingroup$
continuity could be needed to count boxes between the maximum and minimum of the function
$endgroup$
– Javier
Mar 17 at 15:59
$begingroup$
continuity could be needed to count boxes between the maximum and minimum of the function
$endgroup$
– Javier
Mar 17 at 15:59
add a comment |
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$begingroup$
continuity could be needed to count boxes between the maximum and minimum of the function
$endgroup$
– Javier
Mar 17 at 15:59