All Questions
Tagged with fa.functional-analysis fractals
8 questions
0
votes
0
answers
46
views
Fractal dimension using wavelets [closed]
I'm trying to estimate the fractal dimension of a function.
I created the log(energies) Vs log(scales) plot and I'm computing the Fractal Dimension (D) from the slope using the relation
$$
\alpha = -...
4
votes
1
answer
205
views
Existence of an $\alpha$-Hölder continuous function whose graph has positive Hausdorff measure of maximal dimension
It is standard that if $f:[0,1] \rightarrow \mathbb{R}$ is $\alpha$-Hölder continuous, then its graph has Hausdorff dimension at most $2-\alpha$. My naive expectation was that "most" graphs ...
- 45
2
votes
1
answer
280
views
Set operations over iterated function systems
An iterated function system (IFS) is a finite set of contraction mappings on a complete metric space. Symbolically,
$$\{f_i :X \to X \mid i=1,2,..n \},\quad n \in \mathbb{N}$$
is an IFS if each $...
- 21
3
votes
1
answer
383
views
"Nice" functions on infinite-dimensional space of germs of continuous functions at a point
Consider set of all germs of continuous functions at some point.
Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...
- 24.9k
5
votes
1
answer
305
views
boundary density of the Von Koch flake
Given a measurable set $K\subset \mathbb{R}^d$ we consider the occupation ratio $$f_r(x)=vol(K\cap B(x,r))/r^d$$ and especially the asymptotics when $r\to 0$. When $K$ has a fractal boundary and $x$ ...
- 1,352
6
votes
2
answers
476
views
function that is the average of affine transformations of itself
Consider the function $f : \mathbb{R} \to [-1,1]$ with
$$
f(x) = \begin{cases}
-1 & x \le -1 \\
+1 & x \ge +1 \\
\frac{f(\frac32 (x-\frac13)) + f(\frac32 (x+\...
- 165
1
vote
0
answers
215
views
Classification of Self similar sets
I am looking at self similar sets in $\mathbb{C}$ defined as the fixed set or a sequence of contractions or an iterated function system. I am currently trying to classify these sets by how they are ...
- 11
4
votes
1
answer
1k
views
Hausdorff dimension of graphs .
Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
- 3,630