Inverse problem for one-dimensional fractal measures via iterated function systems and the moment method

1990 ◽  
Vol 6 (6) ◽  
pp. 885-896 ◽  
Author(s):  
S Abenda
Keyword(s):  
Inverse Problem ◽  
Moment Method ◽  
Iterated Function Systems ◽  
One Dimensional ◽  
Iterated Function ◽  
Fractal Measures ◽  
The Moment

SOLVING THE INVERSE PROBLEM FOR FUNCTION/IMAGE APPROXIMATION USING ITERATED FUNCTION SYSTEMS I: THEORETICAL BASIS

Fractals ◽  
1994 ◽  
Vol 02 (03) ◽  
pp. 325-334 ◽  
Author(s):  
BRUNO FORTE ◽  
EDWARD R. VRSCAY
Keyword(s):  
Inverse Problem ◽  
Function Approximation ◽  
Image Representation ◽  
Iterated Function Systems ◽  
Rigorous Solution ◽  
Compact Metric Space ◽  
Image Approximation ◽  
Iterated Function ◽  
Contractive Operator ◽  
Contraction Maps

We are concerned with function approximation and image representation using Iterated Function Systems (IFS) over ℒp (X, µ): An N-map IFS with grey level maps (IFSM), to be denoted as (w, Φ), is a set w of N contraction maps wi: X → X over a compact metric space (X, d) (the "base space") with an associated set Φ of maps ϕi: R → R. Associated with each IFSM is a contractive operator T with fixed point [Formula: see text]. We provide a rigorous solution to the following inverse problem: Given a target υ ∈ ℒp(X, µ) and an ∊ > 0, find an IFSM whose attractor satisfies [Formula: see text].

Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps

2011 ◽  
Vol 63 (3) ◽  
pp. 648-688 ◽  
Author(s):  
Sze-Man Ngai
Keyword(s):  
Golden Ratio ◽  
Iterated Function Systems ◽  
Spectral Dimension ◽  
Open Set Condition ◽  
Bernoulli Convolution ◽  
One Dimensional ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar ◽  
Set Up

AbstractWe set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

Inverse problem for fractal sets on the real line via the moment method

1989 ◽  
Vol 104 (2) ◽  
pp. 213-227 ◽  
Author(s):  
S. Abenda ◽  
G. Turchetti
Keyword(s):  
Inverse Problem ◽  
Real Line ◽  
Moment Method ◽  
The Real ◽  
Fractal Sets ◽  
The Moment
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