The singularity spectrum is a function used in Multifractal analysis to describe the fractal dimension of a subset of points of a function belonging to a group of points that have the same Hölder exponent. Intuitively, the singularity spectrum gives a value for how "fractal" a set of points are in a function.
More formally, the singularity spectrum \( D(\alpha ) \) of a function, f(x), is defined as:
\( D(\alpha )=D_{F}\{x,\alpha (x)=\alpha \} \)
Where \( \alpha (x) \) is the function describing the Hölder exponent, \( \alpha (x) \) of f(x) at the point x. \( D_{F}\{\cdot \} \) is the Hausdorff dimension of a point set.
See also
Multifractal analysis
Holder exponent
Hausdorff dimension
Fractal
Fractional Brownian motion
References
van den Berg, J. C. (2004), Wavelets in Physics, Cambridge, ISBN 978-0-521-53353-9.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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