singularity spectrum

{{Short description|Mathematical function}}

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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 $D(\backslash alpha)$ of a function, $f(x)$, is defined as:

:$D(\backslash alpha)\; =\; D\_F\backslash \{x,\; \backslash alpha(x)\; =\; \backslash alpha\backslash \}$

Where $\backslash alpha(x)$ is the function describing the Hölder exponent, $\backslash alpha(x)$ of $f(x)$ at the point $x$. $D\_F\backslash \{\backslash cdot\backslash \}$ is the Hausdorff dimension of a point set.