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Some new characterizations of Olsen’s multifractal functions. (English) Zbl 1451.28008
Summary: The aim of this work is to provide a representation of the multifractal Hausdorff and packing dimensions of compact sets in the Euclidean space in terms of the lower and upper local \((q,\mu)\)-dimensions of a probability measure on \(\mathbb{R}^n\), respectively. In addition, a relationship is established allowing to determine the Olsen’s multifractal functions by means of the corresponding measures’ versions. The multifractal functions are defined as the supremum over the lower and upper multifractal dimensions of all Borel probability measures.
MSC:
28A80 Fractals
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
28A75 Length, area, volume, other geometric measure theory
28A78 Hausdorff and packing measures
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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