arXiv subjects

Ursula M. Molter

Publications and source records attributed to Ursula M. Molter.

3 recordsLinked to original sources

arXiv2025

Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions

In this paper, we define a family of dimensions for Borel measures that lie between the Hausdorff and Minkowski dimensions for measures, analogous to the intermediate dimensions of sets. Previously, Hare et. al. in [11] defined families of dimensions that interpolate between the Minkowski and Assouad dimensions for measures. Additionally, Fraser, in [8] introduced an additional family of dimensions that interpolate between the Fourier and Sobolev dimensions of measures. Our results address a "gap" in the study of dimension interpolation for measures, almost completing the spectrum of intermediate dimensions for measures: from Fourier to Assouad dimensions. Furthermore, Theorem 3.13 can be interpreted as a "reverse Frostman" lemma for intermediate dimensions. We also obtain a capacity-theoretic definition that enables us to estimate the intermediate dimensions of pushforward measures by projections.

math.CA↗

arXiv2010

Classifying Cantor Sets by their Fractal Dimensions

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovitch and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-Packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.

math.CA↗

arXiv2007

Wavelets on Irregular Grids with Arbitrary Dilation Matrices, and Frames Atoms for L^2(R^d)

In this article, we develop a general method for constructing wavelets {|det A_j|^{1/2} g(A_jx-x_{j,k}): j in J, k in K}, on irregular lattices of the form X={x_{j,k} in R^d: j in J, k in K}, and with an arbitrary countable family of invertible dxd matrices {A_j in GL_d(R): j in J} that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L^2(R^d) developed in this article, that allow other time frequency decompositions such as non-harmonic Gabor frames with non-uniform covering of the Euclidean space R^d. Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.

math.CA↗