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Research Article. Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Matthew Badger, Raanan Schul (2017)

Analysis and Geometry in Metric Spaces

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems...

Spaces of sequences, sampling theorem, and functions of exponential type

Rodolfo Torres (1991)

Studia Mathematica

We introduce certain spaces of sequences which can be used to characterize spaces of functions of exponential type. We present a generalized version of the sampling theorem and a "nonorthogonal wavelet decomposition" for the elements of these spaces of sequences. In particular, we obtain a discrete version of the so-called φ-transform studied in [6] [8]. We also show how these new spaces and the corresponding decompositions can be used to study multiplier operators on Besov spaces.

Summation processes viewed from the Fourier properties of continuous unimodular functions on the circle

Jean-Pierre Kahane (2011)

Banach Center Publications

The main purpose of this article is to give a new method and new results on a very old topic: the comparison of the Riemann processes of summation (R,κ) with other summation processes. The motivation comes from the study of continuous unimodular functions on the circle, their Fourier series and their winding numbers. My oral presentation in Poznań at the JM-100 conference exposed the ways by which this study was developed since the fundamental work of Brézis and Nirenberg on the topological degree...

Ultrarigid tangents of sub-Riemannian nilpotent groups

Enrico Le Donne, Alessandro Ottazzi, Ben Warhurst (2014)

Annales de l’institut Fourier

We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In particular, such spaces are not locally bi-Lipschitz homeomorphic. The result is based on the study of Carnot groups that are rigid in the sense that their only quasiconformal maps...

Wavelet analysis of the multivariate fractional brownian motion

Jean-François Coeurjolly, Pierre-Olivier Amblard, Sophie Achard (2013)

ESAIM: Probability and Statistics

The work developed in the paper concerns the multivariate fractional Brownian motion (mfBm) viewed through the lens of the wavelet transform. After recalling some basic properties on the mfBm, we calculate the correlation structure of its wavelet transform. We particularly study the asymptotic behaviour of the correlation, showing that if the analyzing wavelet has a sufficient number of null first order moments, the decomposition eliminates any possible long-range (inter)dependence. The cross-spectral...

Weighted Theorems on Fractional Integrals in the Generalized Hölder Spaces via Indices mω and Mω

Karapetyants, Nikolai, Samko, Natasha (2004)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A16, 26A33, 46E15.There are known various statements on weighted action of one-dimensional and multidimensional fractional integration operators in spaces of continuous functions, such as weighted generalized Hölder spaces Hω0(ρ) of functions with a given dominant ω of their continuity modulus.

Currently displaying 61 – 80 of 88