Related papers: Fourier spectra of measures associated with algori…
We study the Fourier transform of polynomials in an orthogonal family, taken with respect to the orthogonality measure. Mastering the asymptotic properties of these transforms, that we call Fourier--Bessel functions, in the argument, the…
Can one characterise the Fourier decay of a product measure in terms of the Fourier decay of its marginals? We make inroads on this question by describing the Fourier spectrum of a product measure in terms of the Fourier spectrum of its…
We numerically investigate the implementation of Haar-random unitarity transformations and Fourier transformations in photonic devices consisting of beam splitters and phase shifters, which are used for integrated photonics implementations…
Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show…
This paper is concerned mainly with the macroscopic fractal behavior of various random sets that arise in modern and classical probability theory. Among other things, it is shown here that the macroscopic behavior of Boolean coverage…
We consider finite approximations of a fractal generated by an iterated function system of affine transformations on $\mathbb{R}^d$ as a discrete set of data points. Considering a signal supported on this finite approximation, we propose a…
A novel approach towards the spectral analysis of stationary random bivariate signals is proposed. Using the Quaternion Fourier Transform, we introduce a quaternion-valued spectral representation of random bivariate signals seen as…
Fractional Brownian motion belongs to a class of long memory Gaussian processes that can be represented as linear functionals of an infinite dimensional Markov process. This representation leads naturally to: - An efficient algorithm to…
It is shown that Schr\"odinger maximal inequalities over fractals are equivalent to the $L^2$ decay rates of Fourier transforms of fractal measures over the paraboloid. A similar connection is shown between the wave equation and cone…
Continuum soft robots, composed of flexible materials, exhibit theoretically infinite degrees of freedom, enabling notable adaptability in unstructured environments. Cosserat Rod Theory has emerged as a prominent framework for modeling…
The paper gives a new representation for the fractional Brownian motion that can be applied to simulate this self-similar random process in continuous time. Such a representation is based on the spectral form of mathematical description and…
For spherical and parabolic averages of the Fourier transform of fractal measures, we obtain new upper bounds on rates of decay by an "intermediate dimension" trick.
We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian…
Geometric Brownian motion is an exemplary stochastic processes obeying multiplicative noise, with widespread applications in several fields, e.g. in finance, in physics and biology. The definition of the process depends crucially on the…
We construct first a class of Moran fractals in R^d with countably many generators and non-stationary contraction rates; at each step n, the contractions depend on n-truncated sequences, and are related to asymptotic letter frequencies. In…
Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions…
We introduce the notion of boundary representation for fractal Fourier expansions, starting with a familiar notion of spectral pairs for affine fractal measures. Specializing to one dimension, we establish boundary representations for these…
This article presents extensions of the Cram{\'e}r-Wold theorem to measures that may have infinite mass near the origin. Corresponding results for sequences of measures are presented together with examples showing that the assumptions…
We derive an asymptotic expansion for the quadratic variation of a stochastic process satisfying a stochastic differential equation driven by a fractional Brownian motion, based on the theory of asymptotic expansion of Skorohod integrals…
In this paper we characterize the Fourier transformability of a strongly almost periodic measure in terms of an integrability condition for its Fourier Bohr series. We also provide a necessary and sufficient condition for a strongly almost…