Related papers: Fourier transforms and iterated function systems
Several papers have been published recently, particularly in the field of topological insulators, that use a Fourier-transform technique in a way that can be misleading or otherwise inaccurate, and the processing is not always described in…
We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps $\{\mathcal{F}_k\}_{k\in \mathbb{N}}$ where each $\mathcal{F}_k$ maps $\mathcal{H}(X)\to…
This work studies the Fourier transform of the characteristic function of planar convex bodies averaged over affine transformations. We establish lower and upper bounds on the latter quantities in terms of the geometric properties of the…
The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized…
Computing accurate estimates of the Fourier transform of analog signals from discrete data points is important in many fields of science and engineering. The conventional approach of performing the discrete Fourier transform of the data…
Transition from Fourier series to Fourier integrals is considered and error introduced by ordinary substitution of integration for summing is estimated. Ambiguity caused by transition from discrete function to continuous one is examined and…
We study Fourier bases on invariant measures generated by affine iterated function systems in ${\mathbb R}^d$ with integer coefficients. We show that, for simple digit sets, these systems satisfy the open set condition and have no overlap.…
Let $\mu$ be a probability measure on $\mathbb{R}$. We give conditions on the Fourier transform of its density for functionals of the form $H(a)=\int_{\mathbb{R}^n}h(\langle a,x\rangle)\mu^n(dx)$ to be Schur monotone. As applications, we…
Consider an n-fold integrated Brownian motion. We show that a simple change in time and scale transforms it into a stationary Gaussian process. The collection of stationary processes so constructed not only constitutes an interesting family…
In a transformation method, the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. Therefore, a transformation method, like a shooting method, is an initial value method.…
The Fourier transform of the indicator function of arbitrary polygons and polyhedra is computed for complex wavevectors. Using the divergence theorem and Stokes' theorem, closed expressions are obtained. Apparent singularities, all…
We introduce the novel concept of hypercomplex iterated function system (IFS) on the complete metric space $(\mathbb{A}_{n+1}^k,d)$ and define its hypercomplex attractor. Systems of hypercomplex function systems arising from hypercomplex…
Understanding generalization in deep learning has been one of the major challenges in statistical learning theory over the last decade. While recent work has illustrated that the dataset and the training algorithm must be taken into account…
We observe a large number of functions differing from each other only by a translation parameter. While the main pattern is unknown, we propose to estimate the shift parameters using $M$-estimators. Fourier transform enables to transform…
In this paper, we consider the stationary measure of the Hadamard walk on the one-dimensional integer lattice. Here all the stationary measures given by solving the eigenvalue problem are completely determined via the transfer matrix…
In many applications data are measured or defined on a spherical manifold; spherical harmonic transforms are then required to access the frequency content of the data. We derive algorithms to perform forward and inverse spin spherical…
Symmetry fractionalization describes the fascinating phenomena that excitations in a 2D topological system can transform under symmetry in a fractional way. For example in fractional quantum Hall systems, excitations can carry fractional…
We develop a methodology to learn finitely generated random iterated function systems from time-series of partial observations using delay embeddings. We obtain a minimal model representation for the observed dynamics, using a hidden…
We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\bf R$ and other more general IFS's. We extend some of the theory recently developed for Bernoulli…
A quantum mechanical observer might be describable as having a reference system that is a superposition of classical inertial reference frames. The present paper suggests a possible weighting function in such superpositions, determined by…