Related papers: Orthogonal Fourier Analysis on Domains
We study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dimension and spectrum. Interestingly, under reasonably mild…
The survey presents developments in the theory of self-similar groups leading to applications to the study of fractal sets and graphs, and their associated spectra.
Recent works have shown that expansion of pseudorandom sets is of great importance. However, all current works on pseudorandom sets are limited only to product (or approximate product) spaces, where Fourier Analysis methods could be…
Despite being the most popular methods of data analysis, Fourier-based techniques suffer from the problem of static resolution that is currently believed to be a fundamental limitation of the Fourier Transform. Although alternative…
A conjecture of Fuglede states that a bounded measurable set D, of measure 1, can tile space by translations if and only if the Hilbert space L^2(D) has an orthonormal basis consisting of exponentials exp(i 2 pi lambda x). If D has the…
We give an example of a set $\Omega \subset \R^5$ which is a finite union of unit cubes, such that $L^2(\Omega)$ admits an orthonormal basis of exponentials $\{\frac{1}{|\Omega|^{1/2}} e^{2\pi i \xi_j \cdot x}: \xi_j \in \Lambda \}$ for…
Based on diffraction theory and the propagation of the light, Fourier optics is a powerful tool allowing the estimation of a visible-range imaging system to transfer the spatial frequency components of an object. The analyses of the imaging…
A recently developed wavelet based approach is employed to characterize the scaling behavior of spectral fluctuations of random matrix ensembles, as well as complex atomic systems. Our study clearly reveals anti-persistent behavior and…
We describe a method to probe the spectral fluctuations of a transition over broad ranges of frequencies and timescales with the high spectral resolution of Fourier spectroscopy, and a temporal resolution as high as the excited state…
In this note we give an example of a set $\W\subset \R^4$ such that $L^2(\W)$ admits an orthonormal basis of exponentials $\{\frac{1}{|\W |^{1/2}}e^{2\pi i x, \xi}\}_{\xi\in\L}$ for some set $\L\subset\R^4$, but which does not tile $\R^4$…
Mathematicians still use Naive Set Theory when generating sets without danger of producing any contradiction. Therefore their working method can be considered as a consistent inference system with an experience of over 100 years. My…
In recent papers a number of authors have considered Borel probability measures $\mu$ in $\br^d$ such that the Hilbert space $L^2(\mu)$ has a Fourier basis (orthogonal) of complex exponentials. If $\mu$ satisfies this property, the set of…
We construct examples of uncountable compact subsets of complex numbers with the property that any Borel measure on the circle group taking values of its Fourier coefficients from this set has natural spectrum. For measures with Fourier…
In this paper we study the solutions of different forms of fractional equations on the unit sphere $\mathbb{S}_{1}^{2}$ $\subset \mathbb{R}^{3}$ possessing the structure of time-dependent random fields. We study the correlation functions of…
This paper introduces a version of empirical likelihood based on the periodogram and spectral estimating equations. This formulation handles dependent data through a data transformation (i.e., a Fourier transform) and is developed in terms…
Finite trigonometric Fourier series on a set of discrete equidistant points are considered. A finite system of orthogonal functions that have interpolation and certain differential properties on the period is introduced. Finite Fourier…
We investigate certain spectral properties of the Bernoulli convolution measures on attractor sets arising from iterated function systems on the real line. In particular, we examine collections of orthogonal exponential functions in the…
Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration and complex numbers. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has…
Any frequency selective device with an ongoing drift will cause observed spectra to be variously and simultaneously scaled in proportion to their source distances. The reason is that detectors after the drifting selection will integrate…
In a previous paper [1] it was discussed the viability of functional analysis using as a basis a couple of generic functions, and hence vectorial decomposition. Here we complete the paradigm exploiting one of the analysis methodologies…