Random Fourier Series Research Articles

Wiener Chaos Random Fourier Series as a Tool to Study the Coupling of Noise With Power System Transient Response

Wiener Chaos Random Fourier Series as a Tool to Study the Coupling of Noise With Power System Transient Response

A Generalization of Pisier Homogeneous Banach Algebra

In 1979, Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra P strictly contained in C(T), the class of continuous functions on the unit circle T and strictly containing the classical Wiener algebra A(T), that is, A(T)⫋P⫋C(T). This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper, we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in C(T). Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ∑ n∈Zfˆ(n)ξnexp(2πint) in the general setting of dependent random variables (ξn).

On the Convergence of Random Fourier-Jacobi Series of Continuous functions

The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $\sum_{n=0}^\infty d_n r_n(\omega)\varphi_n(y)$ in orthogonal polynomials $\varphi_n(y)$ with random variables $r_n(\omega).$ The random coefficients $r_n(\omega)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $\varphi_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $\gamma,\delta$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.

Buckling Analysis of Cylindrical Shells using Stochastic Finite Element Method with Random Geometric Imperfections

AbstractThin‐walled cylindrical shells often exhibit buckling failure and the experimental buckling load is usually lower than calculation results from classical theory and simulation without geometrical imperfection. Besides, test results with carefully conducted similar specimens still have substantial scatter due to imperfection sensitivity. The nonlinear analysis with FEM can obtain a high‐precision result comparing the experiment. However, this is almost impossible in practical engineering. The initial geometric imperfections of cylindrical shells are complex and random properties. Theoretically, these geometric imperfections can be described using a random field or a Fourier series representation. After that a random sample of the buckling bearing capacity of the cylindrical shell can be obtained using the stochastic finite element method within the subsequent analysis. This paper presents the investigation of buckling analysis of cylindrical shells under axial compression considering the randomness of initial geometric imperfections, and the stochastic FEM is employed to calculate the statistical sample of the shell‐buckling load based on a series of random fields and Fourier series representation of geometric imperfections.

Overparameterization and Generalization Error: Weighted Trigonometric Interpolation

Motivated by surprisingly good generalization properties of learned deep neural networks in overparameterized scenarios and by the related double descent phenomenon, this paper analyzes the relation between smoothness and low generalization error in an overparameterized linear learning problem. We study a random Fourier series model, where the task is to estimate the unknown Fourier coefficients from equidistant samples. We derive exact expressions for the generalization error of both plain and weighted least squares estimators. We show precisely how a bias towards smooth interpolants, in the form of weighted trigonometric interpolation, can lead to smaller generalization error in the overparameterized regime compared to the underparameterized regime. This provides insight into the power of overparameterization, which is common in modern machine learning.

The microlocal irregularity of Gaussian noise

The study of random Fourier series, linear combinations of trigonometric functions whose coefficients are independent (in our case Gaussian) random variables with polynomially bounded means and standard deviations, dates back to Norbert Wiener in one of the original constructions of Brownian motion. A geometric generalization -- relevant e.g.\ to Euclidean quantum field theory with an infrared cutoff -- is the study of random Gaussian linear combinations of the eigenfunctions of the Laplace-Beltrami operator on an arbitrary compact Riemannian manifold $(M,g)$, Gaussian noise $\Phi$. I will prove that, when our random coefficients are independent Gaussians whose standard deviations obey polynomial asymptotics and whose means obey a corresponding polynomial upper bound, the resultant random $\mathscr{H}^s$-wavefront set $\operatorname{WF}^s(\Phi)$ (defined as a subset of the cosphere bundle $\mathbb{S}^*M$) is either almost surely empty or almost surely the entirety of $\mathbb{S}^*M$, depending on $s \in \mathbb{R}$, and we will compute the threshold $s$ and the behavior of the wavefront set at it. Consequently, the random $C^\infty$-wavefront set $\operatorname{WF}(\Phi)$ is almost surely the entirety of the cosphere bundle. The method of proof is as follows: using Sazonov's theorem and its converse, it suffices to understand which compositions of microlocal cutoffs and inclusions of $L^2$-based fractional order Sobolev spaces are Hilbert-Schmidt (HS), and the answer follows from general facts about the HS-norms of the elements of the pseudodifferential calculus of Kohn and Nirenberg.

Random Fourier series with Dependent Random Variables

Given a sequence of independent standard Gaussian variables (Zn), the classical Pisier algebra P is the class of all continuous functions f on the unit circle T such that for each t ∈ 𝕋, the random Fourier series $$ {\sum}_{n\in \mathrm{\mathbb{Z}}}{Z}_n\hat{f}(n)\times \exp \left(2\pi \mathrm{i} nt\right) $$ converges in L2 and the corresponding sums constitute a Gaussian process that admits a continuous version. It was constructed by Pisier in 1979 to answer a long-standing question raised by Katznelson. In this paper, we consider the general random Fourier series $$ {\sum}_{n\in \mathrm{\mathbb{Z}}}{Z}_n\hat{f}(n)\times \exp \left(2\pi \mathrm{i} nt\right) $$ where ξ = (ξn) is a discrete Gaussian process of standard Gaussian random variables but with the restriction of independence relaxed and study the corresponding class P(ξ) of continuous functions f on 𝕋. We obtain sufficient conditions (based on some spectral properties of the covariance matrix of (ξn)) for each of the relations 𝒫 ⊂ (ξ), 𝒫(ξ) ⊂ 𝒫, and 𝒫 = 𝒫(ξ). We illustrate these results by the classical fractional Gaussian noise. Whether in general (ξ) is also a Banach algebra is an open problem.

A theorem for random Fourier series on compact quantum groups

Helgason showed that a given measure $f\in M(G)$ on a compact group $G$ should be in $L^2(G)$ automatically if all random Fourier series of $f$ are in $M(G)$. We explore a natural analogue of the theorem in the framework of compact quantum groups and apply the obtained results to study complete representability problem for convolution algebras of compact quantum groups as an operator algebra.

KLOOSTERMAN PATHS OF PRIME POWERS MODULI, II

G. Ricotta and E. Royer (2018) have recently proved that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums $S(a,b;p^n)/p^(n/2) converge in law in the Banach space of complex-valued continuous function on [0,1] to an explicit random Fourier series as (a,b) varies over (Z/p^nZ)^\times\times(Z/p^nZ)^\times, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. This is the analogue of the result obtained by E. Kowalski and W. Sawin (2016) in the prime moduli case. The purpose of this work is to prove a convergence law in this Banach space as only a varies over (Z/p^nZ)^\times, p tends to infinity among the odd prime numbers and n>=31 is a fixed integer.

Kloosterman paths of prime powers moduli

In [12], the authors proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S(a,b_0;p)/p^{1/2} converge in the sense of finite distributions to a specific random Fourier series, as a varies over (\mathbb{Z}/p\mathbb{Z})^\times , b_0 is fixed in (\mathbb{Z}/p\mathbb{Z})^\times and p tends to infinity among the odd prime numbers. This article considers the case of S(a,b_0;p^n)/p^{n/2} , as a varies over (\mathbb{Z}/p^n\mathbb{Z})^\times , b_0 is fixed in (\mathbb{Z}/p^n\mathbb{Z})^\times , p tends to infinity among the odd prime numbers and n\geq 2 is a fixed integer. A convergence in law in the Banach space of complex-valued continuous function on [0,1] is also established, as (a,b) varies over (\mathbb{Z}/p^n\mathbb{Z})^\times\times(\mathbb{Z}/p^n\mathbb{Z})^\times , p tends to infinity among the odd prime numbers and n\geq 2 is a fixed integer. This is the analogue of the result obtained in [12] in the prime moduli case.

Fourier Series Expansion of Stochastic Measures

We consider processes of the form $\mu(t)=\mu((0,t])$, where $\mu$ is a $\sigma$-additive in probability stochastic set function. Convergence of a random Fourier series to $\mu(t)$ is proved, and t...

Recovery of periodicities hidden in heavy‐tailed noise

AbstractWe address a parametric joint detection‐estimation problem for discrete signals of the form , , with an additive noise represented by independent centered complex random variables . The distributions of are assumed to be unknown, but satisfying various sets of conditions. We prove that in the case of a heavy‐tailed noise it is possible to construct asymptotically strongly consistent estimators for the unknown parameters of the signal, i.e., frequencies , their number N, and complex coefficients . For example, one of considered classes of noise is the following: are independent identically distributed random variables with and . The construction of estimators is based on detection of singularities of anti‐derivatives for Z‐transforms and on a two‐level selection procedure for special discretized versions of superlevel sets. The consistency proof relies on the convergence theory for random Fourier series.

Kloosterman paths and the shape of exponential sums

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums$\text{Kl}_{p}(a)$, as$a$varies over$\mathbf{F}_{p}^{\times }$and as$p$tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

The Billard Theorem for Multiple Random Fourier Series

We propose a generalization of a classical result on random Fourier series, namely the Billard Theorem, for random Fourier series over the d-dimensional torus. We provide an investigation of the independence with respect to a choice of a sequence of partial sums (or method of summation). We also study some probabilistic properties of the resulting sum field such as stationarity and characteristics of the marginal distribution.

The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory

We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46-57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron-Martin paths and complementary Young regularity (CYR) of the Cameron-Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also It\^o-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian H\"ormander theory.

Is the Brownian bridge a good noise model on the boundary of a circle?

In this paper we study periodical stochastic processes, and we define the conditions that are needed by a model to be a good noise model on the circumference. The classes of processes that fit the required conditions are studied together with their expansion in random Fourier series in order to provide results about their path regularity. Finally, we discuss a simple and flexible parametric model with prescribed regularity that is used in applications, and we prove the asymptotic properties of the maximum likelihood estimates of model parameters.

An application of [formula omitted]-subgaussian technique to Fourier analysis

We present an application to random Fourier series for φ-subgaussian random variables. In particular, we generalize previously known notions of dependence for random variables by introducing the concept of F-manageable random variables, and consider Fourier series of F-manageable φ-subgaussian random variables. Gaussian series are then a particular case of the series considered in the paper. Conditions for the uniform convergence of such series in probability are established and a rate of convergence is investigated. Moreover, the results are new even for the case of independent random variables.

Reconstruction of gusty wind speed time series from autonomous data logger records

The collection of wind speed time series by means of digital data loggers occurs in many domains, including civil engineering, environmental sciences and wind turbine technology. Since averaging intervals are often significantly larger than typical system time scales, the information lost has to be recovered in order to reconstruct the true dynamics of the system. In the present work we present a simple algorithm capable of generating a real-time wind speed time series from data logger records containing the average, maximum, and minimum values of the wind speed in a fixed interval, as well as the standard deviation. The signal is generated from a generalized random Fourier series. The spectrum can be matched to any desired theoretical or measured frequency distribution. Extreme values are specified through a postprocessing step based on the concept of constrained simulation. Applications of the algorithm to 10-min wind speed records logged at a test site at 60 m height above the ground show that the recorded 10-min values can be reproduced by the simulated time series to a high degree of accuracy.

Thin sets of integers in harmonic analysis and p-stable random fourier series

We investigate the behaviour of some thin sets of integers defined through random trigonometric polynomials when one replaces Gaussian or Rademacher variables with p-stable ones, 1 < p < 2. We show that in one case, this behaviour is essentially the same as in the Gaussian case, whereas in another case, it is entirely different.

Prediction of turbulent gas‐solid flow in a duct with a 90° bend using an Eulerian‐Lagrangian approach

AbstractA dilute, particle‐laden turbulent flow in a square cross‐sectioned duct with a 90° bend is modeled using a three‐dimensional Eulerian‐Lagrangian approach. Predictions are based on a second‐moment turbulence closure, with particles simulated using a Lagrangian particle tracking technique, coupled to a particle‐wall interaction algorithm and a random Fourier series method used to model particle dispersion. The performance of the model is tested for a gas‐solid flow in a horizontal‐to‐vertical duct, with predictions showing good agreement with experimental data. In particular, the consistent use of anisotropic and fully three‐dimensional approaches throughout yields predictions that result in fluctuating particle velocities in acceptable agreement with data. © 2011 American Institute of Chemical Engineers AIChE J, 2012