Wiener Space Research Articles

Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis

Prolate spheroidal wave functions (PSWFs) possess many remarkable properties. They are orthogonal basis of both square integrable space of finite interval and the Paley–Wiener space of bandlimited functions on the real line. No other system of classical orthogonal functions is known to obey this unique property. This raises the question of whether they possess these properties in Clifford analysis. The aim of the article is to answer this question and extend the results to more flexible integral transforms, such as offset linear canonical transform. We also illustrate how to use the generalized Clifford PSWFs (for offset Clifford linear canonical transform) we derive to analyze the energy preservation problems. Clifford PSWFs is new in literature and has some consequences that are now under investigation. Copyright © 2012 John Wiley & Sons, Ltd.

The average errors for hermite interpolation on the 1-Fold integrated wiener space

For the weighted approximation in L p -norm, the authors determine the weakly asymptotic order for the p-average errors of the sequence of Hermite interpolation based on the Chebyshev nodes on the 1-fold integrated Wiener space. By this result, it is known that in the sense of information-based complexity, if permissible information functionals are Hermite data, then the p-average errors of this sequence are weakly equivalent to those of the corresponding sequence of the minimal p-average radius of nonadaptive information.

A Rotation on Wiener Space with Applications

We first investigate a rotation property of Wiener measure on the product of Wiener spaces. Next, using the concept of the generalized analytic Feynman integral, we define a generalized Fourier-Feynman transform and a generalized convolution product for functionals on Wiener space. We then proceed to establish a fundamental result involving the generalized transform and the generalized convolution product.

The Simultaneous Approximation Average Errors for Bernstein Operators on the r -Fold Integrated Wiener Space

For weighted approximation in L p -norm, we determine strongly asymptotic orders for the average errors of both function approximation and derivative approximation by the Bernstein operators sequence on the r -fold integrated Wiener space.

High order recombination and an application to cubature on Wiener space

Particle methods are widely used because they can provide accurate descriptions of evolving measures. Recently it has become clear that by stepping outside the Monte Carlo paradigm these methods can be of higher order with effective and transparent error bounds. A weakness of particle methods (particularly in the higher order case) is the tendency for the number of particles to explode if the process is iterated and accuracy preserved. In this paper we identify a new approach that allows dynamic recombination in such methods and retains the high order accuracy by simplifying the support of the intermediate measures used in the iteration. We describe an algorithm that can be used to simplify the support of a discrete measure and give an application to the cubature on Wiener space method developed by Lyons and Victoir [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 169-198].

Discrete linear differential equations

We propose new constructions of the approximate solutions of discrete linear differential equations and their inverse source problems. This is mainly based on a reproducing kernel Hilbert spaces approach where different types of spaces are naturally considered as a consequence of the method. Here, the major influence will be given by the Paley–Wiener and Sobolev spaces.

Laplace transform identities and measure-preserving transformations on the Lie–Wiener–Poisson spaces

Given a divergence operator δ on a probability space such that the law of δ(h) is infinitely divisible with characteristic exponent(0.1)h↦−12∫0∞ht2dt,or∫0∞(eih(t)−ih(t)−1)dt,h∈L2(R+), we derive a family of Laplace transform identities for the derivative ∂E[eλδ(u)]/∂λ when u is a non-necessarily adapted process. These expressions are based on intrinsic geometric tools such as the Carleman–Fredholm determinant of a covariant derivative operator and the characteristic exponent (0.1), in a general framework that includes the Wiener space, the path space over a Lie group, and the Poisson space. We use these expressions for measure characterization and to prove the invariance of transformations having a quasi-nilpotent covariant derivative, for Gaussian and other infinitely divisible distributions.

Beurling–Landauʼs density on compact manifolds

Given a compact Riemannian manifold M, we consider the subspace of L2(M) generated by the eigenfunctions of the Laplacian of eigenvalue less than L⩾1. This space behaves like a space of polynomials and we have an analogy with the Paley–Wiener spaces. We study the interpolating and Marcinkiewicz–Zygmund (M–Z) families and provide necessary conditions for sampling and interpolation in terms of the Beurling–Landau densities. As an application, we prove the equidistribution of the Fekete arrays on some compact manifolds.

A sequential analytic Feynman integral of functionals in L 2(C 0[0, T

In this paper, we define a sequential analytic Feynman integral of functionals on Wiener space. We then give the existence of the sequential analytic Feynman integral of functionals in L 2(C 0[0, T]). The expansion of functionals in L 2(C 0[0, T]) in terms of Fourier–Hermite functionals plays a key role in the study.

Gap probabilities for the cardinal sine

We study the zero sets of random analytic functions generated by a sum of the cardinal sine functions which form an orthonormal basis for the Paley–Wiener space. As a model case, we consider real-valued Gaussian coefficients. It is shown that the asymptotic probability that there is no zero in a bounded interval decays exponentially as a function of the length.

A MULTIPLE GENERALIZED FOURIER–FEYNMAN TRANSFORM VIA A ROTATION ON WIENER SPACE

In this paper we use a rotation property of Wiener measure to define a very general multiple Fourier–Feynman transform on Wiener space. We then proceed to establish its many algebraic properties as well as to establish several relationships between this generalized multiple transform and the corresponding generalized convolution product.

FGK INEQUALITY ON THE ANALOGUE OF WIENER SPACE

This work aims at proving the FKG inequalities for the analogue of Wiener functionals with special orders on the analogue of Wiener space.

Generalised Clark-Ocone formulae for differential forms

We generalise the Clark-Ocone formula for functions to give analogous representations for differential forms on the classical Wiener space. Such formulae provide explicit expressions for closed and co-closed differential forms and, as a by-product, a new proof of the triviality of the L^2 de Rham cohomology groups on the Wiener space, alternative to Shigekawa's approach [16] and the chaos-theoretic version [18]. This new approach has the potential of carrying over to curved path spaces, as indicated by the vanishing result for harmonic one-forms in [6]. For the flat path group, the generalised Clark-Ocone formulae can be proved directly using the It\^o map.

Wick product and Clark-Ocone formula in abstract Wiener space

Wick product and Clark-Ocone formula in abstract Wiener space

FOURIER-TYPE FUNCTIONALS ON WIENER SPACE

In this paper we define the Fourier-type functionals via the Fourier transform on Wiener space. We investigate some properties of the Fourier-type functionals. Finally, we establish integral transform of the Fourier-type functionals which also can be expressed by other Fourier-type functionals.

Unboundedness of thresholding and quantization for bandlimited signals

It is well-known that the Shannon sampling series is locally uniformly convergent for all signals in the Paley–Wiener space PWπ1. An interesting question is how this good local approximation behavior is affected if the samples are disturbed by the non-linear threshold operator. This operator, which sets to zero all samples with absolute value smaller than some threshold, arises in the modeling of many applications, e.g., in wireless sensor networks. Moreover, it constitutes an essential part of a large class of quantizers, and consequently is important for all digital signal processing applications that involve conversion between analog and digital domains. In this paper, the approximation behavior of the Shannon sampling series that only uses the samples with absolute value larger than or equal to some threshold is analyzed. It is shown that there exists a signal in PWπ1 such that the local approximation error increases unboundedly as the threshold tends to zero. Moreover, for a fixed threshold, the local approximation error can grow arbitrarily large on the set of signals whose norm is bounded by one. With this, we generalize results of Butzer et al. that were given in the paper “On quantization, truncation and jitter errors in the sampling theorem and its generalizations,” Signal Processing (2) 1980 [1]. We conclude the paper with a discussion about the differences in the reconstruction behavior between the sampling series which is truncated in the domain of the sampled signal, i.e., time-domain truncation, and the sampling series which is truncated in the range of the sampled signal.

The Logvinenko–Sereda theorem for the Fourier–Bessel transform

The aim of this paper is to establish an analogue of Logvinenko–Sereda's theorem for the Fourier–Bessel transform (or Hankel transform) ℱα of order α>−½. Roughly speaking, if we denote by PWα(b) the Paley–Wiener space of L 2-functions with the Fourier–Bessel transform supported in [0, b], then we show that the restriction map f→f|Ω is essentially invertible on PWα(b) if and only if Ω is sufficiently dense. Moreover, we give an estimate of the norm of the inverse map. As a side result, we prove a Bernstein-type inequality for the Fourier–Bessel transform.

Continuous Limits of Classical Repeated Interaction Systems

We consider the physical model of a classical mechanical system (called “small system”) undergoing repeated interactions with a chain of identical small pieces (called “environment”). This physical setup constitutes an advantageous way of implementing dissipation for classical systems; it is at the same time Hamiltonian and Markovian. This kind of model has already been studied in the context of quantum mechanical systems, where it was shown to give rise to quantum Langevin equations in the limit of continuous time interactions (Attal and Pautrat in Ann Henri Poincare 7:59–104, 2006), but it has never been considered for classical mechanical systems yet. The aim of this article is to compute the continuous limit of repeated interactions for classical systems and to prove that they give rise to particular stochastic differential equations (SDEs) in the limit. In particular, we recover the usual Langevin equations associated with the action of heat baths. In order to obtain these results, we consider the discrete-time dynamical system induced by Hamilton’s equations and the repeated interactions. We embed it into a continuous-time dynamical system and compute the limit when the time step goes to 0. This way, we obtain a discrete-time approximation of SDE, considered as a deterministic dynamical system on the Wiener space, which is not exactly of the usual Euler scheme type. We prove the Lp and almost sure convergence of this scheme. We end up with applications to concrete physical examples such as a charged particle in a uniform electric field or a harmonic interaction. We obtain the usual Langevin equation for the action of a heat bath when considering a damped harmonic oscillator as the small system.

Densité des orbites des trajectoires browniennes sous l’action de la transformation de Lévy

Let T be a measurable transformation of a probability space $(E,\mathcal {E},\pi)$, preserving the measure π. Let X be a random variable with law π. Call K(⋅, ⋅) a regular version of the conditional law of X given T(X). Fix $B\in \mathcal {E}$. We first prove that if B is reachable from π-almost every point for a Markov chain of kernel K, then the T-orbit of π-almost every point X visits B. We then apply this result to the Lévy transform, which transforms the Brownian motion W into the Brownian motion |W| − L, where L is the local time at 0 of W. This allows us to get a new proof of Malric’s theorem which states that the orbit under the Lévy transform of almost every path is dense in the Wiener space for the topology of uniform convergence on compact sets.

The average errors for lagrange interpolation on the Wiener space

For the weighted approximation in Lp-norm, we determine the asymptotic order for the paverage errors of Lagrange interpolation sequence based on the Chebyshev nodes on the Wiener space. We also determine its value in some special case.