65,339 research outputs found
Constructing Sublinear Expectations on Path Space
We provide a general construction of time-consistent sublinear expectations
on the space of continuous paths. It yields the existence of the conditional
G-expectation of a Borel-measurable (rather than quasi-continuous) random
variable, a generalization of the random G-expectation, and an optional
sampling theorem that holds without exceptional set. Our results also shed
light on the inherent limitations to constructing sublinear expectations
through aggregation.Comment: 28 pages; forthcoming in 'Stochastic Processes and their
Applications
Upper bounds for Fourier transforms of exponential functions
Meaningful upper bounds for the Fourier transform of polynomial exponential functions are often hard to come by. Regarding Fourier transforms of rational exponential functions, which are of importance, for example in Campbell's sampling theorem, the purpose of finding significant upper bounds is an even more demanding exercise. In this article, we propose a new approach in order to obtain significant upper bounds for Fourier transforms of general exponential functions. The technique is shown to allow further generalization in order to deal with Fourier-like integrals and rational exponential integrals
Shannon Wavelets for the Solution of Integrodifferential Equations
Shannon wavelets are used to define a method for the solution of integrodifferential equations. This method is based on (1) the Galerking method, (2) the Shannon wavelet representation, (3) the decorrelation of the generalized Shannon sampling theorem, and (4) the definition of connection coefficients. The Shannon sampling theorem is considered in a more general approach suitable for analysing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction ofL2(ℝ)functions. Shannon wavelets areC∞-functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series (connection coefficients)
Spacetime could be simultaneously continuous and discrete in the same way that information can
There are competing schools of thought about the question of whether
spacetime is fundamentally either continuous or discrete. Here, we consider the
possibility that spacetime could be simultaneously continuous and discrete, in
the same mathematical way that information can be simultaneously continuous and
discrete. The equivalence of continuous and discrete information, which is of
key importance in information theory, is established by Shannon sampling
theory: of any bandlimited signal it suffices to record discrete samples to be
able to perfectly reconstruct it everywhere, if the samples are taken at a rate
of at least twice the bandlimit. It is known that physical fields on generic
curved spaces obey a sampling theorem if they possess an ultraviolet cutoff.
Most recently, methods of spectral geometry have been employed to show that
also the very shape of a curved space (i.e., of a Riemannian manifold) can be
discretely sampled and then reconstructed up to the cutoff scale. Here, we
develop these results further, and we here also consider the generalization to
curved spacetimes, i.e., to Lorentzian manifolds
On the computation of directional scale-discretized wavelet transforms on the sphere
We review scale-discretized wavelets on the sphere, which are directional and
allow one to probe oriented structure in data defined on the sphere.
Furthermore, scale-discretized wavelets allow in practice the exact synthesis
of a signal from its wavelet coefficients. We present exact and efficient
algorithms to compute the scale-discretized wavelet transform of band-limited
signals on the sphere. These algorithms are implemented in the publicly
available S2DW code. We release a new version of S2DW that is parallelized and
contains additional code optimizations. Note that scale-discretized wavelets
can be viewed as a directional generalization of needlets. Finally, we outline
future improvements to the algorithms presented, which can be achieved by
exploiting a new sampling theorem on the sphere developed recently by some of
the authors.Comment: 13 pages, 3 figures, Proceedings of Wavelets and Sparsity XV, SPIE
Optics and Photonics 2013, Code is publicly available at http://www.s2dw.org
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