A Guide to Distribution Theory and Fourier Transforms

By Robert S. Strichartz. 1994

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Citations in the corpus

Collaborative Deep Learning for Recommender Systems (2014, background): “Note that if λs goes to infinity, the Gaussian distribution in Equation (1) will become a Dirac delta distribution [31] centered at σ(Xl−1,j∗Wl + bl), where σ(·) is the sigmoid function.”
Infinite propagation speed for a two component Camassa-Holm equation (2009, methodology): “The reverse is also true, we can recover u from m by means of taking Fourier transforms [ 43 ], giving us”
Approximation rates for neural networks with general activation functions (2019, background): “ (7) is a polynomialfor any ε>0. 3. supp(σˆ)⊂{0}. Proof. We begin by proving that (3) and (1) are equivalent. This follows from a characterization of distributions supported at a single point (see [21], section 6.3). In particular, a distribution supported at 0 must be a ﬁnite linear combinationo”
Compactly Supported Solutions of the Camassa-Holm Equation (2005, methodology): “The reverse is also true by means of taking Fourier transforms [21], namely 2u(x) = e ∫ x”
Universally optimal distribution of points on spheres (2006, background): “(See [ St , p. 124] or [R, p. 303].) That property follows immediately from the corresponding property of hN,g, because the set of positive-semidefinite matrices is closed.”
Mathematics of thermoacoustic tomography (2007, background): “One can find simple introduction to the notions of microlocal analysis, such as the wave front set, for instance in [118].”
A Survey on Bayesian Deep Learning (2016, methodology): “Note that if λs goes to infinity, the Gaussian distribution in Equation (5) will become a Dirac delta distribution [106] centered at σ (Xl−1, j∗Wl + bl ), where σ (·) is the sigmoid function, and the model will degenerate into a Bayesian formulation of vanilla SDAE.”
Injective Hilbert Space Embeddings of Probability Measures (2008, background): “Conversely, every positive distribution comes from a locally ﬁnite positive measure [Str03, § 6.4].”
Sampling-50 years after Shannon (2000, background): “From the point of view of computation, the Shannon model has a serious handicap because there are no band-limited functions that are compactly supported—a consequence of the Paley–Wiener theorem [117].”
COSMOS - A Representation Scheme for 3D Free-Form Objects (1997, background): “Strictly speaking, for a suitably chosen set of all “test functions” on the domain of x [31], which places no restriction on us in practice.”
A Note on the Degasperis-Procesi Equation (2005, methodology): “We recall that, by the Paley-Wiener theorem [21], an entire (analytic) function g(ξ), where ξ = η + iζ and η, ζ ∈ R, is the Fourier transform of a smooth function f : R → R with compact support in [−a, a] (for a > 0), namely”
Towards Bayesian Deep Learning: A Survey (2016, background): “Note that if λs goes to infinity, the Gaussian distribution in Equation (8) will become a Dirac delta distribution [52] centered at σ(Xl−1,j∗Wl + bl), where σ(·) is the sigmoid function.”