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Endpoint Strichartz estimates
 Amer. J. Math
, 1998
"... Abstract. We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrödinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Stri ..."
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Cited by 525 (42 self)
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Abstract. We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrödinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartztype estimates for more general dispersive equations and for the kinetic transport equation. 1. Introduction. In
Regularization Theory and Neural Networks Architectures
 Neural Computation
, 1995
"... We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Ba ..."
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Cited by 396 (33 self)
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We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, som...
Semicontinuity problems in the calculus of variations
 ARCH. RATIONAL MECH. ANAL
, 1984
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Data compression and harmonic analysis
 IEEE Trans. Inform. Theory
, 1998
"... In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory... ..."
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Cited by 177 (23 self)
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In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory...
The conformal monogenic signal
, 2001
"... The conformal monogenic signal is a novel rotational invariant approach for analyzing i(ntrinsic)1D and i2D local features of twodimensional signals (e.g. images) without the use of any heuristics. It contains the monogenic signal as a special case for i1D signals and combines scalespace, phase, or ..."
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Cited by 160 (38 self)
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The conformal monogenic signal is a novel rotational invariant approach for analyzing i(ntrinsic)1D and i2D local features of twodimensional signals (e.g. images) without the use of any heuristics. It contains the monogenic signal as a special case for i1D signals and combines scalespace, phase, orientation, energy and isophote curvature in one unified algebraic framework. The conformal monogenic signal will be theoretically illustrated and motivated in detail by the relation of the Radon and the Riesz transform. One of the main ideas is to lift up twodimensional signals to a higher dimensional conformal space where the signal can be analyzed with more degrees of freedom. The most interesting result is that isophote curvature can be calculated in a purely algebraic framework without the need of any derivatives.
Regularity of the obstacle problem for a fractional power of the Laplace operator
 Comm. Pure Appl. Math
"... Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: • u ≥ ϕ in Rn, • (−)su ≥ 0 in Rn, • (−)su(x) = 0 for those x such that u(x)> ϕ(x), • limx →+ ∞ u(x) = 0. We show that when ϕ is C1,s or smoother, the solution u is in the space C1,α for every α & ..."
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Cited by 137 (4 self)
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Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: • u ≥ ϕ in Rn, • (−)su ≥ 0 in Rn, • (−)su(x) = 0 for those x such that u(x)> ϕ(x), • limx →+ ∞ u(x) = 0. We show that when ϕ is C1,s or smoother, the solution u is in the space C1,α for every α < s. In the case where the contact set {u = ϕ} is convex, we prove the optimal regularity result u ∈ C1,s. When ϕ is only C1,β for a β < s, we prove
Sickel: Optimal approximation of elliptic problems by linear and nonlinear mappings III
 Triebel, Function Spaces, Entropy Numbers, Differential Operators
, 1996
"... We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) nterm approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f; d) continuous mappings. We co ..."
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Cited by 135 (28 self)
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We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) nterm approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f; d) continuous mappings. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space Br q(Lp(Ω)) and Ω ⊂ Rd is a bounded Lipschitz domain; the error is always measured in the Hsnorm. The respective widths are the linear widths (or approximation numbers), the nonlinear widths, the Gelfand widths, and the manifold widths. As a technical tool, we also study the Bernstein numbers. Our main results are the following. If p ≥ 2 then the order of convergence is the same for all four classes of approximations. In particular, the best linear approximations are of the same order as the best nonlinear ones. The best linear approximation can be quite difficult to realize as a numerical algorithm since the optimal Galerkin space usually depends on the operator and of the shape of the domain Ω. For p < 2 there is a difference, nonlinear approximations are better than linear ones. However, in this case, it turns out that linear information about the right hand side f is again optimal. Our main theoretical tool is the best nterm approximation with respect to an optimal Riesz basis and related nonlinear widths. These general results are used to study the Poisson equation in a polygonal domain. It turns out that best nterm wavelet approximation is (almost) optimal. The main results of
Modulation Spaces on Locally Compact Abelian Groups
 UNIVERSITY OF VIENNA
, 1983
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