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Networks and the Best Approximation Property
 Biological Cybernetics
, 1989
"... Networks can be considered as approximation schemes. Multilayer networks of the backpropagation type can approximate arbitrarily well continuous functions (Cybenko, 1989# Funahashi, 1989# Stinchcombe and White, 1989). Weprovethatnetworks derived from regularization theory and including Radial Bas ..."
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Cited by 143 (8 self)
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Basis Functions (Poggio and Girosi, 1989), have a similar property.From the point of view of approximation theory, however, the property of approximating continuous functions arbitrarily well is not sufficientforcharacterizing good approximation schemes. More critical is the property of best
Approximation properties of multivariate wavelets
 Math. Comp
, 1998
"... Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of ..."
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Cited by 76 (10 self)
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Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order
APPROXIMATION PROPERTIES
"... the diesel engine dataset (7 refinement steps). Fig. 3. Morse sets and connecting regions for the figure eight model subdivided three times. APPENDIX A ..."
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the diesel engine dataset (7 refinement steps). Fig. 3. Morse sets and connecting regions for the figure eight model subdivided three times. APPENDIX A
Approximation by Superpositions of a Sigmoidal Function
, 1989
"... In this paper we demonstrate that finite linear combinations of compositions of a fixed, univariate function and a set ofaffine functionals can uniformly approximate any continuous function of n real variables with support in the unit hypercube; only mild conditions are imposed on the univariate fun ..."
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Cited by 1248 (2 self)
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continuous sigmoidal nonlinearity. The paper discusses approximation properties of other possible types of nonlinearities that might be implemented by artificial neural networks.
Property Testing and its connection to Learning and Approximation
"... We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the fun ..."
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Cited by 475 (67 self)
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We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query
THE WEAK METRIC APPROXIMATION PROPERTY
"... Abstract. We introduce and investigate the weak metric approximation property of Banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property. Among others, we show that if a Banach space has the approximation propert ..."
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Abstract. We introduce and investigate the weak metric approximation property of Banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property. Among others, we show that if a Banach space has the approximation
Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes
 J. COMP. PHYS
, 1981
"... Several numerical schemes for the solution of hyperbolic conservation laws are based on exploiting the information obtained by considering a sequence of Riemann problems. It is argued that in existing schemes much of this information is degraded, and that only certain features of the exact solution ..."
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Cited by 1010 (2 self)
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are worth striving for. It is shown that these features can be obtained by constructing a matrix with a certain “Property U.” Matrices having this property are exhibited for the equations of steady and unsteady gasdynamics. In order to construct them, it is found helpful to introduce “parameter vectors
Hereditary approximation property
 Annals of Math
"... Abstract If X is a Banach space such that the isomorphism constant to n 2 from ndimensional subspaces grows sufficiently slowly as n → ∞, then X has the approximation property. A consequence of this is that there is a Banach space X with a symmetric basis but not isomorphic to 2 so that all subspa ..."
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Cited by 4 (2 self)
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Abstract If X is a Banach space such that the isomorphism constant to n 2 from ndimensional subspaces grows sufficiently slowly as n → ∞, then X has the approximation property. A consequence of this is that there is a Banach space X with a symmetric basis but not isomorphic to 2 so that all
On the Hamkins Approximation Property
, 2004
"... We give a short proof of a lemma which generalizes both the main lemma from the original construction in the author’s thesis of a model with no ω2Aronszajn trees, and also the “Key Lemma ” in Hamkins’s gap forcing theorems. The new lemma directly yields Hamkins’s newer lemma stating that certain fo ..."
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Cited by 4 (0 self)
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forcing notions have the approximation property. According to Hamkins [2], a partial ordering P satisfies the δapproximation property if, whenever A ∈ V P is a subset of an ordinal µ in V P such that A ∩ x ∈ V for each x ∈ ([µ] <δ) V, we have A ∈ V. In [2, Lemma 13] he proves the following lemma
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