The Generalized Weierstrass Approximation Theorem

By M. Stone. 1948

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Citations in the corpus

Crypto-Nets: Neural Networks over Encrypted Data (2014, methodology): “Since N is a continuous function over a compact space and since the set of polynomials is an Algebra that separates points it follows from the Stone-Weierstrass theorem (Stone, 1948) that there exists a polynomial P such that sup x ∈ X N ( x ) − P ( x ) (cid:107) (cid:15) .”
Approximation rates for neural networks with general activation functions (2019, background): “By Weierstrass’s Theorem [22] it follows that Σd(σ) containsC(K) for each compact K ⊂ R.”
Methodology of Resonant Equiangular Composite Quantum Gates (2016, background): “The famous Stone-Weierstrass theorem [36] assures us that AðxÞ of sufficiently large degree L can”
New upper bounds on sphere packings II (2001, background): “By Theorem 18 of [17], f̃ can be uniformly approximated on [0,∞) by g̃ with g a polynomial.”
Korovkin-type Theorems and Approximation by Positive Linear Operators (2010, methodology): “Before stating the Stone approximation theorem, we recall that a subset M of C0(X) is said to separate strongly the points of X if it separates the points of X and if for every x ∈ X there exists f ∈ M such that f(x) 6= 0.”
Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics (2012, background): “By Stone–Weierstrass approximation theorem (Stone, 1948), in the compact set Ω , given any positive number εi, there exists a large enough positive integer v∗i , such that for any vi ≥ v∗i , one can always find an ideal weight vector Wi and a suitable basis set φi(·), which makes (10) satisfy the…”
A survey on the Weierstrass approximation theorem (2006, background): “It is also necessary to mention here the Kakutani-Stone theorem on the closure of a lattice of functions real-valued and Stone-Weierstrass theorem on the closure of an algebra of functions continuous complex-valued (see [28], [18]).”
Energy flow polynomials: a complete linear basis for jet substructure (2017, methodology): “In cases without a Taylor expansion, the Stone-Weierstrass theorem [76] still guarantees that the observable can be approximated over some bounded energy range by polynomials in the energies.”
Cooperative adaptive control for synchronization of second‐order systems with unknown nonlinearities (2011, methodology): “According to the Weierstrass higher order approximation theorem [37] a polynomial basis set sufﬁces to approximate f i ( x i ) as well as its derivatives, when they exist, and moreover, the approximation error ε i → 0 uniformly as i →∞ .”
Variance reduction properties of the reparameterization trick (2018, methodology): “Despite many log-joint density functions not being polynomials, we can find a reasonable polynomial approximation over the sampling region using the Stone-Weierstrass theorem (Stone, 1948).”
Least squares moving particle semi-implicit method (2014, background): “sion [84,88,89], for a fixed point x̄ ∈ Ω̄ , one should always be able to approximate f (x) by a polynomial series locally.”
Holomorphic Approximation: The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan (2018, unknown): “Stone in 1937, [154, 155].”
New upper bounds on sphere packings I (2001, unknown): “or all mand all gsuch that ˜g∈ L∞([0,∞),xα dx), ||σmg||∞ ≤ C||˜g||∞, where ||· ||∞ denotes the norm on L∞([0,∞),xα dx). We can then imitate the proof of Theorem 2 in [P]. Let ε>0. By Theorem 18 of [Sto], f˜can be uniformly approximated on [0,∞) by ˜gwith ga polynomial. Choose gso that f˜− ˜g ∞ &l”
Distributed adaptive control for synchronization of unknown nonlinear networked systems (2010, unknown): “That is ∀ξ > 0 there exist ν̄i, i = 1,N such that νi > ν̄i, ∀i implies supx∈Ω ‖ε(x)‖ ξ (Stone, 1948).”
Expected Policy Gradients for Reinforcement Learning (2018, background): “This result is signiﬁcant since polynomials can approximate any continuous function on a bounded interval with arbitrary accuracy (Weierstrass, 1885; Stone, 1948).”
Networks and the best approximation property (1990, methodology): “The proof is a simple application of the Stone-Weierstrass theorem, which is the generalization given by Stone of the Weierstrass approximation theorem ( Stone 1937, 1948 ).”
Energy flow networks: deep sets for particle jets (2018, methodology): “[63], makes use of the Stone-Weierstrass polynomial approximation theorem [92], which applies for compact spaces.”