Stone–Weierstrass theorems revisited

Timofte, Vlad

doi:10.1016/j.jat.2005.05.004

Cited by 18 publications

(21 citation statements)

References 11 publications

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“…If T 0 and T 1 are defined by Theorem 1 then T (y, y 2 ) can be considered as an approximation to an 'idealistic' transform P such that x = P(y). On the basis of the Stone-Weierstrass theorem [16,21], T (y, y 2 ) should seemingly provide a better associated approximation accuracy of P(y) than that of the first degree polynomial T (y) = T 0 y. Nevertheless, the constraint of the reduced ranks implies a deterioration of P(y) approximation by T (y, y 2 ) = T 0 y + T 1 y 2 .…”

Section: Case 1 Choice Of W On the Basis Of Weierstrass Theoremmentioning

confidence: 99%

Extended Principal Component Analysis

Soto-Quiros¹,

Torokhti²

2021

Preprint

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Principal Component Analysis (PCA) is a transform for finding the principal components (PCs) that represent features of random data. PCA also provides a reconstruction of the PCs to the original data. We consider an extension of PCA which allows us to improve the associated accuracy and diminish the numerical load, in comparison with known techniques. This is achieved due to the special structure of the proposed transform which contains two matrices T 0 and T 1 , and a special transformation F of the so called auxiliary random vector w. For this reason, we call it the three-term PCA. In particular, we show that the three-term PCA always exists, i.e. is applicable to the case of singular data.Both rigorous theoretical justification of the three-term PCA and simulations with real-world data are provided.

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Section: Case 1 Choice Of W On the Basis Of Weierstrass Theoremmentioning

confidence: 99%

Extended Principal Component Analysis

Soto-Quiros¹,

Torokhti²

2021

Preprint

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“…It can be remarked, that, in general, spaces of random variables endowed with the topology of stochastic convergence are not locally convex spaces, so that generalizations of Stone-Weierstrass theorems for locally convex spaces (see, e.g. Timofte [11]) are not applicable in this situation.…”

Section: Stone-weierstrass Theorems For Random Functions and Random Vmentioning

confidence: 99%

Stone-Weierstrass theorems for random functions

Starkloff

2019

Stud. Univ. Babes-Bolyai Math.

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We present several generalizations of the Stone-Weierstrass theorem concerning the approximation of continuous functions on a compact set by using functions from a subalgebra to the case of random functions and random variables in the space of continuous functions. The continuity of the random functions is allowed to be only with respect to a metric, hence including the case of stochastically continuous random functions. These results could be cornerstones for the general theory of approximation for random functions.

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“…A number of fundamental papers have appeared which established significant advances in this research area. Some relevant references can be found, in particular, in [11], [5], [6], [14], [2], [10], [3], [1], [4], [13], [7], [8,9], [12].…”

Section: Motivationmentioning

confidence: 99%

“…While the theory of operator approximation with any given accuracy is well elaborated (see, e.g., [11], [5], [6], [14]), [2], [10], [3], [1], [4], [13], [8], [7], [9], [12]), the theory of best constrained constructive operator approximation is still not so well developed, although this is an area of intensive recent research (see, e.g., [31][32][33][34][35][36]). Despite increasing demands from applications [17][18][19][21][22][23][25][26][27][28][30][31][32][33][34][36][37][38][39][40][41][42][43][44][45][46] this subject is hardly tractable because of intrinsic difficulties in best approximation techniques, especially when the approximating operator should have a specific structure implied by the underlying problem.…”

Section: Motivationmentioning

confidence: 99%

Best approximations of non-linear mappings: Method of optimal injections

Torokhti¹,

Soto-Quiros²

2018

Preprint

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While the theory of operator approximation with any given accuracy is well elaborated, the theory of best constrained constructive operator approximation is still not so well developed. Despite increasing demands from applications this subject is hardly tractable because of intrinsic difficulties in associated approximation techniques. This paper concerns the best constrained approximation of a non-linear operator in probability spaces. We propose and justify a new approach and technique based on the following observation. Methods for best approximation are aimed at obtaining the best solution within a certain class; the accuracy of the solution is limited by the extent to which the class is suitable. By contrast, iterative methods are normally convergent but the convergence can be quite slow. Moreover, in practice only a finite number of iteration loops can be carried out and therefore, the final approximate solution is often unsatisfactorily inaccurate. A natural idea is to combine the methods for best approximation and iterative techniques to exploit their advantageous features. Here, we present an approach which realizes this.The proposed approximating operator has several degrees of freedom to minimize the associated error. In particular, one of the specific features of the approximating technique we develop is special random vectors called injections. They are determined in the way that allows us to further minimize the associated error.

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Stone–Weierstrass theorems revisited

Cited by 18 publications

References 11 publications

Extended Principal Component Analysis

Extended Principal Component Analysis

Stone-Weierstrass theorems for random functions

Best approximations of non-linear mappings: Method of optimal injections

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