Stone-Weierstrass Research Articles

On weighted uniform boundedness and convergence of the Bernstein–Chlodovsky operators

A recent article of the first author presented a constructive proof for the Weierstrass approximation theorem for a wide class of weighted spaces of continuous functions defined on [ 0 , ∞ ) , using Chlodovsky's generalization of the Bernstein polynomial operators. Here, we investigate necessary and sufficient conditions related to the uniform boundedness of a sequence of such operators and to their convergence properties when the weight is a classical Freud weight of the form w ( x ) = e − x α , α ≥ 1 .

A flexible application of the Stone-Weierstrass Theorem in General and Application in Probability Theory

When applying the classical Stone-Weierstrass common version in Probability Theory for example but in other fields also, some problems may arise if all points of the compact set are not separated. A solution may consist in going back to the proof and finding alternative versions. In this note, we did it and come back with two flexible versions which are easily used for the needs of Probability Theory.

Tracking Control of Electrically Driven Robots Using a Model-free Observer

SummaryThis paper presents a robust tracking controller for electrically driven robots, without the need for velocity measurements of joint variables. Many observers require the system dynamics or nominal models, while a model-free observer is presented in this paper. The novelty of this paper is presenting a new observer–controller structure based on function approximation techniques and Stone–Weierstrass theorem using differential equations. In fact, it is assumed that the lumped uncertainty can be modeled by linear differential equations. Then, using Stone–Weierstrass theorem, it is verified that these differential equations are universal approximators. The advantage of proposed approach in comparison with previous related works is simplicity and reducing the dimensions of regressor matrices without the need for any information of the systems’ dynamic. Simulation results on a 6-degrees of freedom robot manipulator driven by geared permanent magnet DC motors indicate the satisfactory performance of the proposed method in overcoming uncertainties and reducing the tracking error. To evaluate the performance of proposed controller in practical implementations, experimental results on an SCARA manipulator are presented.

Stone–Weierstrass and extension theorems in the nonlocally convex case

In the general (nonlocally convex) case we prove a Stone–Weierstrass-type theorem for sets of continuous vector-valued functions on Hausdorff topological spaces whose compact subsets have finite Lebesgue covering dimension (topological dimension). For such topological space T and Hausdorff topological vector space X (real or complex), in the presence of a separating set of multipliers, the theorem characterizes the closure of a subset in: C(T,X) (resp. C0(T,X) and Cb(T,X)), endowed with the compact-open topology (respectively, the uniform convergence topology and the strict topology). Applications include a Stone–Weierstrass theorem for vector subspaces, range-support uniform approximation results (under constraints on both the range and the support of the approximant function), extension theorems for vector-valued functions, and a short proof of a Schauder-type fixed point theorem. Our noncompact version of the Stone–Weierstrass theorem has significant consequences, among which we mention the extension theorem for vector-valued functions defined on closed subsets of paracompact spaces.

101.33 The application of the Weierstrass approximation theorem in the Riemann-Lebesgue lemma

101.33 The application of the Weierstrass approximation theorem in the Riemann-Lebesgue lemma

On a Constructive Proof of the Weierstrass Theorem With a Weight Function On Unbounded Intervals

This article presents a constructive proof for the Weierstrass approximation theorem in weighted spaces of continuous functions defined on $$[0, \infty )$$ or on $$(-\infty , \infty )$$ , using Chlodovsky’s generalization of Bernstein’s approximation operators.

On the representation by sums of algebras of continuous functions

We give a necessary condition for the representation of the space of continuous functions by sums of its k closed subalgebras. A sufficient condition for this representation problem was first obtained by Sternfeld in 1978. In case of two subalgebras (k=2), our necessary condition turns out to be also sufficient. If k=1, our result coincides with a version of the classical Stone–Weierstrass theorem.

A version of the Stone-Weierstrass theorem in fuzzy analysis

A version of the Stone-Weierstrass theorem in fuzzy analysis

Robust impedance control of robot manipulators using differential equations as universal approximator

ABSTRACTThis paper presents a robust impedance controller for robot manipulators using function approximation techniques (FATs). Recently, some FAT-based robust impedance control approaches have been presented using Fourier series expansion or Legendre polynomials for uncertainty estimation. However, the dimensions of regressor matrices in these approaches are relatively large. This problem becomes hypersensitive especially for higher degree of freedom robot manipulators. In this paper, a simpler and less computational FAT-based robust controller is presented without considering discontinuous nonlinearities. It is assumed that the lumped uncertainty can be modelled by a linear differential equation with unknown coefficients. Then, using the Stone–Weierstrass theorem, it is verified that these differential equations are universal approximators. The advantage of the proposed controller in comparison with previous related works is reducing the dimensions of regressor matrices. Simulation results on a Puma560 robot manipulator indicate the efficiency of the proposed method.

Adelic versions of the Weierstrass approximation theorem

Let E_=∏p∈PEp be a compact subset of Zˆ=∏p∈PZp and denote by C(E_,Zˆ) the ring of continuous functions from E_ into Zˆ. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring IntQ(E_,Zˆ):={f(x)∈Q[x]|f(E_)⊆Zˆ} is dense in the product ∏p∈PC(Ep,Zp) for the uniform convergence topology. We also obtain an analogous statement for general compact subsets of Zˆ.Secondly, under the hypothesis that, for each n≥0, #(Ep(modp))>n for all but finitely many primes p, we prove the existence of regular bases of the Z-module IntQ(E_,Zˆ), and show that, for such a basis {fn}n≥0, every function φ_ in ∏p∈PC(Ep,Zp) may be uniquely written as a series ∑n≥0c_nfn where c_n∈Zˆ and limn→∞⁡c_n→0. Moreover, we characterize the compact subsets E_ for which the ring IntQ(E_,Zˆ) admits a regular basis in Pólya's sense by means of an adelic notion of ordering which generalizes Bhargava's p-ordering.

On a generalization of the Stone–Weierstrass theorem

Assume X is a compact Hausdorff space and C(X) is the space of real-valued continuous functions on X. A version of the Stone–Weierstrass theorem states that a closed subalgebra \(A\subset C(X)\), which contains a nonzero constant function, coincides with the whole space C(X) if and only if A separates points of X. In this paper, we generalize this theorem to the case in which two subalgebras of C(X) are involved.

BIBO-Stable Recursive Functional Link Polynomial Filters

The preeminent merit of linear or nonlinear recursive filters is their ability to represent unknown systems with far fewer coefficients than their nonrecursive counterparts. Unfortunately, nonlinear recursive filters become, in general, unstable for large amplitudes of the input signal. In this paper, we introduce a novel subclass of the linear-in-the-parameters nonlinear recursive filters whose members are stable according to the bounded-input bounded-output criterion. It is shown that these filters, called stable recursive functional link polynomial filters, are universal approximators for causal, time-invariant, infinite-memory, continuous, nonlinear systems according to the Stone-Weierstrass theorem. To further reduce the filter complexity, simplified structures are also investigated. Even though simplified filters are no more universal approximators, good performance is demonstrated exploiting either data measured on real systems or available in the literature as benchmarks for nonlinear system identification.

FAT-based robust adaptive control of electrically driven robots without velocity measurements

Recently, regressor-free control approach has been presented in which uncertainties are estimated using function approximation techniques (FAT) such as the Fourier series expansion or Legendre polynomials. However, FAT-based observer design remains as an open problem. With this in mind, this paper presents a robust adaptive controller for electrically driven robots, without any need for velocity measurements. The mixed observer/control design procedure is based on universal approximation theory and using Stone–Weierstrass theorem. To highlight the contribution of the paper, it should be emphasized that in comparison with previous related FAT-based controllers, the proposed controller is simpler and less computational. In addition, the number of required Fourier series expansions, control laws, and also adaptation rules has been reduced. Moreover, the observer design is free of model. Simulation results of the controller on a 6-DOF industrial robot manipulator have been presented which proves robustness of the proposed controller against various uncertainties. The results are also compared to those obtained from Chebyshev neural network.

Approximation of functions by a new family of generalized Bernstein operators

The main object of this paper is to construct a new generalization of the Bernstein operator, depending on a non-negative real parameter. We investigate some elementary properties of this operator, such as end point interpolation, linearity and positivity, etc. By using these generating operators, we provide another proof of the Weierstrass Approximation Theorem. We give the detailed proofs to the rate of convergence and Voronovskaja type asymptotic estimate formula for the operators. Moreover, an upper bound for the error is obtained in terms of the usual modulus of continuity. Shape preserving properties of the generalized Bernstein operators are also studied. It is proved that monotonic or convex functions produce monotonic or convex generalized Bernstein polynomials.

Robust task-space control of robot manipulators using differential equations for uncertainty estimation

SUMMARYMost control algorithms for rigid-link electrically driven robots are given in joint coordinates. However, since the task to be accomplished is expressed in Cartesian coordinates, inverse kinematics has to be computed in order to implement the control law. Alternatively, one can develop the necessary theory directly in workspace coordinates. This has the disadvantage of a more complex robot model. In this paper, a robust control scheme is given to achieve exact Cartesian tracking without the knowledge of the manipulator kinematics and dynamics, actuator dynamics and nor computing inverse kinematics. The control design procedure is based on a new form of universal approximation theory and using Stone–Weierstrass theorem, to mitigate structured and unstructured uncertainties associated with external disturbances and actuated manipulator dynamics. It has been assumed that the lumped uncertainty can be modeled by linear differential equations. As the method is Model-Free, a broad range of manipulators can be controlled. Numerical case studies are developed for an industrial robot manipulator.

Recursive least squares algorithm for a nonlinear additive system with time delay

This paper proposes a recursive least squares algorithm for a nonlinear additive system with time delay. By the Weierstrass approximation theorem and the key term separation principle, the model can be simplified as an identification model. Based on the identification model, a recursive least squares identification algorithm is used to estimate all the unknown parameters of the time-delayed additive system. An example is provided to show the effectiveness of the proposed algorithm.

New Stone-Weierstrass Theorem

Without the successful work of Professor Kakutani on representing a unit vector space as a dense vector sub-lattice of in 1941, where X is a compact Hausdorff space and C(X) is the space of real continuous functions on X. Professor M. H. Stone would not begin to work on “The generalized Weierstrass approximation theorem” and published the paper in 1948. Latter, we call this theorem as “Stone-Weierstrass theorem” which provided the sufficient and necessary conditions for a vector sub-lattice V to be dense in . From the theorem, it is not clear and easy to see whether 1) “the vector sub-lattice V of C(X) contains constant functions” is or is not a necessary condition; 2) Is there any clear example of a vector sub-lattice V which is dense in , but V does not contain constant functions. This implies that we do need some different version of “Stone-Weierstrass theorem” so that we will be able to understand the “Stone-Weierstrass theorem” clearly and apply it to more places where they need this wonderful theorem.

On the Dini and Stone–Weierstrass properties in pointfree topology

For a topological space X, it is customary to equip the f-ring C(X) or its bounded part C⁎(X) with the well-known topologies or uniformities of uniform convergence or pointwise convergence, which thank their importance to several fundamental theorems like a.o. the Stone–Weierstrass theorem or Dini's theorem. These theorems are classically proved subject to possible supplementary (often compactness) conditions on X. Alternatively, one can also in many cases characterize exactly those X for which the conclusion of such a theorem holds, i.e. those X that have the Stone–Weierstrass or Dini property. In pointfree topology, for a frame L, one encounters as a counterpart to C(X) (resp. C⁎(X)), the well-studied f-ring RL of real-valued continuous functions on L and its bounded part R⁎L. A pointfree Stone–Weierstrass theorem has been proved in B. Banaschewski [3,4]. It is the aim of this note to discuss some topological properties of the f-ring RL or its bounded part which are pointfree counterparts of the Stone–Weierstrass and Dini-type properties for spaces.

Rakotch type contractive maps, square roots and uniform convexity

In this paper we introduce a family of weakly contractive maps on the space $B(S)$ of bounded real valued functions and use it to show that a fundamental step in the proof of the well-known Stone-Weierstrass approximation theorem can be achieved via Rakotch’s fixed point theorem for weakly contractive maps. With the same technique, we obtain Zemanek’s theorem on the existence of square roots in certain Banach subalgebras of $B(S)$ , and, finally, in the context of abstract Banach algebras, we exhibit some relationship between weakly contractive maps on the closed unit ball and the geometry of the spheres.

A study about Chebyshev nonlinear filters

The paper studies a novel family of nonlinear filters based on Chebyshev polynomials of the first kind, the Chebyshev nonlinear filters. This family shares many of the characteristics of the recently introduced Legendre and even mirror Fourier nonlinear filters, but has also peculiar properties. Chebyshev nonlinear filters belong to the class of linear-in-the-parameters nonlinear filters. Their basis functions are polynomials, specifically, products of Chebyshev polynomial expansions of the input signal samples. According to the Stone–Weierstrass theorem, they are universal approximators for causal, time-invariant, finite-memory, continuous, nonlinear systems. Their basis functions are mutually orthogonal for white input signals with a particular nonuniform distribution. They admit perfect periodic sequences, i.e., periodic input sequences that guarantee the mutual orthogonality of the basis functions on a finite period. Using perfect periodic input signals, an unknown nonlinear system and its most relevant basis functions can be identified with the cross-correlation method. It is shown in the paper that the perfect periodic sequences of Chebyshev nonlinear filters are simply related to those of even mirror Fourier nonlinear systems. Experimental results involving a real nonlinear system illustrate the potentialities of these filters.