Approximation Of Continuous Functions Research Articles

On the order of weighted approximation by positive linear operators

An estimation of approximation of continuous functions by positive linear operators in weighted norm using the weighted modulus of continuity is established. Application of the main result to the known Gadjiyev-Ibragimov operators is given.

Optimal Design of a Thomson-Coil Actuator Utilizing a Mixed-Integer-Discrete-Continuous Variables Global Optimization Algorithm

To improve the performance of a Thomson-coil actuator, a new global optimization algorithm is proposed with mixed integer-discrete-continuous variables. In the algorithm, an augmented objective function is constructed by introducing a penalty function and an extended function to treat the discrete quantities as continuous ones. For the discrete objective function, two kinds of continuous function approximation methods, step function and linear function approximations, are suggested. The coupling particles swarm optimization method is then applied to the augmented objective function to find all the global and local optimal points. Finally, the validity of the proposed method is proven by comparing obtained optimization results with those from the enumeration algorithm.

Combining Learned Discrete and Continuous Action Models

Action modeling is an important skill for agents that must perform tasks in novel domains. Previous work on action modeling has focused on learning STRIPS operators in discrete, relational domains. There has also been a separate vein of work in continuous function approximation for use in optimal control in robotics. Most real world domains are grounded in continuous dynamics but also exhibit emergent regularities at an abstract relational level of description. These two levels of regularity are often difficult to capture using a single action representation and learning method. In this paper we describe a system that combines discrete and continuous action modeling techniques in the Soar cognitive architecture. Our system accepts a continuous state representation from the environment and derives a relational state on top of it using spatial relations. The dynamics over each representation is learned separately using two simple instance-based algorithms. The predictions from the individual models are then combined in a way that takes advantage of the information captured by each representation. We empirically show that this combined model is more accurate and generalizable than each of the individual models in a spatial navigation domain.

A review of stochastic algorithms with continuous value function approximation and some new approximate policy iteration algorithms for multidimensional continuous applications

We review the literature on approximate dynamic programming, with the goal of better understanding the theory behind practical algorithms for solving dynamic programs with continuous and vector-valued states and actions and complex information processes. We build on the literature that has addressed the well-known problem of multidimensional (and possibly continuous) states, and the extensive literature on model-free dynamic programming, which also assumes that the expectation in Bellman’s equation cannot be computed. However, we point out complications that arise when the actions/controls are vector-valued and possibly continuous. We then describe some recent research by the authors on approximate policy iteration algorithms that offer convergence guarantees (with technical assumptions) for both parametric and nonparametric architectures for the value function.

A note on the rates of uniform approximation of fuzzy systems

For the fuzzy systems with the kernel-shaped fuzzy sets of if part, we estimate the rates of the uniform approximation for continuous functions. Results are given associatively with the rates of convergence of the sequence (logk/k)a.

Uniform approximation by (quantum) polynomials

We show that quantum algorithms can be used to re-prove a classical theorem in approximation theory, Jackson's Theorem, which gives a nearly-optimal quantitative version of Weierstrass's Theorem on uniform approximation of continuous functions by polynomials. We provide two proofs, based respectively on quantum counting and on quantum phase estimation.

Uniform approximation by (quantum) polynomials

We show that quantum algorithms can be used to re-prove a classical theorem in approx-imation theory, Jackson's Theorem, which gives a nearly-optimal quantitative version ofWeierstrass's Theorem on uniform approximation of continuous functions by polynomi-als. We provide two proofs, based respectively on quantum counting and on quantumphase estimation.

On approximation of continuous functions by trigonometric polynomials

We generalize the classical Jackson–Bernstein constructive description of Hölder classes of periodic functions on the interval [ − π , π ] . We approximate by trigonometric polynomials continuous functions defined on a compact set E ⊂ [ − π , π ] . This set may consist of an infinite number of intervals.

Uniform Approximation by the Highest Defect Continuous Polynomial Splines: Necessary and Sufficient Optimality Conditions and Their Generalisations

In this paper necessary and sufficient optimality conditions for uniform approximation of continuous functions by polynomial splines with fixed knots are derived. The obtained results are generalisations of the existing results obtained for polynomial approximation and polynomial spline approximation. The main result is two-fold. First, the generalisation of the existing results to the case when the degree of the polynomials, which compose polynomial splines, can vary from one subinterval to another. Second, the construction of necessary and sufficient optimality conditions for polynomial spline approximation with fixed values of the splines at one or both borders of the corresponding approximation interval.

Piecewise linear approximations of multivariate functions: A multiresolution-based compression algorithm suitable for circuit implementation

This paper is concerned with a multiresolution approach to the piecewise-linear approximation of multivariate nonlinear continuous functions. The proposed technique has no restrictions on the number of variables the functions depend on, and is based on the use of piecewise-linear “hat” functions. The approximation levels are related to nested function spaces. The multiresolution approach allows one to define a simple model reduction strategy that is based on a proper error definition. The interest in the piecewise-linear approximations and in the hat functions is motivated by the simplicity of their circuit implementations. The efficiency of the method is tested via two benchmark examples, one of which concerns the approximation of the vector field of a nonlinear dynamical system.

The Chebyshev rank of multivariate polynomials in [formula omitted]-approximation

Weighted best L 1 -approximation of multivariate continuous real-valued functions by multivariate polynomials is studied. The largest possible dimension of the sets of best L 1 -approximations is determined.

Four-dimensional matrix transformation and the rate of A-statistical convergence of continuous functions

Motivated by our earlier work on the statistical approximation of continuous functions by positive linear operators of two variables, we study rates of A -statistical convergence of a sequence of positive linear operators acting on the space of all continuous real valued functions on any D compact subset of the real two-dimensional space. Furthermore, displaying an example, it is shown that our statistical rates are more efficient than the classical aspects in the approximation theory.

On approximation of continuous functions by determinate functions with delay

We consider determinate functions with delay which are extensions of determinate functions and find some properties of these functions. The problem is posed to approximate continuous functions by functions with delay, and the assertion is proved that it is possible to approximate any continuous function with an arbitrary accuracy. Approximations for some functions are given, including the addition and multiplication functions which are minimal from the delay viewpoint.

On Approximation of Continuous Functions by Entire Functions on Subsets of the Real Line

We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. Approximation of continuous bounded functions by entire functions of exponential type on unbounded closed proper subsets of the real line is studied.

Knowledge Acquisition for Business Intelligent Systems

Knowledge acquisition is a frequent bottleneck in business intelligent systems, and knowledge extraction from trained artificial neural networks (ANNs) provides an excellent way for explaining the functioning of a business connectionist system. This is important for ANNs to gain a wider degree of acceptance in problem domain applications like classification, diagnosis, continuous function approximation, time series prediction, and data mining. All these typical applications could be implemented into hybrid business intelligent systems. This paper presents the new applied research techniques for extracting business knowledge from trained ANNs, and it is organized into four sections that include an introduction, the state of the art, knowledge extraction and representation by rules, and conclusions.

Estimate of the approximation of periodic functions by negative Cesàro means

The problem of approximation of continuous functions by Cesaro (C,α)-means, −1 < α < 0, in terms of Lp and C-modulus of continuity is studied.

Approximation related to quotient functionals

We examine the best approximation of componentwise positive vectors or positive continuous functions f by linear combinations f ˆ = ∑ j α j φ j of given vectors or functions φ j with respect to functionals Q p , 1 ≤ p ≤ ∞ , involving quotients max { f / f ˆ , f ˆ / f } rather than differences | f − f ˆ | . We verify the existence of a best approximating function under mild conditions on { φ j } j = 1 n . For discrete data, we compute a best approximating function with respect to Q p , p = 1 , 2 , ∞ by second order cone programming. Special attention is paid to the Q ∞ functional in both the discrete and the continuous setting. Based on the computation of the subdifferential of our convex functional Q ∞ we give an equivalent characterization of the best approximation by using its extremal set. Then we apply this characterization to prove the uniqueness of the best Q ∞ approximation for Chebyshev sets { φ j } j = 1 n .

Approximation of continuous functions on complex Banach spaces

We prove a complex analog of Kurzweil’s theorem on the approximation of continuous functions on separable Banach spaces admitting a separating polynomial and obtain a complex analog of new results due to Boiso and Hajek.

Integral approximation of the characteristic function of an interval and the Jackson inequality in C( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv39ga % iyaacqWFtcpvaaa!46CC! $$ \mathbb{T} $$ )

We apply the results on the integral approximation of the characteristic function of an interval by the subspace \( \mathcal{T}_{n - 1} \) of trigonometric polynomials of order at most n − 1, which were obtained by the authors earlier, to investigate the Jackson inequality between the best uniform approximation of a continuous periodic function by the subspace \( \mathcal{T}_{n - 1} \) and its modulus of continuity of the second order. The corresponding method of the uniform approximation of continuous periodic functions by trigonometric polynomials is constructed.

Approximation by Stancu–Chlodowsky polynomials

In this paper, we investigate convergence and approximation properties of a Chlodowsky type generalization of Stancu polynomials (we called Stancu–Chlodowsky polynomials).The rates of convergence of this generalization are obtained by means of modulus of continuity and by using the K-functional of Peetre. We also present and prove theorems on weighted approximation and the order of approximation of continuous functions by these operators on all positive semi-axis.