Generalized Bernstein Polynomials Research Articles

Characterization of Deferred Statistical Convergence of Order $$\alpha $$ for Positive Linear Operators and Application to Generalized Bernstein Polynomials

Characterization of Deferred Statistical Convergence of Order $$\alpha $$ for Positive Linear Operators and Application to Generalized Bernstein Polynomials

On the numerical solution of Volterra integral equations on equispaced nodes

In the present paper, a Nyström-type method for second kind Volterra integral equations is introduced and studied. The method makes use of generalized Bernstein polynomials, defined for continuous functions and based on equally spaced points. Stability and convergence are studied in the space of continuous functions. Numerical tests illustrate the performance of the proposed approach.

A product integration rule on equispaced nodes for highly oscillating integrals

This paper provides a product integration rule for highly oscillating integrands of the type ∫−aae−iω(x−y)f(x)dx,a>0,i=−1,y∈[−a,a],ω∈R+,based on the approximation of f by means of the Generalized Bernstein polynomials B̄m,ℓf. The rule involves the samples of f at m+1 equally spaced points of [−a,a], and differently from the classical Bernstein polynomials, the suitable modulation of the parameter ℓ∈N allows to increase the accuracy of the product rule, as the smoothness of f increases. Stability and error estimates are proven for f belonging to the space of continuous functions and their Sobolev-type subspaces. Finally, some numerical tests which confirm such theoretical estimates are shown.

A discretization method for nonlocal diffusion type equations

In this paper we consider a numerical scheme for the treatment of an integro-differential equation. The latter represents the formulation of a nonlocal diffusion type equation. The discretization procedure relies on the application of the line method. However, quadrature formulae are needed for the evaluation of the integral operator. They are based on generalized Bernstein polynomials. Numerical evidence shows that the proposed method is a suitable and reliable approach for the problem.

A certain class of statistical convergence of martingale sequences and its applications to Korovkin-type approximation

In this paper, we investigate and study the notions of statistical product convergence and statistical product summability via deferred Cesàro and deferred Nörlund product means for martingale sequences of random variables. We then establish an inclusion theorem concerning the relation between these two beautiful and definitively useful concepts. Also, based upon our proposed ideas, we demonstrate new thoughtful approximation of Korovkin-type theorems for a martingale sequence over a Banach space. Moreover, we establish that our theorems effectively extend and improve most (if not all) of the previously existing outcomes (in statistical and classical versions). Finally, by using the generalized Bernstein polynomials, we present an illustrative example of a martingale sequence in order to demonstrate that our established theorems are quite stronger than the traditional and statistical versions of different theorems existing in the literature.

Some numerical applications of generalized Bernstein operators

In this paper some recent applications of the so-called Generalized Bernstein polynomials are collected. This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of [0; 1] and depends on an additional parameter which yields the remarkable property of improving the rate of convergence to the function f, according with the smoothness of f. This means that the sequence does not suffer of the saturation phenomena occurring by using the classical Bernstein polynomials or arising in piecewise polynomial approximation. The applications considered here deal with the numerical integration and the simultaneous approximation. Quadrature rules on equidistant nodes of [0; 1] are studied for the numerical computation of ordinary integrals in one or two dimensions, and usefully employed in Nyström methods for solving Fredholm integral equations. Moreover, the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated. For all the applications, some numerical details are given in addition to the error estimates, and the proposed approximation methods have been implemented providing numerical tests which confirm the theoretical estimates. Some open problems are also introduced.

Statistical product convergence of martingale sequences and its applications to Korovkin‐type approximation theorems

In this paper, we investigate and study the concepts of statistical product convergence and statistical product summability via deferred Cesàro and deferred weighted product means for martingale sequences of random variables. We then establish an inclusion theorem concerning the relation between these two nice and potentially useful concepts. Also, based upon our proposed concepts, we state and prove a set of new Korovkin‐type approximation theorems for a martingale sequence over a Banach space. Moreover, we demonstrate that our approximation theorems effectively extend and improve most (if not all) of the previously existing results (both in statistical and classical versions). Finally, by using the generalized Bernstein polynomials, we present an illustrative example of a martingale sequence in order to demonstrate that our established theorems are quite stronger than the traditional and statistical versions of different theorems existing in the literature. We also suggest a direction for future researches on this subject, which are based upon the basic (or q‐) calculus, but not upon the trivial and inconsequential variations involving the so‐called (p, q)‐calculus.

Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes

In the present paper, we propose a numerical method for the simultaneous approximation of the finite Hilbert and Hadamard transforms of a given function f, supposing to know only the samples of f at equidistant points. As reference interval we consider [ − 1 , 1 ] and as approximation tool we use iterated Boolean sums of Bernstein polynomials, also known as generalized Bernstein polynomials. Pointwise estimates of the errors are proved, and some numerical tests are given to show the performance of the procedures and the theoretical results.

A projection method for linear Fredholm–Volterra integro-differential equations

ABSTRACTIn this study, a collocation method, one of the type of projection methods based on the generalized Bernstein polynomials, is developed for the solution of high-order linear Fredholm–Volterra integro-differential equations containing derivatives of unknown function in the integral part. The method is valid for the mixed conditions. The convergence analysis and error bounds of the method are also given. Besides, six examples are presented to demonstrate the applicability and validity of the method.

Personalized quantifier by Bernstein polynomials combined with interpolation spline

A new form of personalized quantifier is developed in this paper, with which to investigate and formalize people's personalities or behavior intentions that have to be considered in increasingly complex situations. As we show in the article, the developed quantifier is realized by generalized Bernstein polynomials combined with interpolation spline, and finally expressed as a sequence of piecewise nonlinear polynomials with an adjustable degree. It is characterized by many excellent properties exemplified by such terms as sufficient smoothness, shape-preserving interpolation, and a high rate of convergence. In particular, the consistency of the ordered weighted averaging (OWA) aggregation under the guidance of the developed quantifier is addressed and proved. This actually provides a sound theoretical basis for practical use. We also experimentally show that the developed quantifier significantly outperforms, in all respects of geometrical characteristics, the other ones presented in previous work, whether from a viewpoint of global approximation of functions or local one. As such, the developed quantifier could be considered as an effective analytical tool for decision making under uncertainty in which different personality traits have to be taken into account.

An improved algorithm for generalized community structure inference in complex networks

In recent years, the research of the community detection is not only on the structure that densely connected internally, but also on the structure of more patterns, such as heterogeneity, overlapping, core–periphery. In this paper, we build the network model based on the random graph models and propose an improved algorithm to infer the generalized community structures. We achieve it by introducing the generalized Bernstein polynomials and computing the latent parameters of vertices. The algorithm is tested both on the computer-generated benchmark networks and the real-world networks. Results show that the algorithm makes better performances on convergence speed and is able to discover the latent continuous structures in networks.

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.

(p, q)-Beta functions and applications in approximation

In the present paper, we consider (p, q)-analogue of the beta operators and using it, we propose the integral modification of the generalized Bernstein polynomials. We estimate some direct results on local and global approximation. Also, we illustrate some graphs for the convergence of (p, q)-Bernstein–Durrmeyer operators for different values of the parameters p and q using Mathematica package.

Analysis and construction of a family of refinable functions based on generalized Bernstein polynomials

In this paper, we construct a new family of refinable functions from generalized Bernstein polynomials, which include pseudo-splines of Type II. A comprehensive analysis of the refinable functions is carried out. We then prove the convergence of cascade algorithms associated with the new masks and construct Riesz wavelets whose dilation and translation form a Riesz basis for $L_{2}(\mathbb{R})$ . Stability of the subdivision schemes, regularity and approximation orders are obtained. We also illustrate the symmetry of the corresponding refinable functions.

A collocation method for linear integral equations in terms of the generalized Bernstein polynomials

A collocation method for linear integral equations in terms of the generalized Bernstein polynomials

Compactly Supported Tight and Sibling Frames Based on Generalized Bernstein Polynomials

We obtain a family of refinable functions based on generalized Bernstein polynomials to provide derived properties. The convergence of cascade algorithms associated with the new masks is proved, which guarantees the existence of refinable functions. Then, we analyze the symmetry, regularity, and approximation order of the refinable functions, which are of importance. Tight and sibling frames are constructed and interorthogonality of sibling frames is demonstrated. Finally, we give numerical examples to explicitly illustrate the construction of the proposed approach.

On a generalization of Bernstein polynomials and Bézier curves based on umbral calculus (II): de Casteljau algorithm

The investigation of the umbral calculus based generalization of Bernstein polynomials and Bézier curves is continued in this paper: First a generalization of the de Casteljau algorithm that uses umbral shift operators is described. Then it is shown that the quite involved umbral shifts can be replaced by a surprisingly simple recursion which in turn can be understood in geometrical terms as an extension of the de Casteljau interpolation scheme. Namely, instead of using only the control points of level r−1 to generate the points on level r as in the ordinary de Casteljau algorithm, one uses also points on level r−2 or more previous levels. Thus the unintuitive parameters in the algebraic definition of generalized Bernstein polynomials get geometric meaning. On this basis a new direct method for the design of Bézier curves is described that allows to adapt the control polygon as a whole by moving a point of the associated Bézier curve.

Approximation properties of ELM-fuzzy systems for smooth functions and their derivatives

In this paper, we utilize generalized Bernstein polynomials to construct fuzzy system. Different from traditional Bernstein polynomials, partition of interval on input variable can be chosen as non-equidistant division. We prove that generalized Bernstein fuzzy systems are universal approximators to a given continuous function and its high-order derivatives. Further, ELM method is used to tune the parameters of generalized Bernstein fuzzy system and Spline fuzzy system. It is proved that ELM-Spline fuzzy system can approximate a function and its derivative. Simulation examples show that the proposed ELM-Bernstein fuzzy system and ELM-spline fuzzy system can achieve high approximation capability for nonlinear model.

The r-monotonicity of generalized Bernstein polynomials

AbstractLet f ∊ C[0, 1] and let the Bn(f, q; x) be generalized Bernstein polynomials based on the q-integers that were introduced by Phillips. We prove that if f is r-monotone, then Bn(f, q; x) is r-monotone, generalizing well-known results when q = 1 and the results when r = 1 and r = 2 by Goodman et al. We also prove a sufficient condition for a continuous function to be r-monotone.

Approximation properties of rth order generalized Bernstein polynomials based on q-calculus

In this paper we introduce a generalization of Bernstein polynomials based on q calculus. With the help of Bohman-Korovkin type theorem, we obtain A-statistical approximation properties of these operators. Also, by using the Modulus of continuity and Lipschitz class, the statistical rate of convergence is established. We also gives the rate of A-statistical convergence by means of Peetre’s type K-functional. At last, approximation properties of a rth order generalization of these operators is discussed.