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In this paper, using operational rules, we derive some explicit expansion formulas associated with Brenke type polynomials. In particular, the corresponding connection and linearization coefficients are expressed either in terms of double sums or integral representations or sometimes in closed hypergeometric terms. The derived results are applied to Generalized Gould–Hopper and q-Laguerre polynomials. Some well-known duplication and convolution formulas are deduced as particular cases. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of n positive real numbers. Together with this interpretation, we prove some relevant properties that will allow us to define numerical approximation methods in several dimensions adapted to discontinuities. [ABSTRACT FROM AUTHOR]

The main object of this paper is to construct a new class of modified (p , q) -Gamma-type operators. For this new class of operators, in section one, the general moments are find; in section two, the Korovkin-type theorem and some direct results are proved by considering the modulus of continuity and modulus of smoothness and their behavior in Lipschitz-type spaces. In section three, some results in the weighted spaces are given, and in the end, some shape-preserving properties are proven. [ABSTRACT FROM AUTHOR]

Linear approximation methods suffer from Gibbs oscillations when approximating functions with jumps. Essentially non oscillatory subcell-resolution (ENO-SR) is a local technique avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper jump or corner (jump in the first order derivative) discontinuity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. This new procedure can be applied as subdivision scheme to design curves and surfaces both in point-value and in cell-average contexts. Using the proposed algorithm, we are able to construct approximations with high precision, with high piecewise regularity, and without smearing nor oscillations in the presence of discontinuities. These are desired properties in real applications as computer aided design or car design, among others. [ABSTRACT FROM AUTHOR]

This paper demonstrates the optimality of an interpolation set employed in derivative-free trust-region methods. This set is optimal in the sense that it minimizes the constant of well-poisedness in a ball centred at the starting point. It is chosen as the default initial interpolation set by many derivative-free trust-region methods based on underdetermined quadratic interpolation, including NEWUOA, BOBYQA, LINCOA, and COBYQA. Our analysis provides a theoretical justification for this choice. [ABSTRACT FROM AUTHOR]

Over the several decades, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples m. Our work focuses on providing theoretical approximation guarantees for the class of so-called (b , ε) -holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of m-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds. [ABSTRACT FROM AUTHOR]

In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity. [ABSTRACT FROM AUTHOR]

We establish two direct estimates by K-functionals of the rate of approximation by the Kantorovich operators in variable exponent Lebesgue spaces. They extend known results in the non-variable exponent Lebesgue spaces. The approach applied heavily relies on the boundedness of the Hardy–Littlewood maximal operator. [ABSTRACT FROM AUTHOR]

In this note, we study some approximation properties on a class of special Lototsky–Bernstein bases. We focus on approximation of | x | on [ − 1 , 1 ] by an approximation process generated from fixed points on Lototsky–Bernstein bases. Our first result shows that the approximation procedure p n (x) to | x | preserves good shapes on [ − 1 , 1 ] . Moreover, some convergence results and inequalities are derived. Our second main result states that the rate convergence of the approximation is O (n − 2) . [ABSTRACT FROM AUTHOR]

Let u be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in H r (u) , the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the L 2 space associated with u . This requires the simultaneous approximation of a function f and its consecutive derivatives up to order N ⩽ r . We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of E n (f (r)) , the error of best approximation of f (r) in L 2 (u) . [ABSTRACT FROM AUTHOR]

We study the solution set to multivariate Chebyshev approximation problem, focussing on the ill-posed case when the uniqueness of solutions can not be established via strict polynomial separation. We obtain an upper bound on the dimension of the solution set and show that nonuniqueness is generic for ill-posed problems on discrete domains. Moreover, given a prescribed set of points of minimal and maximal deviation we construct a function for which the dimension of the set of best approximating polynomials is maximal for any choice of domain. We also present several examples that illustrate the aforementioned phenomena, demonstrate practical application of our results and propose a number of open questions. [ABSTRACT FROM AUTHOR]

The present paper aims to propose an approximation method of Caputo fractional operator using discretization based on quadrature theory to minimize the error function for an Artificial Neural Network (ANN) with higher convergence rate. In the proposed work authors have verified the suitability of multilayer feed forward ANN architecture to get an estimated solution of fractional order differential equations. The back propagation algorithm and an unsupervised learning, was imposed in order to minimize the error function including the optimizing the network parameters such as weights of synapses and the biases. The algorithm uses truncated power series to replace the unknown function in the fractional differential equations. The novelty of the present work is an approximation of a fractional operator using discretization based on quadrature theory to construct an error function to provide an appropriate estimated solution of the nonlinear fractional differential equations. Four illustrative examples with different orders of nonlinear fractional differential equations are unraveled to approve the validity of the model along with demonstration of effectiveness and fast convergence of the proposed method.