Local error estimates for radial basis function interpolation of scattered data

Introducing a suitable variational formulation for the localerror of scattered data interpolation by radial basis functions(r), the error can be bounded by a term depending on the Fouriertransform of the interpolated function f and a certain ‘Krigingfunction’, which allows a formulation as an integral involvingthe Fourier transform of . The explicit construction of locallywell-behaving admissible coefficient vectors makes the Krigingfunction bounded by some power of the local density h of datapoints. This leads to error estimates for interpolation of functionsf whose Fourier transform f is ‘dominated’ by thenonnegative Fourier transform of (x) = (||x||) in the sense . Approximation orders are arbitrarily high for interpolationwith Hardy multiquadrics, inverse multiquadrics and Gaussiankernels. This was also proven in recent papers by Madych andNelson, using a reproducing kernel Hilbert space approach andrequiring the same hypothesis as above on f, which limits thepractical applicability of the results. This work uses a differentand simpler analytic technique and allows to handle the casesof interpolation with (r) = r^s for s R, s > 1, s 2N, and(r) = r^s log r for s 2N, which are shown to have accuracy O(h^s/2)     

On dual Bernstein polynomials and stochastic fractional integro-differential equations

In recent years, random functional or stochastic equations have been reported in a large class of problems. In many cases, an exact analytical solution of such equations is not available and, therefore, is of great importance to obtain their numerical approximation. This study presents a numerical technique based on Bernstein operational matrices for a family of stochastic fractional integro-differential equations (SFIDE) by means of the trapezoidal rule. A relevant feature of this method is the conversion of the SFIDE into a linear system of algebraic equations that can be analyzed by numerical methods. An upper error bound, the convergence, and error analysis of the scheme are investigated. Three examples illustrate the accuracy and performance of the technique.     

Fractals and Bernstein polynomials

Self-similarity of Bernstein polynomials, embodied in their subdivision property is used for construction of an Iterative (hyperbolic) Function System (IFS) whose attractor is the graph of a given algebraic polynomial of arbitrary degree. It is shown that such IFS is of just-touching type, and that it is peculiar to algebraic polynomials. Such IFS is then applied to faster evaluation of Bézier curves and to introduce interactive free-form modeling component into fractal sets.     

Note on Bernstein polynomials and Kantorovich polynomials

We obtain two asymptotic representations of remainder of approximation of derivable functions by Bernstein polynomials and Kantorovich polynomials separately.     

Generalized Bernstein polynomials with Pollaczek weight

Bernstein polynomials are a useful tool for approximating functions. In this paper, we extend the applicability of this operator to a certain class of locally continuous functions. To do so, we consider the Pollaczek weight

which is rapidly decaying at the endpoints of the interval considered. In order to establish convergence theorems and error estimates, we need to introduce corresponding moduli of smoothness and K-functionals. Because of the unusual nature of this weight, we have to overcome a number of technical difficulties, but the equivalence of the moduli and K-functionals is a benefit interesting in itself. Similar investigations have been made in [B. Della Vecchia, G. Mastroianni, J. Szabados, Weighted approximation of functions with endpoint or inner singularities by Bernstein operators, Acta Math. Hungar. 103 (2004) 19–41] in connection with Jacobi weights.     

Error estimates for scattered data interpolation on spheres

We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the -sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.

     

Bernstein polynomials and learning theory

When learning processes depend on samples but not on the order of the information in the sample, then the Bernoulli distribution is relevant and Bernstein polynomials enter into the analysis. We derive estimates of the approximation of the entropy function x log x that are sharper than the bounds from Voronovskaja's theorem. In this way we get the correct asymptotics for the Kullback–Leibler distance for an encoding problem.     

Convexity and Bernstein polynomials onk-simploids

This paper is concerned with Bernstein polynomials onk-simploids by which we mean a cross product ofk lower dimensional simplices. Specifically, we show that if the Bernstein polynomials of a given functionf on ak-simploid form a decreasing sequence thenf +l, wherel is any corresponding tensor product of affine functions, achieves its maximum on the boundary of thek-simploid. This extends recent results in [1] for bivariate Bernstein polynomials on triangles. Unlike the approach used in [1] our approach is based on semigroup techniques and the maximum principle for second order elliptic operators. Furthermore, we derive analogous results for cube spline surfaces.This work was partially supported by NATO Grant No. DJ RG 639/84.     

A new class of polynomials associated with Bernstein and beta polynomials

The purpose of this paper is to define a new class polynomials. Special cases of these polynomials give many famous family of the Bernstein type polynomials and beta polynomials. We also construct generating functions for these polynomials. We investigate some fundamental properties of these functions and polynomials. Using functional equations and generating functions, we derive various identities related to theses polynomials. We also construct interpolation function that interpolates these polynomials at negative integers. Finally, we give a matrix representations of these polynomials. Copyright © 2013 John Wiley & Sons, Ltd.     

-functionals and multivariate Bernstein polynomials

This paper estimates upper and lower bounds for the approximation rates of iterated Boolean sums of multivariate Bernstein polynomials. Both direct and inverse inequalities for the approximation rate are established in terms of a certain K-functional. From these estimates, one can also determine the class of functions yielding optimal approximations to the iterated Boolean sums.     

Asymptotic expansion and extrapolation for Bernstein polynomials with applications

Given a real functionf C ^2k [0,1],k 1 and the corresponding Bernstein polynomials {B _n (f)} _n we derive an asymptotic expansion formula forB _n (f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence thanB _n (f). As a subclass of these sequences we recognize the linear combinations of Bernstein polynomials considered by Butzer, Frentiu, and May [2, 6, 9]. In addition we prove approximation theorems which extend previous results of Butzer and May. Finally we consider some applications to numerical differentiation and quadrature and we perform numerical experiments showing the effectiveness of the considered technique.This work was partially supported by a grant from MURST 40.     

Multivariate generalized Bernstein polynomials: identities for orthogonal polynomials of two variables

We introduce polynomials $B^n_{k}(\boldmath{x};\omega|q)$ of total degree n, where $\boldmath{k} = (k_1,\ldots,k_d)\in\mathbb N_0^d, \; 0\le k_1+\ldots+k_d\le n$ , and $\boldmath{x}=(x_1,x_2,\ldots,x_d)\in\mathbb R^d$ , depending on two parameters q and ω, which generalize the multivariate classical and discrete Bernstein polynomials. For ω=0, we obtain an extension of univariate q-Bernstein polynomials, introduced by Phillips (Ann Numer Math 4:511–518, 1997). Basic properties of the new polynomials are given, including recurrence relations, q-differentiation rules and de Casteljau algorithm. For the case d=2, connections between $B^n_{k}(\boldmath{x};\omega|q)$ and bivariate orthogonal big q-Jacobi polynomials—introduced recently by the first two authors—are given, with the connection coefficients being expressed in terms of bivariate q-Hahn polynomials. As limiting forms of these relations, we give connections between bivariate q-Bernstein and Dunkl’s (little) q-Jacobi polynomials (SIAM J Algebr Discrete Methods 1:137–151, 1980), as well as between bivariate discrete Bernstein and Hahn polynomials.     

Convexity preserving and predicting by Bernstein polynomials

It is known that the Bernstein polynomials of a function f defined on [0, 1 ] preserve its convexity properties, i.e., if f⁽ⁿ⁾ 0 then for m n, (B_mf)⁽ⁿ⁾ 0. Moreover, if f is n-convex then (B_mf)⁽ⁿ⁾ 0. While the converse is not true, we show that if f is bounded on (a, b) and if for every subinterval [α, β] (a, b) the nth derivative of the mth Bernstein polynomial of f on [α, β] is nonnegative then f is n-convex.     

Direct and inverse theorems for Bernstein polynomials with inner singularities

We introduce a new type of Bernstein polynomials, which can be used to approximate the functions with inner singularities. The direct and inverse results of the weighted approximation of this new type of combinations are obtained.     

Random Sampling Scattered Data with Multivariate Bernstein Polynomials

In this paper, the multivariate Bernstein polynomials defined on a simplex are viewed as sampling operators, and a generalization by allowing the sampling operators to take place at scattered sites is studied. Both stochastic and deterministic aspects are applied in the study. On the stochastic aspect, a Chebyshev type estimate for the sampling operators is established. On the deterministic aspect, combining the theory of uniform distribution and the discrepancy method, the rate of approximating continuous function and L ^p convergence for these operators are studied, respectively.     

Numerical solution of stochastic It(o)-Volterra integral equations based on Bernstein multi-scaling polynomials

In this paper,first,Bernstein multi-scaling polynomials (BMSPs) and their prop-erties are introduced.These polynomials are obtained by compressing Bernstein pol...     

On the approximation of differentiable functions by Bernstein polynomials and Kantorovich polynomials

E.V. Voronovskaya and S.N. Bernstein established an asymptotic representation for the deviation of functions from Bernstein polynomials under the condition that the function has an even-order derivative. In the present paper, a similar problem is solved in the case when the function has an odd-order derivative. In addition, analogous representations are obtained for the deviations of functions from Kantorovich polynomials.     

The L2 norm of the approximation error for Bernstein polynomials

Wassily Hoeffding (J. Approximation Theory 4 (1971), 347–356) obtained a convergence rate for the L₁ norm of the approximation error, using Bernstein polynomials for a wide class of functions. Here, by a different method of proof, a similar result is obtained for the L₂ norm.