Spherical Scattered Data Quasi-interpolation by Gaussian Radial Basis Function

Since the spherical Gaussian radial function is strictly positive definite, the authors use the linear combinations of translations of the Gaussian kernel to interpolate the scattered data on spheres in this article. Seeing that target functions are usually outside the native spaces, and that one has to solve a large scaled system of linear equations to obtain combinatorial coefficients of interpolant functions, the authors first probe into some problems about interpolation with Gaussian radial functions. Then they construct quasiinterpolation operators by Gaussian radial function, and get the degrees of approximation. Moreover, they show the error relations between quasi-interpolation and interpolation when they have the same basis functions. Finally, the authors discuss the construction and approximation of the quasi-interpolant with a local support function.     

A shape-preserving quasi-interpolation operator satisfying quadratic polynomial reproduction property to scattered data

In this paper, we construct a univariate quasi-interpolation operator to non-uniformly distributed data by cubic multiquadric functions. This operator is practical, as it does not require derivatives of the being approximated function at endpoints. Furthermore, it possesses univariate quadratic polynomial reproduction property, strict convexity-preserving and shape-preserving of order 3 properties, and a higher convergence rate. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operator with that of Wu and Schaback’s quasi-interpolation scheme.     

The construction and approximation of feedforward neural network with hyp erb olic tangent function

In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tangent activation functionFirstly, an equation of partitions of unity for the hyperbolic tangent function is givenThen, two kinds of quasi-interpolation type neural network operators are constructed to approximate univariate and bivariate functions, respectivelyAlso, the errors of the approximation are estimated by means of the modulus of continuity of functionMoreover, for approximated functions with high order derivatives, the approximation errors of the constructed operators are estimated.     

A Quasi-Interpolation Satisfying Quadratic Polynomial Reproduction with Radial Basis Functions

In this paper,a new quasi-interpolation with radial basis functions which satis- fies quadratic polynomial reproduction is constructed on the infinite set of equally spaced data.A new basis function is constructed by making convolution integral with a constructed spline and a given radial basis function.In particular,for twicely differ- entiable function the proposed method provides better approximation and also takes care of derivatives approximation.     

Limit problems for interpolation by analytic radial basis functions

Interpolation problems for analytic radial basis functions like the Gaussian and inverse multiquadrics can degenerate in two ways: the radial basis functions can be scaled to become increasingly flat, or the data points coalesce in the limit while the radial basis functions stay fixed. Both cases call for a careful regularization, which, if carried out explicitly, yields a preconditioning technique for the degenerating linear systems behind these interpolation problems. This paper deals with both cases. For the increasingly flat limit, we recover results by Larsson and Fornberg together with Lee, Yoon, and Yoon concerning convergence of interpolants towards polynomials. With slight modifications, the same technique can also handle scenarios with coalescing data points for fixed radial basis functions. The results show that the degenerating local Lagrange interpolation problems converge towards certain Hermite–Birkhoff problems. This is an important prerequisite for dealing with approximation by radial basis functions adaptively, using freely varying data sites.     

Approximate approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.     

Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S _h } _h>0 of spaces, each spanned by thehZ ^d -translates of one function, is analyzed. The nearoptimal approximants of [R2] from eachs _h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s _h } is obtained froms ₁ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.     

Minimally supported error representations and approximation by the constants

Summary. Distribution theory is used to construct minimally supported Peano kernel type representations for linear functionals such as the error in multivariate Hermite interpolation. The simplest case is that of representing the error in approximation to f by the constant polynomial f(a) in terms of integrals of the first order derivatives of f. This is discussed in detail. Here it is shown that suprisingly there exist many representations which are not minimally supported, and involve the integration of first order derivatives over multidimensional regions. The distance of smooth functions from the constants in the uniform norm is estimated using our representations for the error. Received June 30, 1997 / Revised version received April 6, 1999 / Published online February 17, 2000     

Approximation inL 2 Sobolev spaces on the 2-sphere by quasi-interpolation

In this article we consider a simple method of radial quasi-interpolation by polynomials on the unit sphere in ℝ³, and present rates of covergence for this method in Sobolev spaces of square integrable functions. We write the discrete Fourier series as a quasi-interpolant and hence obtain convergence rates, in the aforementioned Sobolev spaces, for the discrete Fourier projection. We also discuss some typical practical examples used in the context of spherical wavelets.     

The new approximation operators with sigmoidal functions

The aim of this paper is to investigate approximation operators with logarithmic sigmoidal function of a class of two neural networks weights and a class of quasi-interpolation operators. Using these operators as approximation tools, the upper bounds of estimate errors are estimated for approximating continuous functions.     

B-spline quasi-interpolation on sparse grids

We propose a periodic B-spline quasi-interpolation for multivariate functions on sparse grids and develop a fast scheme for the evaluation of a linear combination of B-splines on sparse grids. We prove that both of these operations require only O(nlog^d−1n) number of multiplications, where n is the number of univariate B-spline basis functions used in each coordinate direction and d is the number of variables of the functions. We also establish the optimal approximation order of the periodic B-spline quasi-interpolation. Numerical examples are presented to confirm the theoretical estimates.     

紧区间上高斯核近似逼近误差估计的改进

本文研究了由高斯核构成的拟插值算子在闭区间上的近似逼近问题.利用函数延拓和近似单位分划的方法,构造了拟插值算子,并得到了一致范数下的逼近阶估计.     

On the convergence of type I Hermite–Padé approximants

Padé approximation has two natural extensions to vector rational approximation through the so-called type I and type II Hermite–Padé approximants. The convergence properties of type II Hermite–Padé approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite–Padé approximants for Nikishin systems of functions.     

Quasi-interpolation for linear functional data

Quasi-interpolation has been studied extensively in the literature. However, most studies of quasi-interpolation are usually only for discrete function values (or a finite linear combination of discrete function values). Note that in practical applications, more commonly, we can sample the linear functional data (the discrete values of the right-hand side of some differential equations) rather than the discrete function values (e.g., remote sensing, seismic data, etc). Therefore, it is more meaningful to study quasi-interpolation for the linear functional data. The main result of this paper is to propose such a quasi-interpolation scheme. Error estimate of the scheme is also given in the paper. Based on the error estimate, one can find a quasi-interpolant that provides an optimal approximation order with respect to the smoothness of the right-hand side of the differential equation. The scheme can be applied in many situations such as the numerical solution of the differential equation, construction of the Lyapunov function and so on. Respective examples are presented in the end of this paper.     

Radial kernels and their reproducing kernel Hilbert spaces

We describe how to use Schoenberg’s theorem for a radial kernel combined with existing bounds on the approximation error functions for Gaussian kernels to obtain a bound on the approximation error function for the radial kernel. The result is applied to the exponential kernel and Student’s kernel. To establish these results we develop a general theory regarding mixtures of kernels. We analyze the reproducing kernel Hilbert space (RKHS) of the mixture in terms of the RKHS’s of the mixture components and prove a type of Jensen inequality between the approximation error function for the mixture and the approximation error functions of the mixture components.     

On quasi-interpolation by radial basis functions with scattered centres

Approximation by radial basis functions with “quasi-uniformly” distributed centres inR ^d is discussed. A construction of new polynomially decaying functions that span the approximation space is presented and the properties of the quasi-interpolation operator with these functions are investigated. It is shown that the quasi-interpolant reproduces polynomials and gives approximation orders identical to those in the uniform square-grid case.     

Nonlinear approximation using Gaussian kernels

It is well known that nonlinear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron (in press) [2] for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore-Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations. As expected and desired, the approximation rates can be as high as needed and are essentially saturated only by the smoothness of the approximand.     

The shape preserving ang high accuracy approximation with multiquadric

This paper discusses the sufficient conditions for the shape preserving quasi-interpolation with multiquadric. Some quasi-interpolation schema is given such that the interpolation as well as its high derivatives is convergent. Supported by the National Natural Science Foundation of China.     

On the approximation power of bivariate splines

We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces S _d ^r (Δ) with d⩾ 3r + 2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the L _p norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the smallest angle in the underlying triangulation and the nature of the boundary of the domain. This revised version was published online in August 2006 with corrections to the Cover Date.     

Multivariate Frobenius–Padé approximants: Properties and algorithms

The aim of this paper is to construct rational approximants for multivariate functions given by their expansion in an orthogonal polynomial system. This will be done by generalizing the concept of multivariate Padé approximation. After defining the multivariate Frobenius–Padé approximants, we will be interested in the two following problems: the first one is to develop recursive algorithms for the computation of the value of a sequence of approximants at a given point. The second one is to compute the coefficients of the numerator and denominator of the approximants by solving a linear system. For some particular cases we will obtain a displacement rank structure for the matrix of the system we have to solve. The case of a Tchebyshev expansion is considered in more detail.