A new continued fraction approximation and inequalities for the Gamma function via the Tri-gamma function

The main objective of this paper is to propose a product approximation for the Gamma function in the form of continued fraction, via the Tri-gamma function. The importance of this new formula is that the convergence of the corresponding asymptotic series is faster than other recently discovered asymptotic series, and some new inequalities related to the new formula are also established.     

在函数逼近论观点下半参数变系数非线性回归函数的估计

在函数逼近论的观点下研究了半参数变系数非线性回归函数的估计问题.采用总体之下L2多项式最佳逼近的方法与样本之下矩估计的方法,独立地分别作出非参数部分与变系数参数部分的解析函数形式的估计,最终得到回归函数的L2与强相合之联合收敛意义下的估计.     

Polynomial function and derivative approximation of Sinc data

Sinc methods consist of a family of one dimensional approximation procedures for approximating nearly every operation of calculus. These approximation procedures are obtainable via operations on Sinc interpolation formulas. Nearly all of these approximations–except that of differentiation–yield exceptional accuracy. The exception: when differentiating a Sinc interpolation formula that gives an approximation over an interval with a finite end-point. In such cases, we obtain poor accuracy in the neighborhood of the finite end-point. In this paper we derive novel polynomial-like procedures for differentiating a function that is known at Sinc points, to obtain an approximation of the derivative of the function that is uniformly accurate on the whole interval, finite or infinite, in the case when the function itself has a derivative on the closed interval.     

An estimate for the three-dimensional discrete Green's function and applications

In this paper we first introduce definitions of the regularized Green's function and the discrete Green's function in three dimensions. With complicated arguments estimates for these two functions are then derived. Finally we give an application of the estimate for the discrete Green's function to the high accuracy analysis of the three-dimensional block finite element approximation.     

Liberating the dimension for function approximation: Standard information

This is a follow-up paper of “Liberating the dimension for function approximation”, where we studied approximation of infinitely variate functions by algorithms that use linear information consisting of finitely many linear functionals. In this paper, we study similar approximation problems, however, now the algorithms can only use standard information consisting of finitely many function values. We assume that the cost of one function value depends on the number of active variables. We focus on polynomial tractability, and occasionally also study weak tractability. We present non-constructive and constructive results. Non-constructive results are based on known relations between linear and standard information for finitely variate functions, whereas constructive results are based on Smolyak’s construction generalized to the case of infinitely variate functions. Surprisingly, for many cases, the results for standard information are roughly the same as for linear information.     

The existence and local behaviour of the quadratic function approximation

This paper analyses the local behaviour of the quadratic function approximation to a function which has a given power series expansion about the origin. It is shown that the quadratic Hermite-Padé form always defines a quadratic function and that this function is analytic in a neighbourhood of the origin. This result holds even if the origin is a critical point of the function (i.e., the discriminant has a zero at the origin). If the discriminant has multiple zeros the order of the approximation will be degraded but only to a limited extent.     

Star-shaped approximation approach for stochastic programming problems with probability function

Star-shaped probability function approximation is suggested. Conditions of log-concavity and differentiability of approximation function are obtained. The method for constructing stochastic estimates of approximation function gradient and stochastic quasi-gradient algorithm for probability function maximization are described in the paper     

Convergence of cubic piecewise function

In this paper we construct a piecewise cubic polynomial function for approximation of a class of functions. From application point of view an explicit representation of the function is obtained. The convergence of constructed function has been discussed. The proofs of the results are simpler and shorter.     

Inequalities for the gamma function

Some inequalities for the gamma function are given. These results refine the classical Stirling approximation and its many recent improvements. Received: 2 May 2008, Revised: 2 September 2008     

Adaptive radial basis function interpolation using an error indicator

In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this paper, we propose a new adaptive algorithm for radial basis function (RBF) interpolation which aims to assess the local approximation quality, and add or remove points as required to improve the error in the specified region. For Gaussian and multiquadric approximation, we have the flexibility of a shape parameter which we can use to keep the condition number of interpolation matrix at a moderate size. Numerical results for test functions which appear in the literature are given for dimensions 1 and 2, to show that our method performs well. We also give a three-dimensional example from the finance world, since we would like to advertise RBF techniques as useful tools for approximation in the high-dimensional settings one often meets in finance.     

An algorithm for the computation of the reversible Randles-Sevcik function in electrochemistry

An algorithm for the numerical approximation of the reversible Randles-Sevcik function has been developed. The algorithm makes use of a piecewise nonlinear approximation that can be employed on a digital computer using FORTRAN.     

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.     

Approximation of functions subharmonic in a disk by the logarithm of the modulus of an analytic function

Yulmukhametov's result concerning the approximation of a function subharmonic in a bounded domain by the logarithm of the modulus of an analytic function is supplemented with an estimate of the exceptional set in the important case of a disk. We show that this approximation is unimprovable in a certain sense.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1080–1083, August, 1994.     

Fourier级数部分和对凸型函数的逼近

The concept of convex type function is introduced in this paper,from which a kind of convex-decomposition approach is proposed. As one of applications of this approach, the approximation of the convex type function by the partial sum of its Fourier series is investigated. Moreover,the order of approximation is described with the 2th continuous modulus.     

非对角二元二次函数逼近的存在性和局部性质

§ 1　IntroductionThis paper is concerned with the properties of the simple off-diagonal bivariatequadratic Hermite-Padé approximation.Thisapproximation may be defined asfollows(see,for example,[1 ] ) .Let f(x,y) be a bivariate function,analytic in some neighbourhood of the origin(0 ,0 ) ,whose series expansion about the origin is known.Let a0 (x,y) ,a1 (x,y) ,a2 (x,y) bebivariate polynomials,a0 (x,y) = ki,j=0 a(0 )ij xiyj,a1 (x,y) = ni,j=0 a(1 )ij xiyj,a2 (x,y) = mi,j=0 a(2 )ij xiyj,such th…     

Continued fraction approximations and inequalities for the gamma function by Burnside

In this paper, based on the Burnside’s formula, some accurate approximation and double inequality for the gamma function are established. Finally, for demonstrating the superiority of our new series over the Burnside’s formula and other classical or recently discovered asymptotic series, some numerical computations are provided.     

Modified clenshaw-curtis method for the computation of Bessel function integrals

The numerical evaluation of Bessel function integrals may be difficult when the Bessel function is rapidly oscillating in the interval of integration. In the method presented here, the smooth factor of the integrand is replaced by a truncated Chebyshev series approximation and the resulting integral is computed exactly. The numerical aspects of this exact integration are discussed.     

Universal bases and greedy algorithms for anisotropic function classes

We suggest a three-step strategy to find a good basis (dictionary) for non-linear m-term approximation. The first step consists of solving an optimization problem of finding a near best basis for a given function class F, when we optimize over a collection D of bases (dictionaries). The second step is devoted to finding a universal basis (dictionary) D _u ∈ D for a given pair (F, D) of collections: F of function classes and D of bases (dictionaries). This means that D_u provides near optimal approximation for each class F from a collection F. The third step deals with constructing a theoretical algorithm that realizes near best m-term approximation with regard to D _u for function classes from F. In this paper we work this strategy out in the model case of anisotropic function classes and the set of orthogonal bases. The results are positive. We construct a natural tensor-product-wavelet-type basis and prove that it is universal. Moreover, we prove that a greedy algorithm realizes near best m-term approximation with regard to this basis for all anisotropic function classes.     

The ridge function representation of polynomials and an application to neural networks

The first goal of this paper is to establish some properties of the ridge function representation for multivariate polynomials, and the second one is to apply these results to the problem of approximation by neural networks. We find that for continuous functions, the rate of approximation obtained by a neural network with one hidden layer is no slower than that of an algebraic polynomial.