MULTIVARIATE PROBABILISTIC APPROXIMATION IN WAVELET STRUCTURE

Let otherwise and F(x,y).be a continuous distribution function on R^2. Then there exist linear wavelet operators L_n(F,x,y)which are also distribution function and where the defining them mother wavelet is(x,y).These approximate F(x,y)in the supnorm.The degree of this approximation is estimated by establishing a Jackson type inequality.Furthermore we give generalizations for the case of a mother wavelet ≠,which is just any distribution function on R^2,also we extend these results in R^r,r>2.

Constructive Approximation by Superposition of Sigmoidal Functions

In this paper, a constructive theory is developed for approximating func- tions of one or more variables by superposition of sigmoidal functions. This is done in the uniform norm as well as in the L^p norm. Results for the simultaneous approx- imation, with the same order of accuracy, of a function and its derivatives （whenever these exist）, are obtained. The relation with neural networks and radial basis func- tions approximations is discussed. Numerical examples are given for the purpose of illustration.

CONVERGENCE IN MEASURE OF MULTIVARIATE PADE APPROXIMANTS

Convergence conclusions of Pade approximants in the univariate case can be found in various papers. However,resuhs in the multivariate case are few.A.Cuyt seems to be the only one who discusses convergence for multivariate Pade approximants,she gives in[2]a de Montessus de Bollore type theorem.In this paper,we will discuss the zero set of a real multivariate polynomial,and present a convergence theorem in measure of multivariate Pade approximant.The proof technique used in this paper is quite different from that used in the univariate case.

Power Flow Analytical Solutions and Multi-dimensional Voltage Stability Boundaries Based on Multivariate Quotient-difference Method

This paper proposes a novel Multivariate Quotient-Difference(MQD)method to obtain the approximate analytical solution for AC power flow equations.Therefore,in the online environment,the power flow solutions covering different operating conditions can be directly obtained by plugging values into multiple symbolic variables,such that the power injections and consumptions of selected buses or areas can be independently adjusted.This method first derives a power flow solution through a Multivariate Power Series(MPS).Next,the MQD method is applied to transform the obtained MPS to a Multivariate Pad´e Approximants(MPA)to expand the Radius of Convergence(ROC),so that the accuracy of the derived analytical solution can be significantly increased.In addition,the hypersurface of the voltage stability boundary can be identified by an analytical formula obtained from the coefficients of MPA.This direct method for power flow solutions and voltage stability boundaries is fast for many online applications,since such analytical solutions can be derived offline and evaluated online by only plugging values into the symbolic variables according to the actual operating conditions.The proposed method is validated in detail on New England 39-bus and IEEE 118-bus systems with independent load variations in multi-regions.

Decomposition and Approximation of Multivariate Functions on the Cube

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jФjψj, where each Фj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each Фjψj is a product of separated variable type and its smoothness is same as f. Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.

THE RELATION BETWEEN nTH MINIMAL ERRORS OF TWO CLASSES OF INFORMATION FOR APPROXIMATION IN HOLDER SPACE

Let be Holder space and G = L2([0, 1]d) with the inner product given byThis paper considers the embedding operator S : H →G,S(f) = f, f ∈ H . We prove thatwhere en(S,Astd ) and en(S, Aall ) denote the nth minimal error of standard and linear information respectively in the worst case, average case and randomized settings, and C is a constant.