APPROXIMATION ORDER AND INTERPOLATION SPACES

In this paper we introduce the two-parameter operators on Abelian group and establish their interpolation theorems of approximation, which are extensions of the interpolation theorems for nonlinear best approximation by R. Devore and are suitable for the approximation of oprators.

The Simultaneous Approximation Order by Hermite Interpolation in a Smooth Domain

Let D be a smooth domain in the complex plane. In D consider the simultaneous ap- proximation to a function and its ith (0≤i≤q) derivatives by Hermite interpolation. The orders of uniform approximation and approximation in the mean, are obtained under some domain boundary conditions. Some known results are included as particular cases of the theorems of this paper.

Two-direction refinable functions and twodirection wavelets with high approximation order and regularity

The concept of two-direction refinable functions and two-direction wavelets is introduced.We investigate the existence of distributional(or L2-stable) solutions of the two-direction refinement equation: φ(x)=∑p+kφ(mx-k)+∑p-kφ(k-mx) where m ≥ 2 is an integer. Based on the positive mask {pk+} and negative mask {p-k}, the conditions that guarantee the above equation has compactly distributional solutions or L2-stable solutions are established. Furthermore, the condition that the L2-stable solution of the above equation can generate a two-direction MRA is given. The support interval of φ(x) is discussed amply. The definition of orthogonal two-direction refinable function and orthogonal two-direction wavelets is presented, and the orthogonality criteria for two-direction refinable functions are established. An algorithm for constructing orthogonal two-direction refinable functions and their two-direction wavelets is presented. Another construction algorithm for two-direction L2-refinable functions, which have nonnegative symbol masks and possess high approximation order and regularity, is presented. Finally, two construction examples are given.

Construction of multiwavelets with high approximation order and symmetry

In this paper, based on existing symmetric multiwavelets, we give an explicit algorithm for constructing multiwavelets with high approximation order and symmetry. Concretely, suppose Φ(x):= (φ1(x), ..., φr(x)) T is a symmetric refinable function vectors with approximation order m. For an arbitrary nonnegative integer n, a new symmetric refinable function vector Φnew(x):= (φ 1 new (x), ..., φ r new (x)) T with approximation order m + n can be constructed through the algorithm mentioned above. Additionally, we reveal the relation between Φ(x) and Φnew(x). To embody our results, we construct a symmetric refinable function vector with approximation order 6 from Hermite cubics which provides approximation order 4.

The Order of Approximation by Complex Rational Type Interpolating Operators at Fejer's Points

In this paper, supose Γ be a boundary of a Jordan domain D and Γ satisfied Альпер condition, the order that rational type interpolating operators at Fejer's points of f(z)∈C(Γ) converge to f(z) in the sense of uniformly convergence is obtained.

ON THE ORDER OF APPROXIMATION BY PERIODIC NEURAL NETWORKS BASED ON SCATTERED NODES

The relationship between the order of approximation by neural network based on scattered threshold value nodes and the neurons involved in a single hidden layer is investigated. The results obtained show that the degree of approximation by the periodic neural network with one hidden layer and scattered threshold value nodes is increased with the increase of the number of neurons hid in hidden layer and the smoothness of excitation function.

ORDER OF MEAN APPROXIMATION BY MIXED QUASI HERMITE-FEJER INTERPOLATION

In this paper we introduce a new kind of the mixed Hermite--Fejér interpolation with boundary condi- tions and obtain the mean approximation order.Our results include a new theorem of Varma and Prasad.Be- sides,we also get some other results about the mean approximation.

Approximation to Continuous Functions by a Kind of Interpolation Polynomials

In this paper, an interpolation polynomial operator F n(f; l,x) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈C b [-1,1] (0≤b≤l) F n(f; l,x) converges to f(x) uniformly, where l is an odd number.

Rational Approximation to |<i>x</i>| at Logarithmic Nodes

Based on a node group , the Newman type rational operator is constructed in the paper. The convergence rate of approximation to a class of non-smooth functions is discussed, which is regarding to X. Moreover, if the operator is constructed based on further subdivision nodes, the convergence rate is . The result in this paper is superior to the approximation results based on equidistant nodes, Chebyshev nodes of the first kind and Chebyshev nodes of the second kind.

APPROXIMATION BY COMPLEX SZSZ-DURRMEYER OPERATORS IN COMPACT DISKS

In the present paper, we deal with the complex Szasz-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks. Also, the exact order of approximation is found.

The essential order of approximation for neural networks

There have been various studies on approximation ability of feedforward neural networks (FNNs). Most of the existing studies are, however, only concerned with density or upper bound estimation on how a multivariate function can be approximated by an FNN, and consequently, the essential approximation ability of an FNN cannot be revealed. In this paper, by establishing both upper and lower bound estimations on approximation order, the essential approximation ability (namely, the essential approximation order) of a class of FNNs is clarified in terms of the modulus of smoothness of functions to be approximated. The involved FNNs can not only approximate any continuous or integrable functions defined on a compact set arbitrarily well, but also provide an explicit lower bound on the number of hidden units required. By making use of multivariate approximation tools, it is shown that when the functions to be approximated are Lipschitzian with order up to 2, the approximation speed of the FNNs is uniquely determined by modulus of smoothness of the functions.

A CLASS OF COMPACTLY SUPPORTED ORTHOGONAL SYMMETRIC COMPLEX WAVELETS WITH DILATION FACTOR 3

When approximation order is an odd positive integer, a simple method is given to construct compactly supported orthogonal symmetric complex scaling function with dilation factor 3. Two corresponding orthogonal wavelets, one is symmetric and the other is antisymmetric about origin, are constructed explicitly. Additionally, when approximation order is an even integer 2, we also give a method to construct compactly supported orthogonal symmetric complex that illustrate the corresponding results. wavelets. In the end, there are several examples

M_n(~R) (a_n, f, x) Apporximation in L_p[0, l]

Generalized Bernstein-Kantorovich polynomials M_n^((k))(a_n, f, x) were introduced in the paper and their order of approximation were estimated in the L_p[0, 1]-spaces.

Subdivision of Uniform ωB-Spline Curves and Two Proofs of Its C^(k-2)-Continuity

ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB-spline curves of arbitrary order ofωB-spline curves and prove its C^k?2-continuity by two kinds of methods.The first method directly prove that the sequence of control polygons of subdivision of order k converges to a C^k?2-continuousωB-spline curve of order k.The second one is based on the theories upon subdivision masks and asymptotic equivalence etc.,which is more convenient to be further extended to the case of surface subdivision.And the problem of approximation order of this non-stationary subdivision scheme is also discussed.Then a uniform ωB-spline curve has both perfect mathematical representation and efficient generation method,which will benefit the application ofωB-splines.

Balance of multi-wavelets

The discrete scalar data need prefiltering when transformed by discrete multi-wavelet, but prefiltering will make some properties of multi-wavelets lost. Balanced multi-wavelets can avoid prefiltering. The sufficient and necessary condition of p-order balance for multi-wavelets in time domain, the interrelation between balance order and approximation order and the sampling property of balanced multi-wavelets are investigated. The algorithms of 1-order and 2-order balancing for multi-wavelets are obtained. The two algorithms both preserve the orthogonal relation between multi-scaling function and multi-wavelets. More importantly, balancing operation doesnt increase the length of filters, which suggests that a relatively short balanced multi-wavelet can be constructed from an existing unbalanced multi-wavelet as short as possible.

Construction of Finite Element Multiscaling Function with Three Coefficients

The aim of this paper is to present construction of finite element multiscaling function with three coefficients. In order to illuminate the result, two examples are given finally.

A Ternary 4-Point Approximating Subdivision Scheme

In this paper, the author presents a class of stationary ternary 4-point approximating symmetrical subdivision algorithm that reproduces cubic polynomials. By these subdivision algorithms at each refinement step, new insertion control points on a finer grid are computed by weighted sums of already existing control points. In the limit of the recursive process, data is defined on a dense set of point, The objective is to find an improved subdivision approximating algorithm which has a smaller support and a higher approximating order. The author chooses a ternary scheme because the best way to get a smaller support is to pass from the binary to ternary or complex algorithm and uses polynomial reproducing propriety to get higher approximation order. Using the cardinal Lagrange polynomials the author has proposed a 4-point approximating ternary subdivision algorithm and found that a higher regularity of limit function does not guarantee a higher approximating order. The proposed 4-point ternary approximation subdivision family algorithms with the mask a have the limit function in C2 and have approximation order 4. Also the author has demonstrated that in this class there is no algorithm whose limit function is in C3. It can be seen that this stationary ternary 4-point approximating symmetrical subdivision algorithm has a lower computational cost than the 6-point binary approximation subdivision algorithm for a greater range of points.

EXTENSION OF CONVEX MODELS AND ITS IMPROVEMENT ON THE APPROXIMATE SOLUTION

In this paper, by means of combining non-probabilistic convex modeling with perturbation theory, an improvement is made on the first order approximate solution in convex models of uncertainties. Convex modeling is extended to largely uncertain and non-convex sets of uncertainties and the combinational convex modeling is developed. The presented method not only extends applications of convex modeling, but also improves its accuracy in uncertain problems and computational efficiency. The numerical example illustrates the efficiency of the proposed method.

COMPLEX RATIONAL TYPE INTERPOLATION AT DISTURBED FEJR POINTS ON A CURVE

The results of accurate order of uniform approximation and simultaneous approximation in the early work ＂Jackson Type Theorems on Complex Curves＂ are improved from Fejer points to disturbed Fejer points in this article. Furthermore, the stability of convergence of Tn,∈（f,z） with disturbed sample values f（z^＊） ＋ Sk are also proved in this article.

The essential order of approximation for nearly exponential type neural networks

For the nearly exponential type of feedforward neural networks （neFNNs）, it is revealed the essential order of their approximation. It is proven that for any continuous function defined on a compact set of Rd, there exists a three-layer neFNNs with fixed number of hidden neurons that attain the essential order. When the function to be approximated belongs to the α-Lipschitz family （0 〈α≤ 2）, the essential order of approxi- mation is shown to be O（n^-α） where n is any integer not less than the reciprocal of the predetermined approximation error. The upper bound and lower bound estimations on approximation precision of the neFNNs are provided. The obtained results not only characterize the intrinsic property of approximation of the neFNNs, but also uncover the implicit relationship between the precision （speed） and the number of hidden neurons of the neFNNs.