Approximation in the mean by rational functions

In this paper we proved that for a large class of compact subsetsK in the complex plane,R(K) is dense inL ^q () if and only if the set of analytic bounded point evaluations forR ^q (K, ) is empty. As a consequence, we showed that this result is true for allK ifR(K) is replaced byA(K). Our main result includes the corresponding result of James Thomson for polynomials approximation as such a special case thatK is a disk.     

Composite mixed finite elements on plane quadrilaterals

Mixed finite elements over a plane convex quadrilateral are obtained by assembling two Raviart-Thomas mixed finite elements over triangles. The macroelement is given by an eliminating procedure of the degrees of freedom related to the common edge to the two triangles. This procedure results in a finite element with a space of interpolating functions containing the polynomials of degree ? l, where l is the greater integer for which the same property is satisfied by the relevant Raviart-Thomas [Mathematical Aspects of Finite Element Methods, Roma 1975, I. Galligani and E. Magenes, Eds., Lecture Notes in Mathematics Vol. 606, Springer-Verlag, Berlin, 1975] mixed finite element. The interpolation error is estimated by means of the technique of almost equivalent affine element as given by Ciavaldini and Nédélec [Rev. Fr. Autom. Inf. Recher. Opérationnelle Ser. Rouge R2 , 29–45 (1974)]. © 1993 John Wiley & Sons, Inc.     

Approximation by rational functions in complex regions

Approximation by rational functions R_n (z) (in the C and L_p metrics) on plane compacta is investigated. The possibility is studied of the coincidence of rational and polynomial approximations for all n, and some functions are described for which this coincidence holds. Approximations on finite sets of points are investigated, and an explanation is given of why there are functions which cannot be approximated by rational functions of degree not higher than n (in the C metric).Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 121–130, February, 1971.Several questions concerning this work were discussed with the author by A. A. Gonchar. The author wishes to thank A. A. Gonchar for his help.     

Approximation by rational operators in Lp spaces

The present paper introduces a kind of Kantorovich type Shepard operators. Complete results including direct and converse results, equivalence results are established. As Della Vecchia and Mastroianni ([4], [7]) did, our results involve a weighted modulus of smoothness related to step‐functions φ (x) vanishing algebraically at the endpoints ±1 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximation by rational functions in Smirnov-Orlicz classes

Let G be a doubly-connected domain bounded by Dini-smooth curves. In this work, the approximation properties of the Faber-Laurent rational series expansions in Smirnov-Orlicz classes EM(G) are studied.     

Approximation by Bernstein type rational functions

Acta Mathematica Hungarica -     

Lp空间Kantorovich-Vertesi算子逼近的Jackson型估计

考虑了Kantorovich-Vertesi有理插值型算子L^*n,s（f,X,x）对L^p[-1,1]（1≤p≤∞）空间函数逼近的Jackson型估计。并获得了如下逼近阶：‖L^*n,s(f,X,x)-f(x)‖L^p[-1,1]≤Cp,sw(f,1/n 2)L^p[-1,1] （s＞2）。     

Approximation and interpolation by convexity-preserving rational splines

A discussion and algorithm for combined interpolation and approximation by convexity-preserving rational splines is given.     

Approximation of the function ¦x¦ by rational functions

We consider the problem of the approximation of the function ¦x¦ by rational functions. We make more precise the best approximation estimate obtained by A. P. Bulanov. We prove that for arbitrary positive integral n where A is an absolute constant.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 163–171, July, 1974.The author thanks E. P. Dolzhenko for his help in preparing this paper.     

Approximation by rational modules in Sobolev and Lipschitz norms

Let X?C be compact, 0>n∈Z, and g a continuous function on X. Let R(n,g,X) be the rational module consisting of the functions on X of the type r₀ + r₁g + ··· + r_ngⁿ, where r_j is a rational function with poles off X, 0 ? j ? n. It is shown that if X is nowhere dense, g is sufficiently smooth, and $\text{}\text{?}\text{t6g(z) ≠ 0, z ∈ X}$, then the restriction to X of each function in C∈(C) is approximable in the Lip(n ? 1, X)-norm, n ? 2, by functions in R(n, g, X). Also dealt with are approximation problems in Sobolev norms by more general types of rational modules.     

Incomplete rational approximation in the complex plane

We consider rational approximations of the form

Weighted rational approximation in the complex plane

Given a triple (G, W, γ) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ(z), which is analytic in G, by weighted rational functions W^m_i+n_i(z)R_mi, n_i(z)_{i = 0}^∞, where R_mi, n_i(z) = P_mi(z)/Q_ni(z) with deg P_mi ≤ m_i and deg Q_ni ≤ n_i for all i ≥ 0 and where m_i + n_i → ∞ as i → ∞ such that lim m_i/[m_i + n_i] = γ. Our main result is a necessary and sufficient condition for such an approximation to be valid. Applications of the result to various classical weights are also included.     

Approximation of values of the Gauss hypergeometric function by rational fractions

We consider a new approach to estimating the irrationality measure of numbers that are values of the Gauss hypergeometric function. Some of the previous results are improved, in particular, those concerning irrationalities of the form $ \sqrt k $ ln(( $ \sqrt k $ + 1)/( $ \sqrt k $ ? 1)) with k ∈ ?.     

Approximation by rational functions in integral metrics and differentiability in the mean

The paper deals with approximations of a functionf of space L_p[0, 1] by rational functions in the metric of this same space (0n(f, p) of functionf of rational functions of degree no higher than n is evidence of the presence inf of derivatives and differentials of a definite order if differentiation is understood as differentiation in the metric of space L_q[0, 1], with 0相似文献