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   

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 of periodic functions by Fejer-Zygmund means in various metrics

Translated from Matematicheskie Zametki, Vol. 48, No. 4, pp. 48–57, October, 1990.     

On approximation of convex functions by rational ones in integral metrics

Пустьf - действительн означная конечная фу нкция на конечном отрезке Δ=[а, b] вещественной оси, |Δ|=b?a, M(f) = sup {|f(x)|: x∈Δ}, R_n(f,p Δ) = inf∥f?r∥_Lp(Δ) (0 < p < ∞), где нижняя грань бере тся по всем рациональ ным функциямr порядка не вышеп, K(М, Δ) класс всех выпуклых на отре зке Δ функцийf, для кот орыхM(f)≦M. Теорема.При любом вещ ественном р, 0<р<∞ и вс ехп=1, 2, ... sup {R_n(f, p, Δ):f∈K(M, Δ)} ≦ C(p)M|Δ|^1/pn^?2,где С(р) - величина, зави сящая лишь от р.     

Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics

We obtain asymptotic equalities for the least upper bounds of approximations of classes of Poisson integrals of periodic functions by a linear approximation method of special form in the metrics of the spaces C and L _p . Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 976–982, July, 2008.     

Some estimates of derivatives of rational functions in integral metrics

Estimates are established in the metric of L_q[0p. Local norms in L_p are also estimated of Taylor remainders of a rational function in terms of its norm in L_p.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 499–510, April, 1973.In conclusion, the author expresses thanks to E. P. Dolzhenko, under whose guidance this work was done.     

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.     

Approximation by Bernstein type rational functions

Acta Mathematica Hungarica -     

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 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.     

Computation of uniform integral means and some similar characteristics of piecewise continuous functions

We obtain formulas for the computation of a certain asymptotic characteristic of piecewise continuous functions defined on the half-line and use it to state the well-known necessary and sufficient condition on a linear homogeneous differential system for the corresponding inhomogeneous system with arbitrary inhomogeneity whose characteristic exponent is nonpositive to have a solution with nonpositive characteristic exponent. We give a new form of this necessary and sufficient condition similar to the Perron-Maizel condition for the exponential dichotomy of the system.     

Relative widths of Sobolev classes in the uniform and integral metrics

Let W_p^r be the Sobolev class consisting of 2π-periodic functions f such that ‖f^(r)‖_p ≤ 1. We consider the relative widths d_n(W_p^r, MW_p^r, L_p), which characterize the best approximation of the class W_p^r in the space L_p by linear subspaces for which (in contrast to Kolmogorov widths) it is additionally required that the approximating functions g should lie in MW_p^r, i.e., ‖g^(r)‖_p ≤ M. We establish estimates for the relative widths in the cases of p = 1 and p = ∞; it follows from these estimates that for almost optimal (with error at most Cn^?r, where C is an absolute constant) approximations of the class W_p^r by linear 2n-dimensional spaces, the norms of the rth derivatives of some approximating functions are not less than cln min(n, r) for large n and r.     

Approximation by means of piecewise linear functions

In the present paper we use piecewise linear functions in order to obtain representations and estimates for the remainder in approximating continuous functions by positive linear operators. Applications of these results for Bernstein and Stancu’s operators are also presented. In addition, we give some partial results concerning the best constant problem for Bernstein operators with respect to the second order modulus of continuity.