Relative widths of smooth functions determined by linear differential operator

Let W and V be centrally symmetric sets in a normed space X. The relative Kolmogorov n-width of W relative to V in X is defined by

     

Jackson-type inequalities and widths of function classes in L 2

The sharp Jackson-type inequalities obtained by Taikov in the space L ₂ and containing the best approximation and the modulus of continuity of first order are generalized to moduli of continuity of kth order (k = 2, 3, ... ). We also obtain exact values of the n-widths of the function classes F(k, r, Φ) and F _k ^r (h), which are a generalization of the classes F(1, r, Φ) and F _k ^r (h) studied by Taikov.     

Estimates for the widths of discrete function classes generated by a two-weight summation operator

Order estimates are obtained for the Kolmogorov, linear, and Gel’fand widths of the image of the unit ball of the space l_p under the action of a two-weight summation operator with weights of special kind. Some limit conditions on the parameters defining the weights are considered.     

Best polynomial approximations and the widths of function classes in L 2

Sharp Jackson-Stechkin type inequalities in which the modulus of continuity of mth order of functions is defined via the Steklov function are obtained. For the classes of functions defined by these moduli of continuity, exact values of various n-widths are derived.     

Optimal recovery and average n-k widths for some function classes in L2(R)*

In this paper the optimal recovery of a linear differential operator on some classes of smooth functions and the average n-k widths of these classes in L₂ R are considered. Supported by National Natural Science Foundation of China     

On certain classes of analytic functions defined by Noor integral operator

Integral operator, introduced by Noor, is defined by using convolution. Let f_n(z)=z/(1−z)ⁿ⁺¹, n∈N₀, and let f be analytic in the unit disc E. Then I_nf=f⁽⁻¹⁾_n★f, where f_n★f⁽⁻¹⁾_n=z/(1−z). Using this operator, certain classes of analytic functions, related with the classes of functions with bounded boundary rotation and bounded boundary radius rotation, are defined and studied in detail. Some basic properties, rate of growth of coefficients, and a radius problem are investigated. It is shown that these classes are closed under convolution with convex functions. Most of the results are best possible in some sense.     

Generalized resolvents of linear relations generated by a nonnegative operator function and a differential elliptic-type expression

We study invertible extensions of the minimal relation generated by a nonnegative operator function and a differential elliptic-type expression. We prove that the operators inverse to such extensions are integral operators and describe such integral operators. We obtain a formula for generalized resolvents of the minimal relation.     

Relative controllability of delay differential systems with impulses and linear parts defined by permutable matrices

This paper investigates the relative controllability of delay differential systems with linear impulses and linear parts defined by permutable matrices. We use the impulsive delay Grammian matrix to discuss the relatively controllability of impulsive linear delay controlled systems and we use the Krasnoselskii's fixed point theorem to discuss the relatively controllability of impulsive semilinear delay controlled systems. Finally, two examples are presented to illustrate our theoretical results.     

由分数次导数所确定的L2(T)中的一元周期函数类在Lq(T)中的相对宽度

作者研究了相对宽度K_n(W₂^α(T), MW₂^β(T), L₂(T)), T=[0,2π], 确定了使等式K_n(W₂^α(T), MW₂^β(T), L₂(T))=d_n(W₂^α(T), L₂(T))成立的最小M值, 得到了相对宽度K_n(W₂^α(T), W₂^α(T), L_q(T))的渐近阶, 其中α≥β>0, 1≤q≤∞, K_n(., ., L_q(T)) 和 d_n(., L_q(T))分别表示Kolmogorov意义下L_q(T)尺度下的相对宽度和宽度, MW_p^α(T), 1≤ p≤∞, 表示有如下卷积表达式的2π 周期函数类, f(t)=c+(B_α* g)(t),c∈ R, B_α*g 表示 B_α 和g 的卷积, g∈L_p(T) 满足∫₀^2πg(τ)dτ=0 和||g||_p≤M, B_α∈ L₁(T) 有如下Fourier展开: B_α(t)=1/2π∑' _{k∈ Z}(ik)^-αe^ikt,∑^'表示去掉 k=0的项.     

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.     

Some classes of functions of a linear closed operator

A linear closed densely defined operator and some domain Ω lying in the regular set of the operator and containing the negative real semiaxis of the real line are specified in a Banach space. We assume that power estimates for the norm of the resolvent operator are known at zero and infinity. We use the Cauchy integral formula to introduce operator functions generated by scalar functions that are analytic in a certain domain not containing the origin and containing the complement of Ω and satisfy power estimates for their absolute values at zero and infinity. We study some properties of operator functions, which were studied by the authors earlier for the case of an operator whose inverse is bounded; in particular, we study the multiplicative property.     

Jackson-Stechkin type inequalities for special moduli of continuity and widths of function classes in the space L 2

We obtain sharp Jackson-Stechkin type inequalities for moduli of continuity of kth order Ω _k in which, instead of the shift operator T _h f, the Steklov operator S _h (f) is used. Similar smoothness characteristic of functions were studied earlier in papers of Abilov, Abilova, Kokilashvili, and others. For classes of functions defined by these characteristics, we calculate the exact values of certain n-widths.     

The best L 1-approximations of classes of functions defined by differential operators in terms of generalized splines from these classes

For classes of periodic functions defined by constraints imposed on the L ₁-norm of the result of action of differential operators with constant coefficients and real spectrum on these functions, we determine the exact values of the best L ₁-approximations by generalized splines from the classes considered. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1443–1451, November, 1998.     

Kolmogorov and linear widths of weighted Sobolev-type classes on a finite interval

Let I be a finite interval, r N and p(t)=dist{t,I}, tI. Denote by W ^r _{p ,}, 0<<, the class of functions x on I with the seminorm x ^(r) p _Lp1. We obtain two-sided estimates of the Kolmogorov widths d _n(W^r _p, )_Lq and of the linear widths d _n(W^r _p,)_Lq ^lin