On an approximation family of discrete polynomial operators

We present a new family of linear discrete polynomial operators giving a Timan type approximation theorem for functions of arbitrary smoothness. Using this we construct two families of operators of this kind to extend Freud type approximation results to functions of higher smoothness.     

On the order of an approximation of functions on sets of positive measure by linear positive polynomial operators

It is proved that at almost all points the order of approximation, even of one of the functions 1, cos x,sin x by means of a sequence of linear positive polynomial operators having uniformly bounded norms, is not higher than 1/n². Refinements of this result are given for operators of convolution type.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 457–468, March, 1973.In conclusion the author expresses thanks to P. P. Korovkin for posing the problem.     

On the approximation of continuous nonlinear operators in normed spaces by polynomial operators

Zusammenfassung Für eine sehr allgemeine Klasse von Funktionen, die eine kompakte TeilmengeX eines BanachraumesE stetig inE abbilden, wird gezeigt, dass sich jede dieser Funktionen aufX beliebig genau gleichmässig durch entartete Polynomoperatoren approximieren lässt.     

Best approximation by polynomial operators in banach spaces

The problem of approximation in the space of bounded linear operators ? (E;G) between normed spaces E and G by compact operators has been extensively studied in the last few years.

Recently Deutsch, Mach and Saatkamp ([2]) have considered the problem of approximating elements of ?(E;G) by the subset K _N(E;G) of operators whose range is at most N dimensional. We consider in this paper the problem of approximating operators (not necessarily linear) beteen normed spaces E and G by continuous homogeneous polynomials, and in particular by such polynomials which have finite-dimensional range.     

On certain positive linear operators in polynomial weight spaces

We consider certain modified Szász-Mirakyan operators A _n (f;r) in polynomial weight spaces of functions of one variable and we study approximation properties of these operators. This revised version was published online in June 2006 with corrections to the Cover Date.     

Statistical approximation theorems by k-positive linear operators

In this paper, using A-statistical convergence we obtain various approximation theorems by means of k-positive linear operators defined on the space of all analytic functions on the unit disk. Received: 17 February 2005     

Order of approximation by linear combinations of positive linear operators

Order of uniform approximation is studied for linear combinations due to May and Rathore of Baskakov-type operators and recent methods of Pethe. The order of approximation is estimated in terms of a higher-order modulus of continuity of the function being approximated.     

On the reducibility of linear differential operators with unbounded operator coefficients

Results on the reducibility of linear differential operators with unbounded operator coefficients to differential operators with a simpler structure are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 587–595, May, 1993.     

On the application of linear positive operators for approximation of functions

For the linear positive Korovkin operator \(f\left( x \right) \to {t_n}\left( {f;x} \right) = \frac{1}{\pi }\int_{ - \pi }^\pi {f\left( {x + t} \right)E\left( t \right)dt} \), where E(x) is the Egervary–Szász polynomial and the corresponding interpolation mean \({t_{n,N}}\left( {f;x} \right) = \frac{1}{N}\sum\limits_{k = - N}^{N - 1} {{E_n}\left( {x - \frac{{\pi k}}{N}} \right)f\left( {\frac{{\pi k}}{N}} \right)} \), the Jackson-type inequalities \(\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant \left( {1 + \pi } \right){\omega _f}\left( {\frac{1}{n}} \right),\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant 2{\omega _f}\left( {\frac{\pi }{{n + 1}}} \right)\), where ωf (x) denotes the modulus of continuity, are proved for N > n/2. For ωf (x) ≤ Mx, the inequality \(\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant \frac{{\pi M}}{{n + 1}}\). is established. As a consequence, an elementary derivation of an asymptotically sharp estimate of the Kolmogorov width of a compact set of functions satisfying the Lipschitz condition is obtained.     

On the approximation of spectra of linear operators on Hilbert spaces

We present several new techniques for approximating spectra of linear operators (not necessarily bounded) on an infinite-dimensional, separable Hilbert space. Our approach is to take well-known techniques from finite-dimensional matrix analysis and show how they can be generalized to an infinite-dimensional setting to provide approximations of spectra of elements in a large class of operators. We conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n-pseudospectrum and argue how that can be used as an approximation to the spectrum.     

The linear factorization of polynomial operator pencils

Sufficient conditions are found for the linear factorization of polynomial operator pencils of arbitrary order in a Banach space. This factorization is generated by the solution of an appropriate operator equation.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 551–559, April, 1973.The author wishes to express his thanks to A. G. Kostyuchenko for the formulation of the problem.