The best approximation of the differentiation operator in the metric of Lp

For Stechkin's problem of the best approximation for the differentiation operator we indicate the necessary and sufficient conditions that E_n be finite. We study some properties of continuous linear operators V from L_p into L_q.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 531–538, November, 1972.     

On the best uniform approximation of differentiation operators

We study the existence and uniqueness of an extremal operator for the problem of S. B. Stechkin on the best approximation of differentiation operators by bounded operators.Translated from Matematicheski Zametki, Vol. 5, No. 3, pp. 273–284, March, 1969.     

On the modulus of continuity of an operator of best approximation

In this paper we characterize spaces with an operator of best approximation uniformly continuous on a class of subspaces.     

The strong derivative of the best approximation operator

ABSTRACT

This article is devoted to the derivation of sampling series associated with eigenvalue problems. No examples of sampling theorems associated with odd order problems are known except when the order is one. Here we give necessary and sufficient conditions for the boundary conditions to be self adjoint. Then sampling series associated with odd order self adjoint and non self adjoint problems are given. The sampling representations associated with non self adjoint odd order problems are derived provided that the boundary condition are regular.     

Lipschitz continuity of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite dimensional Haar subspace of C(X,R^k), the vector-valued continuous functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.     

On approximation of functions on the half-line by algebraic polynomials

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An operator onL pwithout best compact approximation

We construct, for 1<p<∞,p ≠ 2, an operator onL ^pwhose distance to the space of compact operators onL ^pis not attained. We also show that the identity operator onL ^p,p ≠ 1,2, ∞ has a unique best compact approximation. Research partially supported by NSF grant DMS-8201635.     

Positive solutions of operator equations on half-line

This paper is concerned with an operator equation on half-line. Some theorems for the existence of its positive solutions are obtained by using the Krasnosel'skii-Guo theorem on cone expansion and compression in a special function space.     

Lipschitz continuity and Gateaux differentiability of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite-dimensional Haar subspace of C(X,R^k), the vector-valued functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1 and has a Gateaux derivative on a dense set of functions in C(X,R^k).     

Differential properties of the operator of best approximation by rational fractions

Translated from Matematicheskii Zametki, Vol. 45, No. 2, pp. 40–50, February, 1989.     

On a generalization of the extended best polynomial approximation operator in Orlicz–Lorentz spaces

In this paper, we consider the best polynomial approximation operator defined on an Orlicz–Lorentz space ${\mathrm{\Lambda }}_{w,\varphi }$, and its extension to ${\mathrm{\Lambda }}_{w,{\varphi }^{\prime }}$, where w is a non-negative continuous weight function and ${\varphi }^{\prime }$ is the derivative of ϕ, which is not required to be an Orlicz function. Our work generalizes a recent result in this field on an Orlicz–Lorentz space generated by an Orlicz function. In addition, we establish some properties and estimates for any extended best polynomial approximation.     

Common fixed-points for Banach operator pairs in best approximation

The notion of Banach operator pairs is introduced, as a new class of noncommuting maps. Some common fixed-point theorems for Banach operator pairs and the existence of the common fixed-points of best approximation are presented. These results are proved without the assumption of linearity or affinity for either f or g, which shows that the concept about Banach operator pairs is potentially useful in the study of common fixed-points.     

On the modulus of continuity of the operator of best approximation in the space of continuous functions

In the paper we establish estimates for the strong and the weak moduli of continuity of the operator of best approximation in the space of continuous functions.Translated from Matematicheskie Zametki, Vol. 10, No. 6, pp. 601–613, December, 1971.I wish to express my thanks to S. B. Stechkin for suggesting the problem and for his constant interest in the paper.     

On extensions of the Bessel operator on a finite interval and a half-line

The minimum and maximum operators generated by a Bessel differential expression are studied. The boundary triplets for the maximum operator A _??,max are constructed, and the corresponding Weyl functions are found. All non-negative self-adjoint extensions of the minimum operator A _??,min are described.