Approximation of differentiation operator in the space L2 on semiaxis

We establish an upper bound for the error of the best approximation of the first order differentiation operator by linear bounded operators on the set of twice differentiable functions in the space L2 on the half-line. This upper bound is close to a known lower bound and improves the previously known upper bound due to E. E. Berdysheva. We use a specific operator that is introduced and studied in the paper. © Allerton Press, Inc., 2013

On the Best Approximation of the Differentiation Operator

In this paper we give a solution of the problem of the best approximation in the uniform norm of the differentiation operator of order k by bounded linear operators in the class of functions with the property that the Fourier transforms of their derivatives of order n (0 < k <n) are finite measures. We also determine the exact value of the best constant in the corresponding inequality for derivatives

О сопряженности пространства мультипликаторов

A. Figà Talamanca proved (1965) that the space Mr = Mr(G) of bounded linear operators in the space Lr, 1 ≤ r ≤ ∞, on a locally compact group G that are translation invariant (more exactly, invariant under the group operation) is the conjugate space for a space Ar = Ar(G), which he described constructively. In the present paper, for the space Mr = Mr(Rm) of multipliers of the Lebesgue space Lr(Rm), 1 ≤ r &lt; ∞, we present a Banach function space Fr = Fr(Rm) with two properties. The space Mr is conjugate to Fr: Fr ∗ = Mr; actually, it is proved that Fr coincides with Ar = Ar(Rm). The space Fr is described in different terms as compared to Ar. This space appeared and has been used by the author since 1975 in the studies of Stechkin's problem on the best approximation of differentiation operators by bounded linear operators in the spaces Lγ(Rm), 1 ≤ γ ≤ ∞. © 2019 Krasovskii Institute of Mathematics and Mechanics. All right reserved

ON THE BEST APPROXIMATION OF THE DIFFERENTIATION OPERATOR

In this paper we give a solution of the problem of the best approximation in the uniform norm of the differentiation operator of order k by bounded linear operators in the class of functions with the property that the Fourier transforms of their derivatives of order n (0 < k <n) are finite measures. We also determine the exact value of the best constant in the corresponding inequality for derivatives

A CHARACTERIZATION OF EXTREMAL ELEMENTS IN SOME LINEAR PROBLEMS

We give a characterization of elements of a subspace of a complex Banach space with the property that the norm of a bounded linear functional on the subspace is attained at those elements. In particular, we discuss properties of polynomials that are extremal in sharp pointwise Nikol'skii inequalities for algebraic polynomials in a weighted $L_q$-space on a finite or infinite interval

APPROXIMATION OF DIFFERENTIATION OPERATORS BY BOUNDED LINEAR OPERATORS IN LEBESGUE SPACES ON THE AXIS AND RELATED PROBLEMS IN THE SPACES OF (p,q)(p,q)-MULTIPLIERS AND THEIR PREDUAL SPACES

We consider a variant $E_{n,k}(N;r,r;p,p)$ of the four-parameter Stechkin  problem $E_{n,k}(N;r,s;p,q)$ on the best approximation of differentiation operators of order $k$ on the class of $n$ times differentiable functions $(0<k<n)$ in Lebesgue spaces on the real axis. We discuss the state of research in this problem and related problems in the spaces of multipliers of Lebesgue spaces and their predual spaces. We give two-sided estimates for $E_{n,k}(N;r,r;p,p)$. The paper is based on the author's talk at the S.B. Stechkin's International Workshop-Conference on Function Theory (Kyshtym, Chelyabinsk region, August 1–10, 2023)

ON THE BEST APPROXIMATION OF THE DIFFERENTIATION OPERATOR

In this paper we give a solution of the problem of the best approximation in the uniform norm of the differentiation operator of order k by bounded linear operators in the class of functions with the property that the Fourier transforms of their derivatives of order n (0 < k <n) are finite measures. We also determine the exact value of the best constant in the corresponding inequality for derivatives