Hankel operators and best Hankel approximation on the half-plane

The objective of this paper is to give an interrelation between Hankel oeprators on the unit disc and Hankel operators on the half-plane. As an application, the AAK result on the half-plane is established and the rate of best Hankel approximation on the halfplane is derived. Research partially supported by the University Research Grants and Fellowship Committee at UNLV.     

Hankel operators with unbounded symbols

We prove that there are holomorphic functions in the Hardy space of the unit ball or the bidisc such that the big Hankel operator with symbol is bounded and for any holomorphic function the function cannot be bounded.

     

Rates for approximation of unbounded functions by positive linear operators

Let g_a(t) and g_b(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {L_n} be a sequence of linear positive operators, each with domain containing 1, t, g_a(t), and g_b(t). If L_n(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, g_a(t), g_b(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(g_a(t)) (t → a⁺) and ƒ(t) = O(g_b(t)) (t → b⁻). We estimate the convergence rate of L_nƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′.     

Ill-posed problems with unbounded operators

Let A be a linear, closed, densely defined unbounded operator in a Hilbert space. Assume that A is not boundedly invertible. If Eq. (1) Au=f is solvable, and ‖fδ−f‖?δ, then the following results are provided: Problem Fδ(u):=‖Au−fδ^‖2+α‖u^‖2 has a unique global minimizer uα,δ for any fδ, uα,δ=A^*−1(AA^*+αI)fδ. There is a function α=α(δ), limδ→0α(δ)=0 such that limδ→0‖uα(δ),δ−y‖=0, where y is the unique minimal-norm solution to (1). A priori and a posteriori choices of α(δ) are given. Dynamical Systems Method (DSM) is justified for Eq. (1).     

Estimates of best bilinear approximations of functions and approximation numbers of integral operators

Translated from Matematicheskie Zametki, Vol. 51, No. 5, pp. 125–134, May, 1992.     

A generalization of best approximation operators

Summary Let X be a rotund, reflexive Banach space, M⊆X a closed subspace and P_M: X→M the best approximation operator on M. This paper deals with some aspects of the general question of how P_M may be constructed out of simpler operators. A class of operators called B-operators, which generalize best approximation operators, is defined and conditions are given under which P_M is the limit of sequences of B-operators. Several examples and applications are also included. Entrata in Redazione il 20 novembre 1974.     

Hankel convolution operators on entire functions and distributions

In this paper we study the Hankel convolution operators on the space of even and entire functions and on Schwartz distribution spaces. We characterize the Hankel convolution operators as those ones that commute with Hankel translations and with a Bessel operator. Also we prove that the Hankel convolution operators are hypercyclic and chaotic on the spaces under consideration.     

Hankel operators and extremal problems in H1

Integral Equations and Operator Theory -     

Regularization of ill-posed problems with unbounded operators

Variational regularization and the method of quasisolutions are justified for unbounded closed operators.     

Perturbations of best approximation problems

Summary The purpose of this paper is to unify a number of continuity, perturbation, and discretization properties of best approximations in normed linear spaces or in metric or semimetric spaces. It extends work of Aubin and, more specifically, of Daniel.Research supported in part by AEC AT(45-1)-1947     

Doubly nonlinear periodic problems with unbounded operators

The solvability of the evolution system v′(t)+B(t)u(t)∋f(t), v(t)∈A(t)u(t), 0<t<T, with the periodic condition v(0)=v(T) is investigated in the case where are bounded, possibly degenerate, subdifferentials and are unbounded subdifferentials.     

Spectral propeties of functions of commuting unbounded operators

We give generalizations of the joint spectral mapping theorem for certain classes of generally non-closed commuting operators on different complex Hilbert spaces. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 33–35.     

Computing the resolvent of the sum of operators with application to best approximation problems

Optimization Letters - We propose a flexible approach for computing the resolvent of the sum of weakly monotone operators in real Hilbert spaces. This relies on splitting methods where strong...