Operator approach to direct and inverse theorems in the approximation theory of functions
by
Valentyna I. Gorbachuk
Institute of Mathematics, Ukrainian National Academy of Sciences

Let A be a closed linear operator on a Banach space X with norm |.|. For a number a > 0, we denote by C(a, A) the set of all elements x from X satisfying the inequality

|Anx| < can, n = 0, 1, 2, ...

with some constant c depending on x. The linear set C(a, A) is a Banach space with the norm which is the least constant c in the above inequality. Let Exp A be the union of all spaces C(a, A), a > 0. The space ExpA endowed with the inductive limit topology of the spaces C(a, A) is called the space of vectors of exponential type of the operator A. By the type t(x) of a vector x from ExpA we mean the infimum of those a for which x belongs to C(a, A). For an arbitrary vector y from X, we put

E(r, y) = inf |x - y|

where the infimum is taken over the set of all x from ExpA whose type does not exceed r.

In this talk, a general (operator) approach to obtain direct and inverse theorems of the approximation theory of functions is presented. This approach consists in the following. A certain self-adjoint operator on the appropriate Hilbert space is associated with a concrete approximation problem. Vectors of exponential type of this operator (as a rule, they coincide with algebraic or trigonometric polynomials, or entire functions of exponential type) are the vectors that are used for the approximation of various classes of smooth functions belonging to the functional Banach space where the approximation problem is considered. The value E(r, y) is the best approximation. We concentrate also on the further development of this operator approach and its application to the sharp estimation of the approximation error in the variational and power series methods for operator and differential equations in a Banach space.

(T)

Date received: November 26, 1999