Kergin interpolation in Banach spaces

Abstract

We show that Kergin interpolation, a generalized Lagrange-Hermite polynomial interpolation, may be defined on mappings between general Banach spaces. Like its finitedimensional counterpart, Kergin interpolation in this setting is an affine-invariant projector. We obtain an error formula which we use to approximate holomorphic mappings. As an application we give a convergence theorem applicable to, for instance, operators on Banach algebras, such as the algebra of square matrices with complex coefficients. r

On interpolation approximation of differentiable operators in a Hilbert space

“…In the present paper, we continue the investigations carried out in [1][2][3][4][5][6][7][8] and devoted to the construction of interpolation operator approximations in Hilbert spaces and to the examination of their accuracy. The problems of the construction of an operator polynomial of the Lagrange type on a specially selected set of nodes L(m) and of the analysis of the accuracy of interpolation of polynomial and entire operators were considered earlier in [4].…”

Lagrange approximation in Banach spaces

“…Petersson [12], Filipsson [6] and Simon [14] have extended Kergin interpolation and approximation to the Banach space setting. Hence, the results obtained in this paper are not entirely new.…”

Set-valued Hermite interpolation

“…Some of the results achieved in [36,14,34,17,39] for polynomial interpolation in Banach spaces can be applied, since − → D n is itself a Banach space. These pioneering works were aimed at more theoretical results, whereas here we focus on the numerical analysis; in fact, error estimates are not provided in [36] or demand in [34,17,39] too much regularity. We shall point out that, although the regularity assumptions in [14] are rather weak, the conditions (i), (ii), H1 in [14] for deriving error estimates (in − → D n ) still demand research.…”