Thomas Kunkle

Lagrange interpolation on a lattice: bounding derivatives by divided differences

Abstract Fix an integer n > 0. For a multivariate function defined on a (not necessarily rectangular) lattice, an extension is constructed to have, ∀ k ⩽ n , derivatives of total degree k that are bounded by the functions tensor product divided differences of total degree k times a constant independent of the lattice and the function. The extension is locally constructed, can have any prescribed smoothness, and reproduces polynomials of degree n in each variable.

Using quasi-interpolants in a result of Favard

A proof of Favard can be restructured using quasi-interpolants of the type discussed in these proceedings [6] and his result strengthened.

Flat extension and ideal projection

Abstract A generalization of the flat extension theorems of Curto and Fialkow and Laurent and Mourrain is obtained by seeing the problem as one of ideal projection.

Rearrangements of Conditionally Integrable Functions

From an s-variate continuous function whose positive and negative parts both have infinite Lebesgue integral, we construct a continuous rearrangement whose generalized Riemann integral equals any prescribed value. The original function is assumed to be finite a.e., with domain an interval in Rs. A similar result is obtained for functions defined on a ball.