In this short paper, we summarize (without proofs) the constructive method to approximate functions in the uniform (i.e. maximal error) norm, that was recently developed by the authors. [9] This is in contrast to other methods (e.g. back-propagation) that approximate only in the average error norm. We comment on the novelty of the approach and possible extensions. The method includes a ~gradien...

This paper studies inference on nondifferentiable functions using methods based on numerical differentiation. First we show that for an appropriately chosen sequence of step sizes, numerical derivative based delta methods provide consistent inference for functions of parameters that are directionally differentiable. Examples of directionally differentiable functions arise in a variety of econom...

We propose to extend Talenti’s estimates on the L norm of the second order derivatives of the solutions of a uniformly elliptic PDE with measurable coefficients satisfying the Cordes condition to the non-uniformly elliptic case.

We analyze a new Galerkin finite element method for numerically solving a linear convection-dominated convection-diffusion problem in two dimensions. The method is shown to be convergent, uniformly in the perturbation i ii parameter, of order h ' in a global energy norm which is stronger than the L norm. This order is optimal in this norm for our choice of trial functions.

This paper deals with the number of meaningful compositions of higher order of differential operations and the Gateaux directional derivative.

Integral inequalities of Ostrowski type are developed for n−times differentiable mappings, with multiple branches, on the L∞ norm. Some particular inequalities are also investigated, which include explicit bounds for perturbed trapezoid, midpoint, Simpson’s, NewtonCotes and left and right rectangle rules. The results obtained provide sharper bounds than those obtained by Dragomir [5] and Cerone...