Convexity results and sharp error estimates in approximate multivariate integration

On the approximation of strongly convex functions by an upper or lower operator

The aim of this paper is to find a convenient and practical method to approximate a given real-valued function of multiple variables by linear operators, which approximate all strongly convex functions from above (or from below). Our main contribution is to use this additional knowledge to derive sharp error estimates for continuously differentiable functions with Lipschitz continuous gradients...

Approximate Convexity and Submonotonicity in Locally Convex Spaces

Sharp Error Estimates for Interpolatory Approximation on Convex Polytopes

Let P be a convex polytope in the d-dimensional Euclidean space. We consider an interpolation of a function f at the vertices of P and compare it with the interpolation of f and its derivative at a fixed point y ∈ P. The two methods may be seen as multivariate analogues of an interpolation by secants and tangents, respectively. For twice continuously differentiable functions, we establish sharp...

Some Results on Convexity and Concavity of Multivariate Copulas

This paper provides some results on different types of convexity and concavity in the class of multivariate copulas. We also study their properties and provide several examples to illustrate our results.

A Sharp Sufficient Condition for Sparsity Pattern Recovery

Sufficient number of linear and noisy measurements for exact and approximate sparsity pattern/support set recovery in the high dimensional setting is derived. Although this problem as been addressed in the recent literature, there is still considerable gaps between those results and the exact limits of the perfect support set recovery. To reduce this gap, in this paper, the sufficient con...