Strengthened Cauchy–Schwarz inequality for biorthogonal wavelets in Sobolev spaces

Abstract

We prove a strengthened Cauchy-Schwarz inequality for one-dimensional biorthogonal wavelets. The functional frame is given by a class of Hilbert spaces, defined in terms of weighted Fourier transforms, which contain as relevant examples the standard Sobolev spaces H (s) as well as their homogeneous version. Intended applications concern multilevel and hierarchical methods for numerical approximation of partial differential equations.

Coordinate-wise transformation of probability distributions to achieve a Stein-type identity

“…E.1, we first take into account the measures with a non-zero mean as well as coordinate-wise transformations that are constant over an interval. We then present a lower bound of the quantity β(μ) in (8). Subsequent subsections are devoted to finding sufficient conditions for copositivity.…”

“…are said to satisfy the strengthened Cauchy-Schwarz inequality [8]. In our setting, μ is copositive if L 2 0 (μ 1 ) and L 2 0 (μ 2 ) satisfy the strengthened Cauchy-Schwarz inequality.…”