A strengthened Cauchy-Schwarz inequality for biorthogonal wavelets

Abstract

A strengthened Cauchy-Schwarz inequality for spaces of biorthogonal wavelets defined on the real line and on the interval is proved. The strengthened Cauchy-Schwarz inequality is a fundamental tool in the analysis of the multilevel methods and, in particular, plays an important role in the a posteriori error estimates for hierarchical methods.

Strengthened Cauchy–Schwarz inequality for biorthogonal wavelets in Sobolev spaces

“…In the present paper we extend the results of [9] to general Sobolev spaces. Namely we refer to the weighted Sobolev spaces H k , where the weight function k(ξ) is assumed to satisfy conditions similar to those in Hörmander [12,Chapter 10.1].…”

“…In view of some applications to hierarchical methods for numerical approximation of partial differential equations (see [1][2][3]8,11,13,15]) it is interesting to take as H a function space and consider wavelet subspaces V , W . In this connection since now, as far as we know, the only existing result is a strengthened Cauchy-Schwarz inequality for general biorthogonal wavelets in the homogeneous Sobolev space H (1) [9]; see also [10] for a particular family of spline-wavelets in the bi-dimensional case.…”

“…In Section 4 we state and prove our main result. The strengthened Cauchy-Schwarz inequality is obtained by adapting the arguments of [9] to the present general context; crucial point is to impose a certain regularity of the wavelet system, taking into account the order of the zero of k(ξ) at the origin and its growth for ξ → ∞.…”

Some vector inequalities for two operators in Hilbert spaces with applications

“…For other Schwarz and Buzano related inequalities in inner product spaces, see [1]- [4], [5]- [14], [22]- [26], [30]- [39], and the monographs [16], [17] and [18].…”