Advances in Inequalities of the Schwarz, Gruss and Bessel Type in Inner Product Spaces

TL;DR

This paper surveys recent advances in inequalities related to Schwarz, Gruss, and Bessel in inner product spaces, including reverses, generalizations, and applications to Fourier and Mellin transforms.

Contribution

It provides new generalizations and extensions of classical inequalities in inner product spaces, with applications to approximation theory.

Findings

Reverses of Schwarz, triangle, and Bessel inequalities are established.

Generalizations of Boas-Bellman, Bombieri, and other inequalities are presented.

Applications to discrete Fourier and Mellin transforms are demonstrated.

Abstract

The main aim of this monograph is to survey some recent results obtained by the author related to reverses of the Schwarz, triangle and Bessel inequalities. Some Gruss' type inequalities for orthonormal families of vectors in real or complex inner product spaces are presented as well. Generalizations of the Boas-Bellman, Bombieri, Selberg, Heilbronn and Pecaric inequalities for finite sequences of vectors that are not necessarily orthogonal are also provided. Two extensions of the celebrated Ostrowski's inequalities for sequences or real numbers and the generalization of Wagner's inequality in inner product spaces are pointed out. Finally, some Gruss type inequalities for n-tuples of vectors in inner product spaces and their natural applications for the approximation of the discrete Fourier and Mellin transforms are given as well.