On Almost Orthogonal Series

TL;DR

This paper generalizes classical inequalities like Bessel and Riesz-Fisher to Hilbert spaces with respect to sequences, and applies these results to derive inequalities for special functions related to various probability distributions.

Contribution

It introduces generalized versions of Bessel inequality and Riesz-Fisher theorem in Hilbert spaces with respect to sequences, and demonstrates their applications to special functions.

Findings

Generalized Bessel inequality for Hilbert spaces with respect to sequences

Analogues of Riesz-Fisher theorem in this context

Systematic derivation of inequalities for special functions associated with probability distributions

Abstract

In this work we prove analogues of Bessel inequality and Riesz-Fisher theorem in Hilbert spaces with respect to sequences. We apply our generalized Bessel inequality to the Hilbert spaces associated with the Normal, Beta, Gamma and certain discrete probability distributions to show how to generate certain type of inequalities for special functions systematically.