Construction of Higher Orthogonal Polynomials Through a New Inner Product, ‹·,·›p, In a Countable Real Lp-space

Abstract

This research work places a new and consistent inner product ‹·,·›p on a countable family of the real Lp function spaces, proves generalizations of some of the inequalities of the classical inner product for ‹·,·›p provides a construction of a specie of Higher Orthogonal Polynomials in these inner-product–admissible function spaces, and ultimately brings us to a study of the Generalized Fourier Series Expansion in terms of these polynomials. First, the reputation of this new inner product is established by the proofs of various inequalities and identities, all of which are found to be generalizations of the classical inequalities of functional analysis. Thereafter two orthogonalities of ‹·,·›p (which coincide at p = 2) are defined while the Gram-Schmidt orthonormalization procedure is considered and lifted to accommodate this product, out of which emerges a set of higher orthogonal polynomials in Lp[-1,1] that reduce to the Legendre Polynomials at p = 2. We argue that this inner product provides a formidable tool for the investigation of Harmonic Analysis on the real Lp function spaces for p other than p = 2, and a revisit of the various fields where the theory of inner product spaces is indispensable is recommended for further studies.