TITLE:
Onto Orthogonal Projections in the Space of Polynomials Pn[x]
AUTHORS:
Jean-Francois Niglio
KEYWORDS:
Polynomials and Projections, Projections, The Kronecker Product, Idempotent Operators
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.11 No.1,
January
11,
2023
ABSTRACT: In this article, I consider projection groups on function spaces, more specifically the space of polynomials Pn[x]. I will show that a very similar construct of projection operators allows us to project into the subspaces of Pn[x] where the function h ∈ Pn[x] represents the closets function to f ∈ Pn[x] in the least square sense. I also demonstrate that we can generalise projections by constructing operators i.e. in Rn+1 using the metric tensor on Pn[x]. This allows one to project a polynomial function onto another by mapping it to its coefficient vector in Rn+1. This can be also achieved with the Kronecker Product as detailed in this paper.