In this article, I consider projection groups on function spaces, more specifically the space of polynomials P<sub>n</sub>[x]. I will show that a very similar construct of projection operators allows us to...In this article, I consider projection groups on function spaces, more specifically the space of polynomials P<sub>n</sub>[x]. I will show that a very similar construct of projection operators allows us to project into the subspaces of P<sub>n</sub>[x] where the function h ∈P<sub>n</sub>[x] represents the closets function to f ∈P<sub>n</sub>[x] in the least square sense. I also demonstrate that we can generalise projections by constructing operators i.e. in R<sup>n+1</sup> using the metric tensor on P<sub>n</sub>[x]. This allows one to project a polynomial function onto another by mapping it to its coefficient vector in R<sup>n+1</sup>. This can be also achieved with the Kronecker Product as detailed in this paper.展开更多
文摘In this article, I consider projection groups on function spaces, more specifically the space of polynomials P<sub>n</sub>[x]. I will show that a very similar construct of projection operators allows us to project into the subspaces of P<sub>n</sub>[x] where the function h ∈P<sub>n</sub>[x] represents the closets function to f ∈P<sub>n</sub>[x] in the least square sense. I also demonstrate that we can generalise projections by constructing operators i.e. in R<sup>n+1</sup> using the metric tensor on P<sub>n</sub>[x]. This allows one to project a polynomial function onto another by mapping it to its coefficient vector in R<sup>n+1</sup>. This can be also achieved with the Kronecker Product as detailed in this paper.
