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
In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the uni...In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on ?by using the rotation group [3] [4]. It will be proved that the group acts on elements of ?in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation ?in terms of matrix operations using the operator and the Hadamard Product;this construction is consistent with the group operation defined in the first article.展开更多
In this article, we start by a review of the circle group? [1] and its topology induced [1] by the quotient metric, which we later use to define a topological structure on the unit circle . Using points on? under the ...In this article, we start by a review of the circle group? [1] and its topology induced [1] by the quotient metric, which we later use to define a topological structure on the unit circle . Using points on? under the complex exponential map, we can construct orthogonal projection operators. We will show that under this construction, we arrive at a topological group, denoted? of projection matrices. Together with the induced topology, it will be demonstrated that? is Hausdorff and Second Countable forming a topological manifold. Moreover, I will use an example of a group action on? to generate subgroups of?.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.
文摘In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on ?by using the rotation group [3] [4]. It will be proved that the group acts on elements of ?in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation ?in terms of matrix operations using the operator and the Hadamard Product;this construction is consistent with the group operation defined in the first article.
文摘In this article, we start by a review of the circle group? [1] and its topology induced [1] by the quotient metric, which we later use to define a topological structure on the unit circle . Using points on? under the complex exponential map, we can construct orthogonal projection operators. We will show that under this construction, we arrive at a topological group, denoted? of projection matrices. Together with the induced topology, it will be demonstrated that? is Hausdorff and Second Countable forming a topological manifold. Moreover, I will use an example of a group action on? to generate subgroups of?.
