Any orthonormal basis in high dimension is uniformly distributed over the sphere

Uniformly bounded orthonormal polynomials on the sphere

Given any ε > 0, we construct an orthonormal system of nk uniformly bounded polynomials of degree at most k on the unit sphere in R where nk is bigger than 1− ε times the dimension of the space of polynomials of degree at most k. Similarly we construct an orthonormal system of sections of powers A of a positive holomorphic line bundle on a compact Kähler manifold with cardinality bigger than 1−...

Online Inserting Points Uniformly on the Sphere

In many scientific and engineering applications, there are occasions where points need to be inserted uniformly onto a sphere. Previous works on uniform point insertion mainly focus on the offline version, i.e., to compute N positions on the sphere for a given interger N with the objective to distribute these points as uniformly as possible. An example application is the Thomson problem where t...

Building an Orthonormal Basis, Revisited

Frisvad [2012b] describes a widely-used computational method for efficiently augmenting a given single unit vector with two other vectors to produce an orthonormal frame in three dimensions, a useful operation for any physically based renderer. However, the implementation has a precision problem: as the z component of the input vector approaches −1, floating point cancellation causes the frame ...

An Orthonormal Basis for Entailment

Entailment is a logical relationship in which the truth of a proposition implies the truth of another proposition. The ability to detect entailment has applications in IR, QA, and many other areas. This study uses the vector space model to explore the relationship between cohesion and entailment. A Latent Semantic Analysis space is used to measure cohesion between propositions using vector simi...

Modelling with Orthonormal Basis Functions