Polynomial functions on the classical projective spaces

Abstract

Abstract. The polynomial functions on a projective space over a field K = R, C or H come from the corresponding sphere via the Hopf fibration. The main theorem states that every polynomial function φ(x) of degree d is a linear combination of "elementary" functions | x, · | d .1. Spaces and operators. The classical projective spaces are KP m−1 where K is one of three fields R, C, H and m ≥ 2. The quaternion field H is noncommutative in contrast to R and C. However, R ⊂ H and each real number commutes with all quaternions. In general, K is an associative unital algebra over R of dimension δ = δ(K) = 1, 2, 4 for K = R, C, H respectively. The standard conjugation α → α is an involutive automorphism of K (or anti-automorphism if K = H since αβ = βα in this case). For any α ∈ K the real number Re α = 1 2 (α + α) is the real part of α. (If K = R we set α = α.) The space KP m−1 can be constructed starting with K m that consists of all m-tuples x = (ξ i ), ξ i ∈ K. This is an m-dimensional right (for definiteness) linear space over K; the addition in K m is standard, the multiplication by a scalar α ∈ K is xα = (ξ i α). Moreover, K m is a Euclidean space provided with the inner productThe properties of the latter are standard but with the fixed order of factors in the relations xα, y = α x, y , x, yα = x, y α if K = H. The corresponding Euclidean norm on K m is (2) x = x, x = |ξ i | 2 .

On tight projective designs

“…The projective t-designs can be characterized as the averaging sets in the sense of [19] for suitable spaces of functions on FP n . Usually, these spaces are described in terms of harmonic analysis but we prefer a more elementary approach [15], [16].…”

“…Given d ∈ 2N, we denote by Pol F (d) the space of all polynomial functions of degrees ≤ d. It has been proven in [16] that the family {φ d;y : y ∈ FP n } spans the whole space Pol F (d). We apply this result to prove the following Proposition 2.2.…”

Lower bounds for projective designs, cubature formulas and related isometric embeddings

“…As a result, the restriction φ|S(m, K) is well defined on KP m−1 . Accordingly, it is called a polynomial function on KP m−1 [16]. For simplicity we preserve the notation φ for the projective image of φ ∈ ΦK(m, p).…”

“…Now note that the space ΦK(m, p) contains all elementary polynomials φy;p(x) = | x, y | p , y ∈ K m . Moreover, any function φ ∈ ΦK(m, p) is a linear combination of elementary polynomials [16]. For this reason the projective cubature formula (1.7) is equivalent to the identity…”

“…However, there exists an alternative way to calculate dim ΦK(m, p) for all fields K at once, see [14]. In particular, 14], [16]). A projective cubature formula of index p in KP m−1 is an identity…”