Stone-Weierstrass theorem

By G. Laville, I. Ramadanoff. 2004

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It will be shown that the Stone–Weierstrass theorem for Clifford-valued functions is true for the case of even dimension. It remains valid for the odd dimension if we add a stability condition by principal automorphism. Introduction. Recall the classical Stone–Weierstrass theorem: let Y be a metric space, C(Y ; R) the set of all continuous functions from Y in R, B ⊂ C(Y ; R) a subset such that B contains the constant function 1 and separates the points of Y . Then the algebra AB(Y ; R), generated by B is dense in C(Y ; R) for the topology of the uniform convergence on every compact. It is well-known that if one substitutes the field R by C, then an additional hypothesis is needed, namely: B should be stable with respect to complex conjugation. In case we are omitting this hypothesis and if we take, for example, Y to be an open subset of C and Y = {1, z}, then we will get the algebra of holomorphic functions. Let us mention that the case of functions taking values in the quaternion field is known [2] and it is analogous to the real case. Here, we will investigate the situation when R is replaced by Rp,q — a universal Clifford algebra of R, n = p+ q, with a quadratic form of signature (p, q). This study is motivated by the theory of monogenic functions [1]. The present paper is organized as follows: in Section 1 we will recall some notations usually employed in Clifford algebras. Section 2 will deal with some elements of combinatorics. The essential part of the paper is Section 3 in which we give a formula allowing to compute the scalar part of a given Clifford number. As an application of this formula, we are able to prove in Section 4 the following Stone–Weierstrass theorem for C(Y ; Rp,q): Theorem. Let Y be a metric space and C(Y ; Rp,q) the set of all continuous functions from Y to Rp,q. Let B ⊂ C(Y, Rp,q) be such that B contains the constant function 1 1991 Mathematics Subject Classification: Primary 30G35; Secondary 32K99. The paper is in final form and no version of it will be published elsewhere. [189] 190 G. LAVILLE AND I. P. RAMADANOFF and separates the points of Y . If p + q is odd , suppose in addition that B is stable with respect to the principal automorphism ∗. Then, the algebra AB(Y ; Rp,q), generated byB, is dense in C(Y ; Rp,q) for the topology of uniform convergence on compact sets. 1. Notations. In a Clifford algebra Rp,q = C0 ⊕ C1 ⊕ . . . ⊕ Cn,with n = p + q, the spaces C0, C1, . . . , Cn are supposed to be of respective basis {1}, {e1, e2, . . . , en}, {eij}i<j , . . . , {ei1...ik}i1<i2<···<ik , . . . , {e1.2...n}, where (i1, . . . , ik) is a multiindex with i1, . . . , ik ∈ {1, . . . , n}, 1 ≤ i1 < . . . < ik ≤ n. The algebra obeys to the laws:  ei = 1, i = 1, . . . , p, ei = −1, i = p+ 1, . . . , n, eiej = −ejei, i 6= j, ei1...ik = ei1ei2 · · · eik , for i1 < i2 < . . . < ik. We will make use of the decomposition of a Clifford number a in its scalar (real) part 〈a〉0, its 1-vector 〈a〉1 ∈ C1, its bivector part 〈a〉2 ∈ C2, etc . . . up to its pseudo-scalar part 〈a〉n ∈ Cn, i.e: a = 〈a〉0 + 〈a〉1 + · · ·+ 〈a〉n, where, 〈a〉k = ∑

Subfields: Applied Mathematics

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