Mercer’s Theorem for Quaternionic Kernels

the series being uniformly and absolutely convergent in (x,y). A number of generalisations to Mercer’s theorem may be found in the literature, in particular dealing with kernels K : Y × Y → C for various choices of Y . However there would appear to have been (to the best of the author’s knowledge) no attempts made to extend Mercer’s theorem to cover non-complex valued kernels. In the present paper we show how Mercer’s theorem may be extended to cover one such family of kernels, namely the continuous quaternionic valued kernels K : R × R → H. As quaternions provide a powerful tool for describing geometric problems [6] we anticipate that this extension will find applications in geometrical learning problems. Throughout this paper the quaternionic division algebra [7], [3] will be denoted H, the field of complex numbers C and the completely ordered field of reals R. The conjugate and norm of x ∈ H will be denoted x̄ and |x| respectively, where |x| = x̄x ∈ R\R. We define R = (0,∞) to be the set of positive reals, R = (−∞, 0) the set of negative reals, N the set of natural numbers including 0, Zp = {0, 1, . . . , p− 1} the set of integers modulo p (with the extensions Z∞ = N, Z0 = ∅), N ∞ = N ∪ {∞}, and L the set of square integrable quaternionic functions on R: L = {