Dimension Walks on Generalized Spaces
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Let d,k be positive integers. We call generalized spaces the cartesian product of the d-dimensional sphere, 𝕊^d, with the k-dimensional Euclidean space, ℝ^k. We consider the class 𝒫(𝕊^d ×ℝ^k) of continuous functions φ: [-1,1] × [0,∞) →ℝ such that the mapping C: ( 𝕊^d ×ℝ^k )^2 →ℝ, defined as C ( (x,y),(x^',y^') ) = φ ( cosθ(x,x^'), y-y^' ), (x,y), (x^',y^') ∈𝕊^d ×ℝ^k, is positive definite. We propose linear operators that allow for walks through dimension within generalized spaces while preserving positive definiteness.
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