Dimension Walks on Generalized Spaces
Let d,k be positive integers. We call generalized spaces the cartesian product of the d-dimensional sphere, π^d, with the k-dimensional Euclidean space, β^k. We consider the class π«(π^d Γβ^k) of continuous functions Ο: [-1,1] Γ [0,β) ββ such that the mapping C: ( π^d Γβ^k )^2 ββ, defined as C ( (x,y),(x^',y^') ) = Ο ( cosΞΈ(x,x^'), y-y^' ), (x,y), (x^',y^') βπ^d Γβ^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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