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We study multiple notions of Hilbert spaces of functions which, via the respective inner products, reproduce function values, or differences of function values. We do this by extending results from the more familiar settings of reproducing kernel Hilbert spaces, RKHSs. Our main results deal with operations on infinite graphsof vertices and edges, and associated Hilbert spaces. For electrical network models, the differencesrepresent voltage differences for pairs of vertices  x ,  y . In these cases, relative RKHSs will depend on choices of conductance functions  c , where an appropriate function  c  is specified as a positive function defined on the edge-set  E  from  G . Our present study of higher order differences, using choices of relative RKHSs, is motivated in part by existing numerical algorithms for discretization of PDEs. Our approach to higher order differences uses both combinatorial operations on graphs, and operator theory for the respective RKHSs. Starting with a graph, we introduce an induced graphsuch that the vertices inare the edges in  E  from  G , while the edges inare pairs of neighboring edges from  G .

Higher Order Difference Operators and Associated Relative Reproducing Kernel Hilbert Spaces