Higher Order Difference Operators and Associated Relative Reproducing Kernel Hilbert Spaces

AbstractWe 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 graphs G=(V,E) of vertices and edges, and associated Hilbert spaces. For electrical network models, the differences f(x)−f(y) represent 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 G=(V,E), we introduce an induced graph G′ such that the vertices in G′ are the edges in E from G, while the edges in G′ are pairs of neighboring edges from G.KEYWORDS: Conduction functionsdrop operatorgraph Laplacianhigher order differencesinduced graphsisometriesnetwork modelsrelative reproducingreproducing kernel Hilbert spaceresistance distanceMATHEMATICS SUBJECT CLASSIFICATION: Primary: 47B3247B9047N4047N70Secondary: 05C6305C9046C0546E2247B25 Data availability statementThe datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.Disclosure statementThe authors report there are no competing interests to declare.Additional informationFundingNo funding was received to assist with the preparation of this manuscript. The authors have no relevant financial or non-financial interests to disclose.