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To this end, we consider composing different component kernels, each acting on a low-dimensional Euclidean space. Due to Aronszajn (Trans. Am. Math. Soc. <jats:bold>68<\/jats:bold>, 337\u2013404 1950), the product of positive <jats:italic>semi-<\/jats:italic>definite kernel functions is again positive <jats:italic>semi-<\/jats:italic>definite, where, moreover, the corresponding native space is a particular instance of a tensor product, referred to as Hilbert tensor product. We first analyze the general problem of multivariate interpolation by product kernels. Then, we further investigate the tensor product structure, in particular for <jats:italic>grid-like<\/jats:italic> samples. We use this case to show that the product of positive definite kernel functions is again positive definite. Moreover, we develop an efficient computation scheme for the well-known Newton basis. Supporting numerical examples show the good performance of product kernels, especially for their flexibility.<\/jats:p>","DOI":"10.1007\/s10444-025-10226-y","type":"journal-article","created":{"date-parts":[[2025,3,20]],"date-time":"2025-03-20T05:02:35Z","timestamp":1742446955000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Product kernels are efficient and flexible tools for high-dimensional scattered data interpolation"],"prefix":"10.1007","volume":"51","author":[{"given":"Kristof","family":"Albrecht","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Juliane","family":"Entzian","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1743-5484","authenticated-orcid":false,"given":"Armin","family":"Iske","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,3,20]]},"reference":[{"key":"10226_CR1","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/978-3-031-45158-4_4","volume-title":"Modeling, Simulation and Optimization of Fluid Dynamic Applications","author":"K Albrecht","year":"2023","unstructured":"Albrecht, K., Entzian, J., Iske, A.: Anisotropic kernels for particle flow simulation. 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