{"paper":{"title":"Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Differential Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Qi Ye","submitted_at":"2011-09-01T07:29:16Z","abstract_excerpt":"In this paper we introduce a generalization of the classical $\\Leb_2(\\Rd)$-based Sobolev spaces with the help of a vector differential operator $\\mathbf{P}$ which consists of finitely or countably many differential operators $P_n$ which themselves are linear combinations of distributional derivatives. We find that certain proper full-space Green functions $G$ with respect to $L=\\mathbf{P}^{\\ast T}\\mathbf{P}$ are positive definite functions. Here we ensure that the vector distributional adjoint operator $\\mathbf{P}^{\\ast}$ of $\\mathbf{P}$ is well-defined in the distributional sense. We then pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0109","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}