{"paper":{"title":"Fourier series of Jacobi-Sobolev polynomial","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Judit M\\'inguez, \\'Oscar Ciaurri","submitted_at":"2018-06-21T08:28:33Z","abstract_excerpt":"Let $\\{q_n^{(\\alpha,\\beta,m)}(x)\\}_{n\\ge 0}$ be the orthonormal polynomials respect to the Sobolev-type inner product \\begin{equation*} \\langle f,g\\rangle_{\\alpha,\\beta,m}=\\sum_{k=0}^m \\int_{-1}^{1}f^{(k)}(x)g^{(k)}(x)\\, dw_{\\alpha+k,\\beta+k}(x), \\quad \\alpha,\\beta>-1, \\quad m\\ge 1, \\end{equation*} where $dw_{a,b}(x)=(1-x)^{a}(1+x)^b\\, dx$. We obtain necessary and sufficient conditions for the uniform boundedness of the partial sum operators related to this sequence of polynomials in the Sobolev space $W_{\\alpha,\\beta}^{p,m}$. As a consequence we deduce the convergence of such partial sums in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.08105","kind":"arxiv","version":2},"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"}