{"paper":{"title":"Optimal $C^{1,1}$ and Quasi-Optimal $C^2$ Monotone Interpolation with Curvature Control","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Monotone Hermite data admits an explicit quadratic-spline interpolant that achieves the smallest possible L^infty curvature among all C^{1,1} functions.","cross_cats":[],"primary_cat":"math.CA","authors_text":"Fushuai Jiang, Garving K. Luli","submitted_at":"2026-05-14T03:08:39Z","abstract_excerpt":"We study monotone Hermite interpolation on an interval, where both function values and first derivatives are prescribed at the nodes. Among all $C^{1,1}$ interpolants, we seek one with optimal curvature, measured by $\\|F''\\|_{L^\\infty}$. In this paper, we analyze the limitations of some classical techniques, and provide an explicit optimal construction in $C^{1,1}$ given by quadratic splines by studying the optimal velocity profile. Moreover, given $E = \\{x_1,\\cdots,x_N\\}$ and $f: E\\to \\mathbb{R}$ (without derivatives), we also provide a formula to compute the corresponding trace seminorm \\[ \\"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we provide an explicit optimal construction in C^{1,1} given by quadratic splines by studying the optimal velocity profile. Moreover, given E = {x1,…,xN} and f:E→R (without derivatives), we also provide a formula to compute the corresponding trace seminorm inf{||F''||_L^∞ : F(x)=f(x) on E and F'≥0 everywhere}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the global minimizer of ||F''||_L^∞ over monotone C^{1,1} Hermite interpolants is attained inside the subclass of quadratic splines whose pieces are determined by the optimal velocity profile analysis.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Explicit optimal quadratic-spline construction for C^{1,1} monotone Hermite interpolation minimizing ||F''||_infty, plus a formula for the monotone trace seminorm and a controlled mollification to C^2.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Monotone Hermite data admits an explicit quadratic-spline interpolant that achieves the smallest possible L^infty curvature among all C^{1,1} functions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"408d056f313bb85469ffdaeaa296c1afdc58f629788a6d83ee0ce37c1fc8e1d6"},"source":{"id":"2605.14302","kind":"arxiv","version":1},"verdict":{"id":"54149f1f-2ed9-43dc-9bd3-f1ad2faf21bf","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:04:10.045358Z","strongest_claim":"we provide an explicit optimal construction in C^{1,1} given by quadratic splines by studying the optimal velocity profile. Moreover, given E = {x1,…,xN} and f:E→R (without derivatives), we also provide a formula to compute the corresponding trace seminorm inf{||F''||_L^∞ : F(x)=f(x) on E and F'≥0 everywhere}.","one_line_summary":"Explicit optimal quadratic-spline construction for C^{1,1} monotone Hermite interpolation minimizing ||F''||_infty, plus a formula for the monotone trace seminorm and a controlled mollification to C^2.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the global minimizer of ||F''||_L^∞ over monotone C^{1,1} Hermite interpolants is attained inside the subclass of quadratic splines whose pieces are determined by the optimal velocity profile analysis.","pith_extraction_headline":"Monotone Hermite data admits an explicit quadratic-spline interpolant that achieves the smallest possible L^infty curvature among all C^{1,1} functions."},"references":{"count":25,"sample":[{"doi":"","year":2012,"title":"Majid, and Jamaludin Md","work_id":"3a160b9d-2f38-4572-8cd0-17cf59ca77c0","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"Francesc Ar` andiga, Antonio Baeza, and Dionisio F. Y´ a˜ nez. A new class of non-linear monotone hermite interpolants.Advances in Computational Mathematics, 39:289–309, 2013","work_id":"f836e60a-99a9-44cf-8574-2a6407d010f1","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"D. Azagra, E. Le Gruyer, and C. Mudarra. Explicit formulas forC 1,1 andC 1,ω conv extensions of 1-jets in Hilbert and superreflexive spaces.Journal of Functional Analysis, 274(10):3003–3032, 2018","work_id":"d3c24afc-dd2b-494c-bab6-f1e9c7243134","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1965,"title":"Piecewise polynomial interpolation and approximation","work_id":"0e56e64d-fbc7-45ed-b63e-f02d83c781d7","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1978,"title":"Springer, New York, 1978","work_id":"19d4436a-047e-45c1-897c-b9e1684c32d8","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":25,"snapshot_sha256":"b6de2b786deec835b1682d2e10539de062a1745772a59f2b61b62cb3327f27eb","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"6d614436a13ce216b772a57e3268b5a3b31f83f5fa324ec0bb615a82bc0eb607"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}