{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:6SMNYJJ2FZCTJHA3RYZ2GIFUQA","short_pith_number":"pith:6SMNYJJ2","schema_version":"1.0","canonical_sha256":"f498dc253a2e45349c1b8e33a320b480372319952a5913d0c5dfed9045f92b57","source":{"kind":"arxiv","id":"1807.02363","version":1},"attestation_state":"computed","paper":{"title":"Schoenberg coefficients and curvature at the origin of continuous isotropic positive definite kernels on spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","stat.TH"],"primary_cat":"math.ST","authors_text":"Ahmed Arafat, Emilio Porcu, Pablo Gregori","submitted_at":"2018-07-06T11:37:09Z","abstract_excerpt":"We consider the class $\\Psi_d$ of continuous functions $\\psi \\colon [0,\\pi] \\to \\mathbb{R}$, with $\\psi(0)=1$ such that the associated isotropic kernel $C(\\xi,\\eta)= \\psi(\\theta(\\xi,\\eta))$ ---with $\\xi,\\eta \\in \\mathbb{S}^d$ and $\\theta$ the geodesic distance--- is positive definite on the product of two $d$-dimensional spheres $\\mathbb{S}^d$. We face Problems 1 and 3 proposed in the essay Gneiting (2013b). We have considered an extension that encompasses the solution of Problem 1 solved in Fiedler (2013), regarding the expression of the $d$-Schoenberg coefficients of members of $\\Psi_d$ as c"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1807.02363","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-07-06T11:37:09Z","cross_cats_sorted":["math.FA","stat.TH"],"title_canon_sha256":"6acaab01322220c49224ab792fd600686eacd61000ce92c6c25e8daeddefb75a","abstract_canon_sha256":"06220ff258d7ab85912291be1cd81346751bacc371f0ac48aa2529d294d62ee7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:22.310924Z","signature_b64":"0CbmBXW999DMQLR3A52qNd2oWctH8gTqn9vILwJeSPA75bShEKBkQL8Uf1g77RR1BJJD3kDME1/Zlrb1QfpdDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f498dc253a2e45349c1b8e33a320b480372319952a5913d0c5dfed9045f92b57","last_reissued_at":"2026-05-18T00:11:22.310182Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:22.310182Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Schoenberg coefficients and curvature at the origin of continuous isotropic positive definite kernels on spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","stat.TH"],"primary_cat":"math.ST","authors_text":"Ahmed Arafat, Emilio Porcu, Pablo Gregori","submitted_at":"2018-07-06T11:37:09Z","abstract_excerpt":"We consider the class $\\Psi_d$ of continuous functions $\\psi \\colon [0,\\pi] \\to \\mathbb{R}$, with $\\psi(0)=1$ such that the associated isotropic kernel $C(\\xi,\\eta)= \\psi(\\theta(\\xi,\\eta))$ ---with $\\xi,\\eta \\in \\mathbb{S}^d$ and $\\theta$ the geodesic distance--- is positive definite on the product of two $d$-dimensional spheres $\\mathbb{S}^d$. We face Problems 1 and 3 proposed in the essay Gneiting (2013b). We have considered an extension that encompasses the solution of Problem 1 solved in Fiedler (2013), regarding the expression of the $d$-Schoenberg coefficients of members of $\\Psi_d$ as c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.02363","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1807.02363","created_at":"2026-05-18T00:11:22.310298+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.02363v1","created_at":"2026-05-18T00:11:22.310298+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.02363","created_at":"2026-05-18T00:11:22.310298+00:00"},{"alias_kind":"pith_short_12","alias_value":"6SMNYJJ2FZCT","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_16","alias_value":"6SMNYJJ2FZCTJHA3","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_8","alias_value":"6SMNYJJ2","created_at":"2026-05-18T12:32:11.075285+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6SMNYJJ2FZCTJHA3RYZ2GIFUQA","json":"https://pith.science/pith/6SMNYJJ2FZCTJHA3RYZ2GIFUQA.json","graph_json":"https://pith.science/api/pith-number/6SMNYJJ2FZCTJHA3RYZ2GIFUQA/graph.json","events_json":"https://pith.science/api/pith-number/6SMNYJJ2FZCTJHA3RYZ2GIFUQA/events.json","paper":"https://pith.science/paper/6SMNYJJ2"},"agent_actions":{"view_html":"https://pith.science/pith/6SMNYJJ2FZCTJHA3RYZ2GIFUQA","download_json":"https://pith.science/pith/6SMNYJJ2FZCTJHA3RYZ2GIFUQA.json","view_paper":"https://pith.science/paper/6SMNYJJ2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.02363&json=true","fetch_graph":"https://pith.science/api/pith-number/6SMNYJJ2FZCTJHA3RYZ2GIFUQA/graph.json","fetch_events":"https://pith.science/api/pith-number/6SMNYJJ2FZCTJHA3RYZ2GIFUQA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6SMNYJJ2FZCTJHA3RYZ2GIFUQA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6SMNYJJ2FZCTJHA3RYZ2GIFUQA/action/storage_attestation","attest_author":"https://pith.science/pith/6SMNYJJ2FZCTJHA3RYZ2GIFUQA/action/author_attestation","sign_citation":"https://pith.science/pith/6SMNYJJ2FZCTJHA3RYZ2GIFUQA/action/citation_signature","submit_replication":"https://pith.science/pith/6SMNYJJ2FZCTJHA3RYZ2GIFUQA/action/replication_record"}},"created_at":"2026-05-18T00:11:22.310298+00:00","updated_at":"2026-05-18T00:11:22.310298+00:00"}