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Given a spherical cap $$ B(e,\\al)=\\{x\\in\\sph: d(x, e) \\leq \\al\\}, (e\\in\\sph, \\text{$\\al\\in (0,\\pi)$ is bounded away from $\\pi$}),$$ we define the metric $$\\rho(x,y):=\\frac 1{\\al} \\sqrt{(d(x, y))^2+\\al(\\sqrt{\\al-d(x, e)}-\\sqrt{\\al-d(y,e)})^2},\n  $$ where $x, y\\in B(e,\\al)$. It is shown that given any $\\be\\ge 1$, $1\\leq p<\\infty$ and any finite subset $\\Ld$ of $B(e,\\al)$ s"},"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":"math/0703768","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2007-03-26T18:44:05Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"213825e8f99dbbdae7d2fc42ccb42c1504c3871eb7bf0fa8c796366790cf0236","abstract_canon_sha256":"bbbce67b651fb2f7057569c763a59781f2078904b1231691c7e759a9a2fb91f8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T22:06:22.324702Z","signature_b64":"9FBR0c0ILMcJgKaFRSX9faYF0MaqRnXhaJwcEM86gWnNp2jcdIXCUyU1g0sIMLaXc9x1uC9f/y12zwwJhgRZAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7967d09f29c152a6a02b7549a7a082cae79e136ff85c740a5714c60ce2ab8be9","last_reissued_at":"2026-06-03T22:06:22.324284Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T22:06:22.324284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Positive Cubature formulas and Marcinkiewicz-Zygmund inequalities on spherical caps","license":"","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.CA","authors_text":"Feng Dai, Heping Wang","submitted_at":"2007-03-26T18:44:05Z","abstract_excerpt":"Let $\\Pi_n^d$ denote the space of all spherical polynomials of degree at most $n$ on the unit sphere $\\sph$ of $\\mathbb{R}^{d+1}$, and let $d(x, y)$ denote the usual geodesic distance $\\arccos x\\cdot y$ between $x, y\\in \\sph$. Given a spherical cap $$ B(e,\\al)=\\{x\\in\\sph: d(x, e) \\leq \\al\\}, (e\\in\\sph, \\text{$\\al\\in (0,\\pi)$ is bounded away from $\\pi$}),$$ we define the metric $$\\rho(x,y):=\\frac 1{\\al} \\sqrt{(d(x, y))^2+\\al(\\sqrt{\\al-d(x, e)}-\\sqrt{\\al-d(y,e)})^2},\n  $$ where $x, y\\in B(e,\\al)$. It is shown that given any $\\be\\ge 1$, $1\\leq p<\\infty$ and any finite subset $\\Ld$ of $B(e,\\al)$ s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0703768","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0703768/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"math/0703768","created_at":"2026-06-03T22:06:22.324351+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0703768v1","created_at":"2026-06-03T22:06:22.324351+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0703768","created_at":"2026-06-03T22:06:22.324351+00:00"},{"alias_kind":"pith_short_12","alias_value":"PFT5BHZJYFJK","created_at":"2026-06-03T22:06:22.324351+00:00"},{"alias_kind":"pith_short_16","alias_value":"PFT5BHZJYFJKNIBL","created_at":"2026-06-03T22:06:22.324351+00:00"},{"alias_kind":"pith_short_8","alias_value":"PFT5BHZJ","created_at":"2026-06-03T22:06:22.324351+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/PFT5BHZJYFJKNIBLOVE2PIECZL","json":"https://pith.science/pith/PFT5BHZJYFJKNIBLOVE2PIECZL.json","graph_json":"https://pith.science/api/pith-number/PFT5BHZJYFJKNIBLOVE2PIECZL/graph.json","events_json":"https://pith.science/api/pith-number/PFT5BHZJYFJKNIBLOVE2PIECZL/events.json","paper":"https://pith.science/paper/PFT5BHZJ"},"agent_actions":{"view_html":"https://pith.science/pith/PFT5BHZJYFJKNIBLOVE2PIECZL","download_json":"https://pith.science/pith/PFT5BHZJYFJKNIBLOVE2PIECZL.json","view_paper":"https://pith.science/paper/PFT5BHZJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0703768&json=true","fetch_graph":"https://pith.science/api/pith-number/PFT5BHZJYFJKNIBLOVE2PIECZL/graph.json","fetch_events":"https://pith.science/api/pith-number/PFT5BHZJYFJKNIBLOVE2PIECZL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PFT5BHZJYFJKNIBLOVE2PIECZL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PFT5BHZJYFJKNIBLOVE2PIECZL/action/storage_attestation","attest_author":"https://pith.science/pith/PFT5BHZJYFJKNIBLOVE2PIECZL/action/author_attestation","sign_citation":"https://pith.science/pith/PFT5BHZJYFJKNIBLOVE2PIECZL/action/citation_signature","submit_replication":"https://pith.science/pith/PFT5BHZJYFJKNIBLOVE2PIECZL/action/replication_record"}},"created_at":"2026-06-03T22:06:22.324351+00:00","updated_at":"2026-06-03T22:06:22.324351+00:00"}