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We prove the following the direct and inverse theorem for $\\oplus^rJ_{k,s}(f)$: there are constants $C_1$ and $C_2$ such that \\begin{equation*} C_1\\|\\oplus^rJ_{k,s}f-f\\|_p \\leq \\omega^{2r}(f,k^{-1})_p \\leq C_2 \\max_{v\\geq k}\\|\\oplus^rJ_{k,s}f-f\\|_p \\end{equation*} for any positive integer $k$ and any $p$th Lebesgue integrable functions $f$ defined on $\\mathbb S^{n-1}$, where $\\omega^{2r}(f,t)_p$ is the modulus of smoothness of degree $2r$"},"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":"1409.3923","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-13T07:39:33Z","cross_cats_sorted":[],"title_canon_sha256":"da72c550a7fcdffd6f89acbe5abb51cbc9f87b4e64dbfb4f43505d22fef99747","abstract_canon_sha256":"13f734cb0fcf0d48f7ccb2034c104f348da48feb66ab5be7c1b75bd953359ba0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:53.377136Z","signature_b64":"FMffx/ypnnljc4PPQd8ZNdJWbweD2RUqiYiEchEUQzCWm5WmwBhY74h8s4lguK2xjh2cqucY3evitc1TItudAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e4fb34ec72226191ea752fe6ec243fea35a059a86056ef572b11aaf5f1ee1501","last_reissued_at":"2026-05-18T02:42:53.376764Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:53.376764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Approximation by boolean sums of Jackson operators on the sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Feilong Cao, Yuguang Wang","submitted_at":"2014-09-13T07:39:33Z","abstract_excerpt":"This paper concerns the approximation by the Boolean sums of Jackson operators $\\oplus^rJ_{k,s}(f)$ on the unit sphere $\\mathbb S^{n-1}$ of $\\mathbb{R}^{n}$. We prove the following the direct and inverse theorem for $\\oplus^rJ_{k,s}(f)$: there are constants $C_1$ and $C_2$ such that \\begin{equation*} C_1\\|\\oplus^rJ_{k,s}f-f\\|_p \\leq \\omega^{2r}(f,k^{-1})_p \\leq C_2 \\max_{v\\geq k}\\|\\oplus^rJ_{k,s}f-f\\|_p \\end{equation*} for any positive integer $k$ and any $p$th Lebesgue integrable functions $f$ defined on $\\mathbb S^{n-1}$, where $\\omega^{2r}(f,t)_p$ is the modulus of smoothness of degree $2r$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.3923","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":"1409.3923","created_at":"2026-05-18T02:42:53.376827+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.3923v1","created_at":"2026-05-18T02:42:53.376827+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.3923","created_at":"2026-05-18T02:42:53.376827+00:00"},{"alias_kind":"pith_short_12","alias_value":"4T5TJ3DSEJQZ","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"4T5TJ3DSEJQZD2TV","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"4T5TJ3DS","created_at":"2026-05-18T12:28:14.216126+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/4T5TJ3DSEJQZD2TVF7TOYJB75I","json":"https://pith.science/pith/4T5TJ3DSEJQZD2TVF7TOYJB75I.json","graph_json":"https://pith.science/api/pith-number/4T5TJ3DSEJQZD2TVF7TOYJB75I/graph.json","events_json":"https://pith.science/api/pith-number/4T5TJ3DSEJQZD2TVF7TOYJB75I/events.json","paper":"https://pith.science/paper/4T5TJ3DS"},"agent_actions":{"view_html":"https://pith.science/pith/4T5TJ3DSEJQZD2TVF7TOYJB75I","download_json":"https://pith.science/pith/4T5TJ3DSEJQZD2TVF7TOYJB75I.json","view_paper":"https://pith.science/paper/4T5TJ3DS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.3923&json=true","fetch_graph":"https://pith.science/api/pith-number/4T5TJ3DSEJQZD2TVF7TOYJB75I/graph.json","fetch_events":"https://pith.science/api/pith-number/4T5TJ3DSEJQZD2TVF7TOYJB75I/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4T5TJ3DSEJQZD2TVF7TOYJB75I/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4T5TJ3DSEJQZD2TVF7TOYJB75I/action/storage_attestation","attest_author":"https://pith.science/pith/4T5TJ3DSEJQZD2TVF7TOYJB75I/action/author_attestation","sign_citation":"https://pith.science/pith/4T5TJ3DSEJQZD2TVF7TOYJB75I/action/citation_signature","submit_replication":"https://pith.science/pith/4T5TJ3DSEJQZD2TVF7TOYJB75I/action/replication_record"}},"created_at":"2026-05-18T02:42:53.376827+00:00","updated_at":"2026-05-18T02:42:53.376827+00:00"}