{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:JB4F5D4MKZIVZLDIXJYFTMPVJE","short_pith_number":"pith:JB4F5D4M","canonical_record":{"source":{"id":"0704.1865","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2007-04-14T12:50:22Z","cross_cats_sorted":[],"title_canon_sha256":"9e8d0fbe5cf535fa3f0cdf61186f86ac09a39289fac531edeeaecdda00df1124","abstract_canon_sha256":"17ead59582ff6a350a32a8f2dbd710350b2d7890c9af383bc530242ea22b4462"},"schema_version":"1.0"},"canonical_sha256":"48785e8f8c56515cac68ba7059b1f5490c0c8be6e62fc7a418fce898bb251e8b","source":{"kind":"arxiv","id":"0704.1865","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0704.1865","created_at":"2026-07-04T15:00:53Z"},{"alias_kind":"arxiv_version","alias_value":"0704.1865v1","created_at":"2026-07-04T15:00:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0704.1865","created_at":"2026-07-04T15:00:53Z"},{"alias_kind":"pith_short_12","alias_value":"JB4F5D4MKZIV","created_at":"2026-07-04T15:00:53Z"},{"alias_kind":"pith_short_16","alias_value":"JB4F5D4MKZIVZLDI","created_at":"2026-07-04T15:00:53Z"},{"alias_kind":"pith_short_8","alias_value":"JB4F5D4M","created_at":"2026-07-04T15:00:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:JB4F5D4MKZIVZLDIXJYFTMPVJE","target":"record","payload":{"canonical_record":{"source":{"id":"0704.1865","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2007-04-14T12:50:22Z","cross_cats_sorted":[],"title_canon_sha256":"9e8d0fbe5cf535fa3f0cdf61186f86ac09a39289fac531edeeaecdda00df1124","abstract_canon_sha256":"17ead59582ff6a350a32a8f2dbd710350b2d7890c9af383bc530242ea22b4462"},"schema_version":"1.0"},"canonical_sha256":"48785e8f8c56515cac68ba7059b1f5490c0c8be6e62fc7a418fce898bb251e8b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:00:53.438447Z","signature_b64":"GPMAAL9DfNEsI4t4Da27YH1DGnI+Leby7+oIF+YxXSkMV8zMHRJGxsdCDUBGgtA03dSkhjONVhCKjw6oQLIeDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48785e8f8c56515cac68ba7059b1f5490c0c8be6e62fc7a418fce898bb251e8b","last_reissued_at":"2026-07-04T15:00:53.438000Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:00:53.438000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0704.1865","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:00:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ii7kGsLMFH6Oc4A0jqFyT4eLcqRH5a9wPprHjboTAxhxEzBYoYPULt8+bPVdgkehgvq5wCaxOxzaGm5d8S36AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T04:41:02.516028Z"},"content_sha256":"42d74a357d1bafe2dd4f9001365e2b0cc3b9d0b5bbd70971a99f690c3de1f577","schema_version":"1.0","event_id":"sha256:42d74a357d1bafe2dd4f9001365e2b0cc3b9d0b5bbd70971a99f690c3de1f577"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:JB4F5D4MKZIVZLDIXJYFTMPVJE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On $L^{1}$-Convergence of Fourier Series Under $MVBV$ Condition","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Dan Sheng Yu, Ping Zhou, Song Ping Zhou","submitted_at":"2007-04-14T12:50:22Z","abstract_excerpt":"Let $f\\in L_{2\\pi}$ be a real-valued even function with its Fourier series $ \\frac{a_{0}}{2}+\\sum_{n=1}^{\\infty}a_{n}\\cos nx,$ and let $S_{n}(f,x), n\\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence $\\{a_{n}\\}$ is decreasing and $\\lim\\limits_{n\\to \\infty}a_{n}=0$, then $$ \\lim\\limits_{n\\to \\infty}\\Vert f-S_{n}(f)\\Vert_{L}=0 {if and only if} \\lim\\limits_{n\\to \\infty}a_{n}\\log n=0. $$ We weaken the monotone condition in this classical result to the so-called mean value bounded variation ($MVBV$) condition. The generalization of the above "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.1865","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/0704.1865/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:00:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ncuDJiIyeDUgMyd1RHSfOUzyR68AT5NeB0zbIAA4U0biQZIM7vVG1PLQSa4J662gmTCrjlPsxDZu+DDCCYSNAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T04:41:02.516547Z"},"content_sha256":"e5a34d4ebcf32d2418e0175713903e5a1194f65876ed5ed993245f05ce9ef25a","schema_version":"1.0","event_id":"sha256:e5a34d4ebcf32d2418e0175713903e5a1194f65876ed5ed993245f05ce9ef25a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JB4F5D4MKZIVZLDIXJYFTMPVJE/bundle.json","state_url":"https://pith.science/pith/JB4F5D4MKZIVZLDIXJYFTMPVJE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JB4F5D4MKZIVZLDIXJYFTMPVJE/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-12T04:41:02Z","links":{"resolver":"https://pith.science/pith/JB4F5D4MKZIVZLDIXJYFTMPVJE","bundle":"https://pith.science/pith/JB4F5D4MKZIVZLDIXJYFTMPVJE/bundle.json","state":"https://pith.science/pith/JB4F5D4MKZIVZLDIXJYFTMPVJE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JB4F5D4MKZIVZLDIXJYFTMPVJE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:JB4F5D4MKZIVZLDIXJYFTMPVJE","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"17ead59582ff6a350a32a8f2dbd710350b2d7890c9af383bc530242ea22b4462","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"2007-04-14T12:50:22Z","title_canon_sha256":"9e8d0fbe5cf535fa3f0cdf61186f86ac09a39289fac531edeeaecdda00df1124"},"schema_version":"1.0","source":{"id":"0704.1865","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0704.1865","created_at":"2026-07-04T15:00:53Z"},{"alias_kind":"arxiv_version","alias_value":"0704.1865v1","created_at":"2026-07-04T15:00:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0704.1865","created_at":"2026-07-04T15:00:53Z"},{"alias_kind":"pith_short_12","alias_value":"JB4F5D4MKZIV","created_at":"2026-07-04T15:00:53Z"},{"alias_kind":"pith_short_16","alias_value":"JB4F5D4MKZIVZLDI","created_at":"2026-07-04T15:00:53Z"},{"alias_kind":"pith_short_8","alias_value":"JB4F5D4M","created_at":"2026-07-04T15:00:53Z"}],"graph_snapshots":[{"event_id":"sha256:e5a34d4ebcf32d2418e0175713903e5a1194f65876ed5ed993245f05ce9ef25a","target":"graph","created_at":"2026-07-04T15:00:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0704.1865/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $f\\in L_{2\\pi}$ be a real-valued even function with its Fourier series $ \\frac{a_{0}}{2}+\\sum_{n=1}^{\\infty}a_{n}\\cos nx,$ and let $S_{n}(f,x), n\\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence $\\{a_{n}\\}$ is decreasing and $\\lim\\limits_{n\\to \\infty}a_{n}=0$, then $$ \\lim\\limits_{n\\to \\infty}\\Vert f-S_{n}(f)\\Vert_{L}=0 {if and only if} \\lim\\limits_{n\\to \\infty}a_{n}\\log n=0. $$ We weaken the monotone condition in this classical result to the so-called mean value bounded variation ($MVBV$) condition. The generalization of the above ","authors_text":"Dan Sheng Yu, Ping Zhou, Song Ping Zhou","cross_cats":[],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"2007-04-14T12:50:22Z","title":"On $L^{1}$-Convergence of Fourier Series Under $MVBV$ Condition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.1865","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:42d74a357d1bafe2dd4f9001365e2b0cc3b9d0b5bbd70971a99f690c3de1f577","target":"record","created_at":"2026-07-04T15:00:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"17ead59582ff6a350a32a8f2dbd710350b2d7890c9af383bc530242ea22b4462","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"2007-04-14T12:50:22Z","title_canon_sha256":"9e8d0fbe5cf535fa3f0cdf61186f86ac09a39289fac531edeeaecdda00df1124"},"schema_version":"1.0","source":{"id":"0704.1865","kind":"arxiv","version":1}},"canonical_sha256":"48785e8f8c56515cac68ba7059b1f5490c0c8be6e62fc7a418fce898bb251e8b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48785e8f8c56515cac68ba7059b1f5490c0c8be6e62fc7a418fce898bb251e8b","first_computed_at":"2026-07-04T15:00:53.438000Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:00:53.438000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GPMAAL9DfNEsI4t4Da27YH1DGnI+Leby7+oIF+YxXSkMV8zMHRJGxsdCDUBGgtA03dSkhjONVhCKjw6oQLIeDg==","signature_status":"signed_v1","signed_at":"2026-07-04T15:00:53.438447Z","signed_message":"canonical_sha256_bytes"},"source_id":"0704.1865","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42d74a357d1bafe2dd4f9001365e2b0cc3b9d0b5bbd70971a99f690c3de1f577","sha256:e5a34d4ebcf32d2418e0175713903e5a1194f65876ed5ed993245f05ce9ef25a"],"state_sha256":"5fe181a7993aa06eb79a7e5e06ed2cfde4589ac960da7325cfacfe297e0a2df4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sG3xqcsTjO0ljtfQjKsrBOwe+nuv19NzaIHIQY9pAIwxT1On/swP7fm6hLCHYUYMaDzIueH2leKN6liFOU87AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T04:41:02.520214Z","bundle_sha256":"91730806b1d43c95a97f4ab4a12e77964181e81fe5ee9200cbb7868b6ee6990d"}}