{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:R5CHXQ7452XFIJSHWA3ZD3ZK3H","short_pith_number":"pith:R5CHXQ74","canonical_record":{"source":{"id":"1612.04464","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-12-14T02:49:06Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"f5bb7e3d4ece2260c69fe3f6c395149a4e63f5c33c37b10a9af71ac37e4aa9da","abstract_canon_sha256":"2b31f8a0f8d3ab7b29a19f3fb5418c7be10a407ea3b0d6c753802ec0422f9ff6"},"schema_version":"1.0"},"canonical_sha256":"8f447bc3fceeae542647b03791ef2ad9fd29a88bd486a50453bc0d2d9e3e5fd1","source":{"kind":"arxiv","id":"1612.04464","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.04464","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"arxiv_version","alias_value":"1612.04464v4","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.04464","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"pith_short_12","alias_value":"R5CHXQ7452XF","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"R5CHXQ7452XFIJSH","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"R5CHXQ74","created_at":"2026-05-18T12:30:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:R5CHXQ7452XFIJSHWA3ZD3ZK3H","target":"record","payload":{"canonical_record":{"source":{"id":"1612.04464","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-12-14T02:49:06Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"f5bb7e3d4ece2260c69fe3f6c395149a4e63f5c33c37b10a9af71ac37e4aa9da","abstract_canon_sha256":"2b31f8a0f8d3ab7b29a19f3fb5418c7be10a407ea3b0d6c753802ec0422f9ff6"},"schema_version":"1.0"},"canonical_sha256":"8f447bc3fceeae542647b03791ef2ad9fd29a88bd486a50453bc0d2d9e3e5fd1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:30.688448Z","signature_b64":"8FGdKNUHjIKWkDxd2UhtQCU8byabH30c+eIuaxsHRuR0GahdEAb7Q9VV3FoAbDnJhGEBMS/S0w7eDc/rcJ91DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f447bc3fceeae542647b03791ef2ad9fd29a88bd486a50453bc0d2d9e3e5fd1","last_reissued_at":"2026-05-18T00:01:30.688041Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:30.688041Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.04464","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9CkC+pjoPfN3A5Qt/a8rHEHfqGAoeldZ0ECrUCNKPKkp13aS6ZP2cYC/9I1uBnIiS3TLc9xOsF46Gys091FpAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T04:42:43.918673Z"},"content_sha256":"00ff9a42330650dd37f6b0f1b1d94f6dc7d0545693c7d6c971415c2cc93090d6","schema_version":"1.0","event_id":"sha256:00ff9a42330650dd37f6b0f1b1d94f6dc7d0545693c7d6c971415c2cc93090d6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:R5CHXQ7452XFIJSHWA3ZD3ZK3H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Frames and numerical approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Ben Adcock, Daan Huybrechs","submitted_at":"2016-12-14T02:49:06Z","abstract_excerpt":"Functions of one or more variables are usually approximated with a basis: a complete, linearly-independent system of functions that spans a suitable function space. The topic of this paper is the numerical approximation of functions using the more general notion of frames: that is, complete systems that are generally redundant but provide infinite representations with bounded coefficients. While frames are well-known in image and signal processing, coding theory and other areas of applied mathematics, their use in numerical analysis is far less widespread. Yet, as we show via a series of examp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04464","kind":"arxiv","version":4},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rBi3x8xdCZo/XO/4h2uGyioQTeIEaFYJlYQeQrKTxJg6USyAOIgnScl5cGFtCGgZINrVfAooUNTWkS91k+xCBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T04:42:43.919310Z"},"content_sha256":"f9d5ac0cce4f624e0b51c57fbf628582fc61bd66c2f59f8605a7ed91c367764b","schema_version":"1.0","event_id":"sha256:f9d5ac0cce4f624e0b51c57fbf628582fc61bd66c2f59f8605a7ed91c367764b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R5CHXQ7452XFIJSHWA3ZD3ZK3H/bundle.json","state_url":"https://pith.science/pith/R5CHXQ7452XFIJSHWA3ZD3ZK3H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R5CHXQ7452XFIJSHWA3ZD3ZK3H/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":"2026-06-11T04:42:43Z","links":{"resolver":"https://pith.science/pith/R5CHXQ7452XFIJSHWA3ZD3ZK3H","bundle":"https://pith.science/pith/R5CHXQ7452XFIJSHWA3ZD3ZK3H/bundle.json","state":"https://pith.science/pith/R5CHXQ7452XFIJSHWA3ZD3ZK3H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R5CHXQ7452XFIJSHWA3ZD3ZK3H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:R5CHXQ7452XFIJSHWA3ZD3ZK3H","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":"2b31f8a0f8d3ab7b29a19f3fb5418c7be10a407ea3b0d6c753802ec0422f9ff6","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-12-14T02:49:06Z","title_canon_sha256":"f5bb7e3d4ece2260c69fe3f6c395149a4e63f5c33c37b10a9af71ac37e4aa9da"},"schema_version":"1.0","source":{"id":"1612.04464","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.04464","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"arxiv_version","alias_value":"1612.04464v4","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.04464","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"pith_short_12","alias_value":"R5CHXQ7452XF","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"R5CHXQ7452XFIJSH","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"R5CHXQ74","created_at":"2026-05-18T12:30:41Z"}],"graph_snapshots":[{"event_id":"sha256:f9d5ac0cce4f624e0b51c57fbf628582fc61bd66c2f59f8605a7ed91c367764b","target":"graph","created_at":"2026-05-18T00:01:30Z","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"},"paper":{"abstract_excerpt":"Functions of one or more variables are usually approximated with a basis: a complete, linearly-independent system of functions that spans a suitable function space. The topic of this paper is the numerical approximation of functions using the more general notion of frames: that is, complete systems that are generally redundant but provide infinite representations with bounded coefficients. While frames are well-known in image and signal processing, coding theory and other areas of applied mathematics, their use in numerical analysis is far less widespread. Yet, as we show via a series of examp","authors_text":"Ben Adcock, Daan Huybrechs","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-12-14T02:49:06Z","title":"Frames and numerical approximation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04464","kind":"arxiv","version":4},"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":"sha256:00ff9a42330650dd37f6b0f1b1d94f6dc7d0545693c7d6c971415c2cc93090d6","target":"record","created_at":"2026-05-18T00:01:30Z","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":"2b31f8a0f8d3ab7b29a19f3fb5418c7be10a407ea3b0d6c753802ec0422f9ff6","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-12-14T02:49:06Z","title_canon_sha256":"f5bb7e3d4ece2260c69fe3f6c395149a4e63f5c33c37b10a9af71ac37e4aa9da"},"schema_version":"1.0","source":{"id":"1612.04464","kind":"arxiv","version":4}},"canonical_sha256":"8f447bc3fceeae542647b03791ef2ad9fd29a88bd486a50453bc0d2d9e3e5fd1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8f447bc3fceeae542647b03791ef2ad9fd29a88bd486a50453bc0d2d9e3e5fd1","first_computed_at":"2026-05-18T00:01:30.688041Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:30.688041Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8FGdKNUHjIKWkDxd2UhtQCU8byabH30c+eIuaxsHRuR0GahdEAb7Q9VV3FoAbDnJhGEBMS/S0w7eDc/rcJ91DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:30.688448Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.04464","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:00ff9a42330650dd37f6b0f1b1d94f6dc7d0545693c7d6c971415c2cc93090d6","sha256:f9d5ac0cce4f624e0b51c57fbf628582fc61bd66c2f59f8605a7ed91c367764b"],"state_sha256":"f647a7fd10a9d9894fefd011ce2bf9bfbfd55965c360cc38cc799a3b1f0d795a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x0oThA0uHs9r5DtPXjipZc8T18wK8P39LtFaUZ+a+Fmb3frwCsRd6GjQ2EVnKfF0d/oLL0opyRSTyEliUNPLAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T04:42:43.922999Z","bundle_sha256":"e80400c27cfef0320f830b3395449d5183a8d99ed153290c0d4d69ccbc288716"}}