{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:37UJHEHR3WDXPQJ3KMV52NUBAC","short_pith_number":"pith:37UJHEHR","canonical_record":{"source":{"id":"1701.08226","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-01-28T00:11:14Z","cross_cats_sorted":[],"title_canon_sha256":"938223bb800b84875f047e4b701720b10fb2b67b6c03ee23de0b3b25fd2128fa","abstract_canon_sha256":"8232ee4a8e00f80fba104b38200bc75cce4316053bf846380f49de69d9a548de"},"schema_version":"1.0"},"canonical_sha256":"dfe89390f1dd8777c13b532bdd368100a1a90d685e952babf58d78b1c8c6db62","source":{"kind":"arxiv","id":"1701.08226","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.08226","created_at":"2026-05-18T00:02:50Z"},{"alias_kind":"arxiv_version","alias_value":"1701.08226v2","created_at":"2026-05-18T00:02:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.08226","created_at":"2026-05-18T00:02:50Z"},{"alias_kind":"pith_short_12","alias_value":"37UJHEHR3WDX","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"37UJHEHR3WDXPQJ3","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"37UJHEHR","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:37UJHEHR3WDXPQJ3KMV52NUBAC","target":"record","payload":{"canonical_record":{"source":{"id":"1701.08226","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-01-28T00:11:14Z","cross_cats_sorted":[],"title_canon_sha256":"938223bb800b84875f047e4b701720b10fb2b67b6c03ee23de0b3b25fd2128fa","abstract_canon_sha256":"8232ee4a8e00f80fba104b38200bc75cce4316053bf846380f49de69d9a548de"},"schema_version":"1.0"},"canonical_sha256":"dfe89390f1dd8777c13b532bdd368100a1a90d685e952babf58d78b1c8c6db62","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:50.040250Z","signature_b64":"lf5G3PpfBiBhPul23GT14PoJDraOh6QoU4aBzITUtU+lhwIwUD/qsqgCb8fP80NQ6Ne5w/or8sgQ80f83v57Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfe89390f1dd8777c13b532bdd368100a1a90d685e952babf58d78b1c8c6db62","last_reissued_at":"2026-05-18T00:02:50.039638Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:50.039638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1701.08226","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:02:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ukmO/MxYaN6T+20lmDscA5cdgp19bxqyI19whPYXQokDQtwnZ0TsfyOVfgddhUKDmiAqQtzaVqVE4HpFQQlIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T17:05:52.048163Z"},"content_sha256":"f4b87273b8cb55f480608d25e1d4b62ed251f3b2948cbb3a08a50dc10f539c87","schema_version":"1.0","event_id":"sha256:f4b87273b8cb55f480608d25e1d4b62ed251f3b2948cbb3a08a50dc10f539c87"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:37UJHEHR3WDXPQJ3KMV52NUBAC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximation by crystal-refinable function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alejandro Quintero, Maria del Carmen Moure, Ursula Molter","submitted_at":"2017-01-28T00:11:14Z","abstract_excerpt":"Let $\\Gamma$ be a crystal group in $\\mathbb R^d$. A function $\\varphi:\\mathbb R^d\\longrightarrow \\mathbb C$ is said to be {\\em crystal-refinable} (or $\\Gamma-$refinable) if it is a linear combination of finitely many of the rescaled and translated functions $\\varphi(\\gamma^{-1}(ax))$, where the {\\em translations} $\\gamma$ are taken on a crystal group $\\Gamma$, and $a$ is an expansive dilation matrix such that $a\\Gamma a^{-1}\\subset\\Gamma.$ A $\\Gamma-$refinable function $\\varphi: \\mathbb R^d \\rightarrow \\mathbb C$ satisfies a refinement equation $\\varphi(x)=\\sum_{\\gamma\\in\\Gamma}d_\\gamma \\varph"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08226","kind":"arxiv","version":2},"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":"2026-05-18T00:02:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZOBaAvL+iMy9GYeHZyAiahV/oFB2Ib531Xa3ValSqiY4CPRqX+JuiXUOFX8Qmvyg21JllrNJm8d6Sf2pe6IaCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T17:05:52.048526Z"},"content_sha256":"15d72c80e9f19b1b6f9330f3f19c9ea29265098004c4beb7446f7850e33da78f","schema_version":"1.0","event_id":"sha256:15d72c80e9f19b1b6f9330f3f19c9ea29265098004c4beb7446f7850e33da78f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/37UJHEHR3WDXPQJ3KMV52NUBAC/bundle.json","state_url":"https://pith.science/pith/37UJHEHR3WDXPQJ3KMV52NUBAC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/37UJHEHR3WDXPQJ3KMV52NUBAC/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":"2026-05-23T17:05:52Z","links":{"resolver":"https://pith.science/pith/37UJHEHR3WDXPQJ3KMV52NUBAC","bundle":"https://pith.science/pith/37UJHEHR3WDXPQJ3KMV52NUBAC/bundle.json","state":"https://pith.science/pith/37UJHEHR3WDXPQJ3KMV52NUBAC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/37UJHEHR3WDXPQJ3KMV52NUBAC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:37UJHEHR3WDXPQJ3KMV52NUBAC","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":"8232ee4a8e00f80fba104b38200bc75cce4316053bf846380f49de69d9a548de","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-01-28T00:11:14Z","title_canon_sha256":"938223bb800b84875f047e4b701720b10fb2b67b6c03ee23de0b3b25fd2128fa"},"schema_version":"1.0","source":{"id":"1701.08226","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.08226","created_at":"2026-05-18T00:02:50Z"},{"alias_kind":"arxiv_version","alias_value":"1701.08226v2","created_at":"2026-05-18T00:02:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.08226","created_at":"2026-05-18T00:02:50Z"},{"alias_kind":"pith_short_12","alias_value":"37UJHEHR3WDX","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"37UJHEHR3WDXPQJ3","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"37UJHEHR","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:15d72c80e9f19b1b6f9330f3f19c9ea29265098004c4beb7446f7850e33da78f","target":"graph","created_at":"2026-05-18T00:02:50Z","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":"Let $\\Gamma$ be a crystal group in $\\mathbb R^d$. A function $\\varphi:\\mathbb R^d\\longrightarrow \\mathbb C$ is said to be {\\em crystal-refinable} (or $\\Gamma-$refinable) if it is a linear combination of finitely many of the rescaled and translated functions $\\varphi(\\gamma^{-1}(ax))$, where the {\\em translations} $\\gamma$ are taken on a crystal group $\\Gamma$, and $a$ is an expansive dilation matrix such that $a\\Gamma a^{-1}\\subset\\Gamma.$ A $\\Gamma-$refinable function $\\varphi: \\mathbb R^d \\rightarrow \\mathbb C$ satisfies a refinement equation $\\varphi(x)=\\sum_{\\gamma\\in\\Gamma}d_\\gamma \\varph","authors_text":"Alejandro Quintero, Maria del Carmen Moure, Ursula Molter","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-01-28T00:11:14Z","title":"Approximation by crystal-refinable function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08226","kind":"arxiv","version":2},"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:f4b87273b8cb55f480608d25e1d4b62ed251f3b2948cbb3a08a50dc10f539c87","target":"record","created_at":"2026-05-18T00:02:50Z","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":"8232ee4a8e00f80fba104b38200bc75cce4316053bf846380f49de69d9a548de","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-01-28T00:11:14Z","title_canon_sha256":"938223bb800b84875f047e4b701720b10fb2b67b6c03ee23de0b3b25fd2128fa"},"schema_version":"1.0","source":{"id":"1701.08226","kind":"arxiv","version":2}},"canonical_sha256":"dfe89390f1dd8777c13b532bdd368100a1a90d685e952babf58d78b1c8c6db62","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dfe89390f1dd8777c13b532bdd368100a1a90d685e952babf58d78b1c8c6db62","first_computed_at":"2026-05-18T00:02:50.039638Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:02:50.039638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lf5G3PpfBiBhPul23GT14PoJDraOh6QoU4aBzITUtU+lhwIwUD/qsqgCb8fP80NQ6Ne5w/or8sgQ80f83v57Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:02:50.040250Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.08226","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f4b87273b8cb55f480608d25e1d4b62ed251f3b2948cbb3a08a50dc10f539c87","sha256:15d72c80e9f19b1b6f9330f3f19c9ea29265098004c4beb7446f7850e33da78f"],"state_sha256":"449727b5ab443361ea4231074f985fa9276bb760dfbc05246320ca66d11a58f4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CTqbzx+/2aMqK8Cj7j2O1lVoYoTWxTEmcSNCIR6V0bHJPLuIiXEl9kxOiuBhm+PFscXKfz0NcpmQDy4RCjV7DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T17:05:52.050542Z","bundle_sha256":"1efa471f76d6c21fed29411bc6291ec7a5556c55e00ffd4308cbd84a921ba5d0"}}