{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:6WDZI73EINQ6LGPMUIPYUZJKNS","short_pith_number":"pith:6WDZI73E","canonical_record":{"source":{"id":"2003.13113","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2020-03-29T19:07:35Z","cross_cats_sorted":[],"title_canon_sha256":"2b09c8092b6c6c15e6ce02e0bcda3a177cb261d4df978df90dda1f29141ba92e","abstract_canon_sha256":"09d658e055028eb1b4cc9e85c940a162b1e3d36183883eb5b8215330745992af"},"schema_version":"1.0"},"canonical_sha256":"f587947f644361e599eca21f8a652a6c9b6b53297d2a7191c806965acb93f37c","source":{"kind":"arxiv","id":"2003.13113","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2003.13113","created_at":"2026-07-05T02:12:40Z"},{"alias_kind":"arxiv_version","alias_value":"2003.13113v4","created_at":"2026-07-05T02:12:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2003.13113","created_at":"2026-07-05T02:12:40Z"},{"alias_kind":"pith_short_12","alias_value":"6WDZI73EINQ6","created_at":"2026-07-05T02:12:40Z"},{"alias_kind":"pith_short_16","alias_value":"6WDZI73EINQ6LGPM","created_at":"2026-07-05T02:12:40Z"},{"alias_kind":"pith_short_8","alias_value":"6WDZI73E","created_at":"2026-07-05T02:12:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:6WDZI73EINQ6LGPMUIPYUZJKNS","target":"record","payload":{"canonical_record":{"source":{"id":"2003.13113","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2020-03-29T19:07:35Z","cross_cats_sorted":[],"title_canon_sha256":"2b09c8092b6c6c15e6ce02e0bcda3a177cb261d4df978df90dda1f29141ba92e","abstract_canon_sha256":"09d658e055028eb1b4cc9e85c940a162b1e3d36183883eb5b8215330745992af"},"schema_version":"1.0"},"canonical_sha256":"f587947f644361e599eca21f8a652a6c9b6b53297d2a7191c806965acb93f37c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:12:40.944078Z","signature_b64":"q2dD3h05Pv8a8WHzhosllnrD+VQZCqRVsyGEldQs6kwZv2CGEVzL4IIo0nI0cKgrXKhLLmp3hrtLlTDBk0dQBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f587947f644361e599eca21f8a652a6c9b6b53297d2a7191c806965acb93f37c","last_reissued_at":"2026-07-05T02:12:40.943732Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:12:40.943732Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2003.13113","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":"2026-07-05T02:12:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6LzL5n96w93qX4pjnb2B8KexSLE2TTsKsYQBGOyIrUlkADWl42bufLoQzDSeHndC42zPnj8f7lhLXaOxD8DaCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T03:15:39.199686Z"},"content_sha256":"3cfcbbbfb65122237452606091eb3acca674da7032227f1994cf3ba88b639f64","schema_version":"1.0","event_id":"sha256:3cfcbbbfb65122237452606091eb3acca674da7032227f1994cf3ba88b639f64"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:6WDZI73EINQ6LGPMUIPYUZJKNS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discrete Frames For $L^2({\\mathbb R}^{n^2})$ Arising From Tiling Systems On ${\\rm GL}_n({\\mathbb R})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Kris Hollingsworth, Mahya Ghandehari","submitted_at":"2020-03-29T19:07:35Z","abstract_excerpt":"A discrete frame for $L^2({\\mathbb R}^d)$ is a countable sequence $\\{e_j\\}_{j\\in J}$ in $L^2({\\mathbb R}^d)$ together with real constants $0<A\\leq B< \\infty$ such that $$ A\\|f\\|_2^2 \\leq \\sum_{j\\in J}|\\langle f,e_j \\rangle |^2 \\leq B\\|f\\|_2^2,$$ for all $f\\in L^2(\\mathbb{R}^d)$. We present a method of sampling continuous frames, which arise from square-integrable representations of affine-type groups, to create discrete frames for high-dimensional signals. Our method relies on partitioning the ambient space by using a suitable \"tiling system\". We provide all relevant details for constructions "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2003.13113","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2003.13113/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":"2026-07-05T02:12:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"POGvvQcotv4b3q+Msnr8/IW+Du6tkSeLfP+W/dDDrposw3RWJT2VJ78vkYBpPR6IhuWTOSEAkGzahchgeASzDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T03:15:39.200201Z"},"content_sha256":"f89cd63405b44e82b9a17ce561cec4da3f4c992a2c6f5c8693d6f88a023a4b18","schema_version":"1.0","event_id":"sha256:f89cd63405b44e82b9a17ce561cec4da3f4c992a2c6f5c8693d6f88a023a4b18"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6WDZI73EINQ6LGPMUIPYUZJKNS/bundle.json","state_url":"https://pith.science/pith/6WDZI73EINQ6LGPMUIPYUZJKNS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6WDZI73EINQ6LGPMUIPYUZJKNS/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":"2026-08-11T03:15:39Z","links":{"resolver":"https://pith.science/pith/6WDZI73EINQ6LGPMUIPYUZJKNS","bundle":"https://pith.science/pith/6WDZI73EINQ6LGPMUIPYUZJKNS/bundle.json","state":"https://pith.science/pith/6WDZI73EINQ6LGPMUIPYUZJKNS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6WDZI73EINQ6LGPMUIPYUZJKNS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:6WDZI73EINQ6LGPMUIPYUZJKNS","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":"09d658e055028eb1b4cc9e85c940a162b1e3d36183883eb5b8215330745992af","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2020-03-29T19:07:35Z","title_canon_sha256":"2b09c8092b6c6c15e6ce02e0bcda3a177cb261d4df978df90dda1f29141ba92e"},"schema_version":"1.0","source":{"id":"2003.13113","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2003.13113","created_at":"2026-07-05T02:12:40Z"},{"alias_kind":"arxiv_version","alias_value":"2003.13113v4","created_at":"2026-07-05T02:12:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2003.13113","created_at":"2026-07-05T02:12:40Z"},{"alias_kind":"pith_short_12","alias_value":"6WDZI73EINQ6","created_at":"2026-07-05T02:12:40Z"},{"alias_kind":"pith_short_16","alias_value":"6WDZI73EINQ6LGPM","created_at":"2026-07-05T02:12:40Z"},{"alias_kind":"pith_short_8","alias_value":"6WDZI73E","created_at":"2026-07-05T02:12:40Z"}],"graph_snapshots":[{"event_id":"sha256:f89cd63405b44e82b9a17ce561cec4da3f4c992a2c6f5c8693d6f88a023a4b18","target":"graph","created_at":"2026-07-05T02:12:40Z","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/2003.13113/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A discrete frame for $L^2({\\mathbb R}^d)$ is a countable sequence $\\{e_j\\}_{j\\in J}$ in $L^2({\\mathbb R}^d)$ together with real constants $0<A\\leq B< \\infty$ such that $$ A\\|f\\|_2^2 \\leq \\sum_{j\\in J}|\\langle f,e_j \\rangle |^2 \\leq B\\|f\\|_2^2,$$ for all $f\\in L^2(\\mathbb{R}^d)$. We present a method of sampling continuous frames, which arise from square-integrable representations of affine-type groups, to create discrete frames for high-dimensional signals. Our method relies on partitioning the ambient space by using a suitable \"tiling system\". We provide all relevant details for constructions ","authors_text":"Kris Hollingsworth, Mahya Ghandehari","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2020-03-29T19:07:35Z","title":"Discrete Frames For $L^2({\\mathbb R}^{n^2})$ Arising From Tiling Systems On ${\\rm GL}_n({\\mathbb R})$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2003.13113","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":"sha256:3cfcbbbfb65122237452606091eb3acca674da7032227f1994cf3ba88b639f64","target":"record","created_at":"2026-07-05T02:12:40Z","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":"09d658e055028eb1b4cc9e85c940a162b1e3d36183883eb5b8215330745992af","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2020-03-29T19:07:35Z","title_canon_sha256":"2b09c8092b6c6c15e6ce02e0bcda3a177cb261d4df978df90dda1f29141ba92e"},"schema_version":"1.0","source":{"id":"2003.13113","kind":"arxiv","version":4}},"canonical_sha256":"f587947f644361e599eca21f8a652a6c9b6b53297d2a7191c806965acb93f37c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f587947f644361e599eca21f8a652a6c9b6b53297d2a7191c806965acb93f37c","first_computed_at":"2026-07-05T02:12:40.943732Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:12:40.943732Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q2dD3h05Pv8a8WHzhosllnrD+VQZCqRVsyGEldQs6kwZv2CGEVzL4IIo0nI0cKgrXKhLLmp3hrtLlTDBk0dQBg==","signature_status":"signed_v1","signed_at":"2026-07-05T02:12:40.944078Z","signed_message":"canonical_sha256_bytes"},"source_id":"2003.13113","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3cfcbbbfb65122237452606091eb3acca674da7032227f1994cf3ba88b639f64","sha256:f89cd63405b44e82b9a17ce561cec4da3f4c992a2c6f5c8693d6f88a023a4b18"],"state_sha256":"07c1a9d3a9f12f470af350fe2fabd1292695f41ff91f96187c8cf6b995786a78"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6HAIys+hvFUaIoyyNCrAoLgL1gGSWMiWSOPEi36ZZ/OJjtrOLZUprJ+Qtu1L0BlgBI7SYeDB6M6Q0IXDnUoSBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T03:15:39.205396Z","bundle_sha256":"48c8c69285b247693d663ef5daef956657288c6361164bc193e670fb5939bb56"}}