{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JTME4UZF6XJXO5LOV32QSLTXTA","short_pith_number":"pith:JTME4UZF","canonical_record":{"source":{"id":"1505.00631","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-05-04T13:41:05Z","cross_cats_sorted":[],"title_canon_sha256":"95b3c0b92e91a952c33a099577d56bb28dd27d0251d8ce376756771131b17af2","abstract_canon_sha256":"bacce7235b561dd9665b128e19e91c40efc789309bc6c2988331b0289caeeacf"},"schema_version":"1.0"},"canonical_sha256":"4cd84e5325f5d377756eaef5092e7798300d8a072af3e32993a25846cfbf7a6f","source":{"kind":"arxiv","id":"1505.00631","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.00631","created_at":"2026-05-18T01:04:50Z"},{"alias_kind":"arxiv_version","alias_value":"1505.00631v3","created_at":"2026-05-18T01:04:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.00631","created_at":"2026-05-18T01:04:50Z"},{"alias_kind":"pith_short_12","alias_value":"JTME4UZF6XJX","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JTME4UZF6XJXO5LO","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JTME4UZF","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JTME4UZF6XJXO5LOV32QSLTXTA","target":"record","payload":{"canonical_record":{"source":{"id":"1505.00631","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-05-04T13:41:05Z","cross_cats_sorted":[],"title_canon_sha256":"95b3c0b92e91a952c33a099577d56bb28dd27d0251d8ce376756771131b17af2","abstract_canon_sha256":"bacce7235b561dd9665b128e19e91c40efc789309bc6c2988331b0289caeeacf"},"schema_version":"1.0"},"canonical_sha256":"4cd84e5325f5d377756eaef5092e7798300d8a072af3e32993a25846cfbf7a6f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:50.604206Z","signature_b64":"gNXkoPEyLgLQM+UEAI/5WTibD3nii4Jc9iB+WgHQFFVmeofgPdQdbGS68hHPlfK+NC9yB7x4DMYBK7p7pnlhAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4cd84e5325f5d377756eaef5092e7798300d8a072af3e32993a25846cfbf7a6f","last_reissued_at":"2026-05-18T01:04:50.603825Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:50.603825Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.00631","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:04:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XdevaorY86CDE/+xjQPcLpgj3UQDypOadVhgi3S393SFs12XkgkXyFs2NEFvC0Ent1qbzTxru7rkrsTjQbW2DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T12:34:15.857732Z"},"content_sha256":"e786683244b22144e2b97d7313314495971271f536b1295ae56d4452917643a8","schema_version":"1.0","event_id":"sha256:e786683244b22144e2b97d7313314495971271f536b1295ae56d4452917643a8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JTME4UZF6XJXO5LOV32QSLTXTA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting via entropy: new preasymptotics for the approximation numbers of Sobolev embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Sebastian Mayer, Thomas K\\\"uhn, Tino Ullrich","submitted_at":"2015-05-04T13:41:05Z","abstract_excerpt":"In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm $\\|\\cdot\\|$ on $\\mathbb{R}^d$, and entropy numbers of the embedding $\\textrm{id}: \\ell_{\\|\\cdot\\|}^d \\to \\ell_\\infty^d$. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in $L_2$ and $H^1$, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.00631","kind":"arxiv","version":3},"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":"2026-05-18T01:04:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KT13acSh/7BtRTIV1B5fyzn8upF0G58RLeTW1Fz9HMKaxe4YqDi0CiTgvrNhPgeMJ3qkTKATTJ1Wmf9jCmybBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T12:34:15.858609Z"},"content_sha256":"96500f25cef5f7129b107c795037977c6ecfdcc451756bff79dd1e1a0341e0cf","schema_version":"1.0","event_id":"sha256:96500f25cef5f7129b107c795037977c6ecfdcc451756bff79dd1e1a0341e0cf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JTME4UZF6XJXO5LOV32QSLTXTA/bundle.json","state_url":"https://pith.science/pith/JTME4UZF6XJXO5LOV32QSLTXTA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JTME4UZF6XJXO5LOV32QSLTXTA/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":"2026-05-31T12:34:15Z","links":{"resolver":"https://pith.science/pith/JTME4UZF6XJXO5LOV32QSLTXTA","bundle":"https://pith.science/pith/JTME4UZF6XJXO5LOV32QSLTXTA/bundle.json","state":"https://pith.science/pith/JTME4UZF6XJXO5LOV32QSLTXTA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JTME4UZF6XJXO5LOV32QSLTXTA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JTME4UZF6XJXO5LOV32QSLTXTA","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":"bacce7235b561dd9665b128e19e91c40efc789309bc6c2988331b0289caeeacf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-05-04T13:41:05Z","title_canon_sha256":"95b3c0b92e91a952c33a099577d56bb28dd27d0251d8ce376756771131b17af2"},"schema_version":"1.0","source":{"id":"1505.00631","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.00631","created_at":"2026-05-18T01:04:50Z"},{"alias_kind":"arxiv_version","alias_value":"1505.00631v3","created_at":"2026-05-18T01:04:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.00631","created_at":"2026-05-18T01:04:50Z"},{"alias_kind":"pith_short_12","alias_value":"JTME4UZF6XJX","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JTME4UZF6XJXO5LO","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JTME4UZF","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:96500f25cef5f7129b107c795037977c6ecfdcc451756bff79dd1e1a0341e0cf","target":"graph","created_at":"2026-05-18T01:04:50Z","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":"In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm $\\|\\cdot\\|$ on $\\mathbb{R}^d$, and entropy numbers of the embedding $\\textrm{id}: \\ell_{\\|\\cdot\\|}^d \\to \\ell_\\infty^d$. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in $L_2$ and $H^1$, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers","authors_text":"Sebastian Mayer, Thomas K\\\"uhn, Tino Ullrich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-05-04T13:41:05Z","title":"Counting via entropy: new preasymptotics for the approximation numbers of Sobolev embeddings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.00631","kind":"arxiv","version":3},"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:e786683244b22144e2b97d7313314495971271f536b1295ae56d4452917643a8","target":"record","created_at":"2026-05-18T01:04:50Z","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":"bacce7235b561dd9665b128e19e91c40efc789309bc6c2988331b0289caeeacf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-05-04T13:41:05Z","title_canon_sha256":"95b3c0b92e91a952c33a099577d56bb28dd27d0251d8ce376756771131b17af2"},"schema_version":"1.0","source":{"id":"1505.00631","kind":"arxiv","version":3}},"canonical_sha256":"4cd84e5325f5d377756eaef5092e7798300d8a072af3e32993a25846cfbf7a6f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4cd84e5325f5d377756eaef5092e7798300d8a072af3e32993a25846cfbf7a6f","first_computed_at":"2026-05-18T01:04:50.603825Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:50.603825Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gNXkoPEyLgLQM+UEAI/5WTibD3nii4Jc9iB+WgHQFFVmeofgPdQdbGS68hHPlfK+NC9yB7x4DMYBK7p7pnlhAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:50.604206Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.00631","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e786683244b22144e2b97d7313314495971271f536b1295ae56d4452917643a8","sha256:96500f25cef5f7129b107c795037977c6ecfdcc451756bff79dd1e1a0341e0cf"],"state_sha256":"cabcbadfc2e2bbb82f30874950362dd6840e83304e24a6ec6d760b05423418fc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QjjCsnAwNz6BOKIxjygoP8GdxI9z8kGNxVYceX2aFJEoF3YneuB6Ml5H4LK3T1q8/5n+1EQ3RZNM8yrPkE2eDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T12:34:15.864269Z","bundle_sha256":"307af6596ec909e33b48deed0501244ab2a71fcc70c34700d2b95559652b62b3"}}