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Thus, $d_N(\\mathcal{M})$ sets a limit to any projection-based approximation such as determined by the reduced basis method. While it is known that $d_N(\\mathcal{M})$ decays exponentially fast for many linear coercive parametrized partial differential equations, i.e., $d_N(\\mathcal{M})=\\mathcal{O}(e^{-\\beta N})$, we show in this note, that only "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1903.08488","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-03-20T12:54:02Z","cross_cats_sorted":[],"title_canon_sha256":"484dd1a8f5bd6655125ecb2ff2f8ef8bf2a29d3f212502a10da341f0c64e9dc4","abstract_canon_sha256":"193a7c3ba0d154bac1f8fe86914c9cd6978798b718404d256304c83da376b804"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:54.819583Z","signature_b64":"CWziWjCCVq6PcLiDTgV5H1zNce9hnt6N/9h+r3I8T52/34Mpc3HbqmzECl09FHav5hpkduRSI55pNCe/Uws6AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4713425a009db3c50b9f0c4ff096ac062b36421f9db7b62c95046fbe1a09222","last_reissued_at":"2026-05-17T23:48:54.819130Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:54.819130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decay of the Kolmogorov $N$-width for wave problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Constantin Greif, Karsten Urban","submitted_at":"2019-03-20T12:54:02Z","abstract_excerpt":"The Kolmogorov $N$-width $d_N(\\mathcal{M})$ describes the rate of the worst-case error (w.r.t.\\ a subset $\\mathcal{M}\\subset H$ of a normed space $H$) arising from a projection onto the best-possible linear subspace of $H$ of dimension $N\\in\\mathbb{N}$. Thus, $d_N(\\mathcal{M})$ sets a limit to any projection-based approximation such as determined by the reduced basis method. While it is known that $d_N(\\mathcal{M})$ decays exponentially fast for many linear coercive parametrized partial differential equations, i.e., $d_N(\\mathcal{M})=\\mathcal{O}(e^{-\\beta N})$, we show in this note, that only "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.08488","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1903.08488","created_at":"2026-05-17T23:48:54.819202+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.08488v2","created_at":"2026-05-17T23:48:54.819202+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.08488","created_at":"2026-05-17T23:48:54.819202+00:00"},{"alias_kind":"pith_short_12","alias_value":"WRYTIJNABHNT","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_16","alias_value":"WRYTIJNABHNTYUFZ","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_8","alias_value":"WRYTIJNA","created_at":"2026-05-18T12:33:30.264802+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WRYTIJNABHNTYUFZ6DCP6CLKYB","json":"https://pith.science/pith/WRYTIJNABHNTYUFZ6DCP6CLKYB.json","graph_json":"https://pith.science/api/pith-number/WRYTIJNABHNTYUFZ6DCP6CLKYB/graph.json","events_json":"https://pith.science/api/pith-number/WRYTIJNABHNTYUFZ6DCP6CLKYB/events.json","paper":"https://pith.science/paper/WRYTIJNA"},"agent_actions":{"view_html":"https://pith.science/pith/WRYTIJNABHNTYUFZ6DCP6CLKYB","download_json":"https://pith.science/pith/WRYTIJNABHNTYUFZ6DCP6CLKYB.json","view_paper":"https://pith.science/paper/WRYTIJNA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.08488&json=true","fetch_graph":"https://pith.science/api/pith-number/WRYTIJNABHNTYUFZ6DCP6CLKYB/graph.json","fetch_events":"https://pith.science/api/pith-number/WRYTIJNABHNTYUFZ6DCP6CLKYB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WRYTIJNABHNTYUFZ6DCP6CLKYB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WRYTIJNABHNTYUFZ6DCP6CLKYB/action/storage_attestation","attest_author":"https://pith.science/pith/WRYTIJNABHNTYUFZ6DCP6CLKYB/action/author_attestation","sign_citation":"https://pith.science/pith/WRYTIJNABHNTYUFZ6DCP6CLKYB/action/citation_signature","submit_replication":"https://pith.science/pith/WRYTIJNABHNTYUFZ6DCP6CLKYB/action/replication_record"}},"created_at":"2026-05-17T23:48:54.819202+00:00","updated_at":"2026-05-17T23:48:54.819202+00:00"}