{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:TEHR24YUBRDTIWP6AEDESQN576","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":"8393c23c1f9505c1f5db4bfe79aec9c7a9fd1b97294e7c10ae42cb7da34726c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-06-01T05:50:11Z","title_canon_sha256":"2f4dc7bb2006ecbe0c6db085a4a7add7bf9da421dde8e0bf18dbd672266977e8"},"schema_version":"1.0","source":{"id":"1606.00123","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.00123","created_at":"2026-05-18T00:03:37Z"},{"alias_kind":"arxiv_version","alias_value":"1606.00123v3","created_at":"2026-05-18T00:03:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.00123","created_at":"2026-05-18T00:03:37Z"},{"alias_kind":"pith_short_12","alias_value":"TEHR24YUBRDT","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"TEHR24YUBRDTIWP6","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"TEHR24YU","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:1187f541956aa9c623528453a243b2b0255383f1112927ecba1b4514e9e96267","target":"graph","created_at":"2026-05-18T00:03:37Z","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":"Given $\\beta>0$ and $\\delta>0$, the function $t^{-\\beta}$ may be approximated for $t$ in a compact interval $[\\delta,T]$ by a sum of terms of the form $we^{-at}$, with parameters $w>0$ and $a>0$. One such an approximation, studied by Beylkin and Monz\\'on, is obtained by applying the trapezoidal rule to an integral representation of $t^{-\\beta}$, after which Prony's method is applied to reduce the number of terms in the sum with essentially no loss of accuracy. We review this method, and then describe a similar approach based on an alternative integral representation. The main difference is tha","authors_text":"William McLean","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-06-01T05:50:11Z","title":"Exponential sum approximations for $t^{-\\beta}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.00123","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":"sha256:5c5de0fef7d3990b53d5216736b4d7eb9421c99646d33aa7f425826bf437bd56","target":"record","created_at":"2026-05-18T00:03:37Z","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":"8393c23c1f9505c1f5db4bfe79aec9c7a9fd1b97294e7c10ae42cb7da34726c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-06-01T05:50:11Z","title_canon_sha256":"2f4dc7bb2006ecbe0c6db085a4a7add7bf9da421dde8e0bf18dbd672266977e8"},"schema_version":"1.0","source":{"id":"1606.00123","kind":"arxiv","version":3}},"canonical_sha256":"990f1d73140c473459fe01064941bdff8cb4a0833a1fd0be01319d05b72d700f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"990f1d73140c473459fe01064941bdff8cb4a0833a1fd0be01319d05b72d700f","first_computed_at":"2026-05-18T00:03:37.499672Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:03:37.499672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yzH601ua0pYONmrDZGGT0qvj6R849I2FwmBO6RY9+WjQlC+bW5h/em21EMVFm/gHwo7fMAO7yuSahLCNug3GDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:03:37.500441Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.00123","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c5de0fef7d3990b53d5216736b4d7eb9421c99646d33aa7f425826bf437bd56","sha256:1187f541956aa9c623528453a243b2b0255383f1112927ecba1b4514e9e96267"],"state_sha256":"69b21841ea09be4e378e0ee22cef595086910276b15eb784160106663a884dec"}