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Given such a sequence and a parameter $0<h\\leq1$, we consider interpolation of functions $g\\in W_2^k(\\mathbb{R})$ at the set $(hx_j)_{j\\in\\mathbb{Z}}$ via translates of the Gaussian kernel. Existence is shown of an interpolant of the form $$I^{hX}(g)(x):=\\underset{j\\in\\mathbb{Z}}{\\sum}a_je^{-(x-hx_j)^2},\\quad x\\in\\mathbb{R},$$ which is continuous and square-"},"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":"1310.2892","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-10-10T17:20:59Z","cross_cats_sorted":[],"title_canon_sha256":"62c399077c5ff76fbd4e814ca05c13ee7d7cc49879aaa9b22eb71bb0457b1f5c","abstract_canon_sha256":"1fd0627de1483e57ddd67a752e20c504b295546e198cdc056282f2fd5d870788"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:29.984833Z","signature_b64":"Wdois/bbtgZXmbqtBYNIH4XLoLwgTX9PneWJ37Vhx1C7QXjJis8UHKEmy0yc1zgAFz+F68RRTHOpg4wFsJYcAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b25f6edefe5f13db8cdf83ebb928948616fd5700b5deb87adfb6a4f6bf98e276","last_reissued_at":"2026-05-18T01:23:29.984288Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:29.984288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Approximation Rates for Interpolation of Sobolev Functions via Gaussians and Allied Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Keaton Hamm","submitted_at":"2013-10-10T17:20:59Z","abstract_excerpt":"A \\Riesz-basis sequence for $L_2[-\\pi,\\pi]$ is a strictly increasing sequence $X:=(x_j)_{j\\in\\mathbb{Z}}$ in $\\mathbb{R}$ such that the set of functions $\\left(e^{-ix_j(\\cdot)}\\right)_{j\\in\\mathbb{Z}}$ is a Riesz basis for $L_2[-\\pi,\\pi]$. Given such a sequence and a parameter $0<h\\leq1$, we consider interpolation of functions $g\\in W_2^k(\\mathbb{R})$ at the set $(hx_j)_{j\\in\\mathbb{Z}}$ via translates of the Gaussian kernel. Existence is shown of an interpolant of the form $$I^{hX}(g)(x):=\\underset{j\\in\\mathbb{Z}}{\\sum}a_je^{-(x-hx_j)^2},\\quad x\\in\\mathbb{R},$$ which is continuous and square-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2892","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1310.2892","created_at":"2026-05-18T01:23:29.984385+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.2892v3","created_at":"2026-05-18T01:23:29.984385+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.2892","created_at":"2026-05-18T01:23:29.984385+00:00"},{"alias_kind":"pith_short_12","alias_value":"WJPW5XX6L4J5","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"WJPW5XX6L4J5XDG7","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"WJPW5XX6","created_at":"2026-05-18T12:28:04.890932+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/WJPW5XX6L4J5XDG7QPV3SKEUQY","json":"https://pith.science/pith/WJPW5XX6L4J5XDG7QPV3SKEUQY.json","graph_json":"https://pith.science/api/pith-number/WJPW5XX6L4J5XDG7QPV3SKEUQY/graph.json","events_json":"https://pith.science/api/pith-number/WJPW5XX6L4J5XDG7QPV3SKEUQY/events.json","paper":"https://pith.science/paper/WJPW5XX6"},"agent_actions":{"view_html":"https://pith.science/pith/WJPW5XX6L4J5XDG7QPV3SKEUQY","download_json":"https://pith.science/pith/WJPW5XX6L4J5XDG7QPV3SKEUQY.json","view_paper":"https://pith.science/paper/WJPW5XX6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.2892&json=true","fetch_graph":"https://pith.science/api/pith-number/WJPW5XX6L4J5XDG7QPV3SKEUQY/graph.json","fetch_events":"https://pith.science/api/pith-number/WJPW5XX6L4J5XDG7QPV3SKEUQY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WJPW5XX6L4J5XDG7QPV3SKEUQY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WJPW5XX6L4J5XDG7QPV3SKEUQY/action/storage_attestation","attest_author":"https://pith.science/pith/WJPW5XX6L4J5XDG7QPV3SKEUQY/action/author_attestation","sign_citation":"https://pith.science/pith/WJPW5XX6L4J5XDG7QPV3SKEUQY/action/citation_signature","submit_replication":"https://pith.science/pith/WJPW5XX6L4J5XDG7QPV3SKEUQY/action/replication_record"}},"created_at":"2026-05-18T01:23:29.984385+00:00","updated_at":"2026-05-18T01:23:29.984385+00:00"}