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For each $s\\geq 0$ we put \\begin{eqnarray*} p(s)=-\\log \\int_{|t|\\geq s}|\\phi (t)|dt. \\end{eqnarray*} If $\\phi$ satisfies \\begin{equation} \\lim_{s\\to \\infty}\\frac{p(s)}{s}=\\infty ,\\label{170506.1} \\end{equation} we obtain asymptotic estimates of the size of small-valued sets $B_{\\epsilon}=\\{x\\in\\RR : |\\hat{\\phi}(x)|\\leq \\epsilon, |x|\\leq R_{\\epsilon}\\}$ of Fourier transform \\begin{eqnarray*} \\hat{\\phi}(x)=\\int_{-\\infty}^{\\infty}e^{-ixt}\\phi (t)dt, x\\in \\RR, \\end{eqnarray*} in terms of $p(s)$ or in terms of its Young dual function \\begin{eqnarra"},"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":"math/0610046","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.NA","submitted_at":"2006-10-01T14:47:05Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"668a455b05eacfb3a6505fff01b5b70bb26887ccc04aa1f6902b00e502bd7ac3","abstract_canon_sha256":"fe34c3ac47ab42ffcfea7ed5151a218a2f89d3870ea21e2b38dd15e72fc1dd53"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T22:06:20.671725Z","signature_b64":"vQKLgLSIWnwWOu9vetgCNK4Q3Fozq/UZo0KM5aNAiCIn5nc+4q3OJh4pqE3u3jmsUw4inI15SJK+FoRJuLpEDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3e8fb28d0bbdd83a26f45f2fab70a9b6a34e52bc92ee728cf6ec4464335022c","last_reissued_at":"2026-06-03T22:06:20.671386Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T22:06:20.671386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Error of Tikhonov's regularization for integral convolution equations","license":"","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Dang Duc Trong, Truong Trung Tuyen","submitted_at":"2006-10-01T14:47:05Z","abstract_excerpt":"Let $\\phi$ be a nontrivial function of $L^1(\\RR)$. For each $s\\geq 0$ we put \\begin{eqnarray*} p(s)=-\\log \\int_{|t|\\geq s}|\\phi (t)|dt. \\end{eqnarray*} If $\\phi$ satisfies \\begin{equation} \\lim_{s\\to \\infty}\\frac{p(s)}{s}=\\infty ,\\label{170506.1} \\end{equation} we obtain asymptotic estimates of the size of small-valued sets $B_{\\epsilon}=\\{x\\in\\RR : |\\hat{\\phi}(x)|\\leq \\epsilon, |x|\\leq R_{\\epsilon}\\}$ of Fourier transform \\begin{eqnarray*} \\hat{\\phi}(x)=\\int_{-\\infty}^{\\infty}e^{-ixt}\\phi (t)dt, x\\in \\RR, \\end{eqnarray*} in terms of $p(s)$ or in terms of its Young dual function \\begin{eqnarra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610046","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0610046/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0610046","created_at":"2026-06-03T22:06:20.671447+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0610046v2","created_at":"2026-06-03T22:06:20.671447+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610046","created_at":"2026-06-03T22:06:20.671447+00:00"},{"alias_kind":"pith_short_12","alias_value":"6PUPWKGQXPOY","created_at":"2026-06-03T22:06:20.671447+00:00"},{"alias_kind":"pith_short_16","alias_value":"6PUPWKGQXPOYHITP","created_at":"2026-06-03T22:06:20.671447+00:00"},{"alias_kind":"pith_short_8","alias_value":"6PUPWKGQ","created_at":"2026-06-03T22:06:20.671447+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/6PUPWKGQXPOYHITPIXZPVNYKTN","json":"https://pith.science/pith/6PUPWKGQXPOYHITPIXZPVNYKTN.json","graph_json":"https://pith.science/api/pith-number/6PUPWKGQXPOYHITPIXZPVNYKTN/graph.json","events_json":"https://pith.science/api/pith-number/6PUPWKGQXPOYHITPIXZPVNYKTN/events.json","paper":"https://pith.science/paper/6PUPWKGQ"},"agent_actions":{"view_html":"https://pith.science/pith/6PUPWKGQXPOYHITPIXZPVNYKTN","download_json":"https://pith.science/pith/6PUPWKGQXPOYHITPIXZPVNYKTN.json","view_paper":"https://pith.science/paper/6PUPWKGQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0610046&json=true","fetch_graph":"https://pith.science/api/pith-number/6PUPWKGQXPOYHITPIXZPVNYKTN/graph.json","fetch_events":"https://pith.science/api/pith-number/6PUPWKGQXPOYHITPIXZPVNYKTN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6PUPWKGQXPOYHITPIXZPVNYKTN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6PUPWKGQXPOYHITPIXZPVNYKTN/action/storage_attestation","attest_author":"https://pith.science/pith/6PUPWKGQXPOYHITPIXZPVNYKTN/action/author_attestation","sign_citation":"https://pith.science/pith/6PUPWKGQXPOYHITPIXZPVNYKTN/action/citation_signature","submit_replication":"https://pith.science/pith/6PUPWKGQXPOYHITPIXZPVNYKTN/action/replication_record"}},"created_at":"2026-06-03T22:06:20.671447+00:00","updated_at":"2026-06-03T22:06:20.671447+00:00"}