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Let $k=const>0$ be fixed, $S^2$ be the unit sphere in $\\mathbb{R}^3$, $D$ be a connected bounded domain with $C^2-$smooth boundary $S$, $j_0(r)$ be the spherical Bessel function. The harmonic analysis symmetry problems are stated in the following theorems:\n  {\\bf Theorem A.} {\\em Assume that $\\int_S e^{ik\\beta \\cdot s}ds=0$ for all $\\beta\\in S^2$. Then $S$ is a sphere of radius $a$, where $j_0(ka)=0$. }\n  {\\bf Theore"},"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":"1904.11363","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-21T18:25:31Z","cross_cats_sorted":[],"title_canon_sha256":"83b61c560ff50531d379d49b729425257205dc6cda31df792b2a12524f7b92f1","abstract_canon_sha256":"b9e66af2119b11995b2b440aa3f0e7a689fe47e6255e44216aabc92024d5ef2d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:45.129661Z","signature_b64":"pSSNNAsZ+KCUT+IWM5O1gHVpfCSFGOxP1lxS4gfz4jXzZlYk/x4N0GM28//puvZWsNewgDzbvdiNjvNP7iVGBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bcda7595186298ee95d22ac392fa8ce8f1d1c205812dd575a1f9c4671fa59dda","last_reissued_at":"2026-05-17T23:47:45.129232Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:45.129232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Symmetry problems in harmonic analysis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexander G. Ramm","submitted_at":"2019-04-21T18:25:31Z","abstract_excerpt":"Symmetry problems in harmonic analysis are formulated and solved. One of these problems is equivalent to the refined Schiffer's conjecture which was recently proved by the author. Let $k=const>0$ be fixed, $S^2$ be the unit sphere in $\\mathbb{R}^3$, $D$ be a connected bounded domain with $C^2-$smooth boundary $S$, $j_0(r)$ be the spherical Bessel function. The harmonic analysis symmetry problems are stated in the following theorems:\n  {\\bf Theorem A.} {\\em Assume that $\\int_S e^{ik\\beta \\cdot s}ds=0$ for all $\\beta\\in S^2$. Then $S$ is a sphere of radius $a$, where $j_0(ka)=0$. }\n  {\\bf Theore"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.11363","kind":"arxiv","version":1},"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":"1904.11363","created_at":"2026-05-17T23:47:45.129295+00:00"},{"alias_kind":"arxiv_version","alias_value":"1904.11363v1","created_at":"2026-05-17T23:47:45.129295+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.11363","created_at":"2026-05-17T23:47:45.129295+00:00"},{"alias_kind":"pith_short_12","alias_value":"XTNHLFIYMKMO","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_16","alias_value":"XTNHLFIYMKMO5FOS","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_8","alias_value":"XTNHLFIY","created_at":"2026-05-18T12:33:33.725879+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/XTNHLFIYMKMO5FOSFLBZF6UM5D","json":"https://pith.science/pith/XTNHLFIYMKMO5FOSFLBZF6UM5D.json","graph_json":"https://pith.science/api/pith-number/XTNHLFIYMKMO5FOSFLBZF6UM5D/graph.json","events_json":"https://pith.science/api/pith-number/XTNHLFIYMKMO5FOSFLBZF6UM5D/events.json","paper":"https://pith.science/paper/XTNHLFIY"},"agent_actions":{"view_html":"https://pith.science/pith/XTNHLFIYMKMO5FOSFLBZF6UM5D","download_json":"https://pith.science/pith/XTNHLFIYMKMO5FOSFLBZF6UM5D.json","view_paper":"https://pith.science/paper/XTNHLFIY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1904.11363&json=true","fetch_graph":"https://pith.science/api/pith-number/XTNHLFIYMKMO5FOSFLBZF6UM5D/graph.json","fetch_events":"https://pith.science/api/pith-number/XTNHLFIYMKMO5FOSFLBZF6UM5D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XTNHLFIYMKMO5FOSFLBZF6UM5D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XTNHLFIYMKMO5FOSFLBZF6UM5D/action/storage_attestation","attest_author":"https://pith.science/pith/XTNHLFIYMKMO5FOSFLBZF6UM5D/action/author_attestation","sign_citation":"https://pith.science/pith/XTNHLFIYMKMO5FOSFLBZF6UM5D/action/citation_signature","submit_replication":"https://pith.science/pith/XTNHLFIYMKMO5FOSFLBZF6UM5D/action/replication_record"}},"created_at":"2026-05-17T23:47:45.129295+00:00","updated_at":"2026-05-17T23:47:45.129295+00:00"}