{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:BQYQ7YPZRHTEIHTOUTXP37MZJ4","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":"83ebc696e506b6b0e24c51729e8d9be010f3465f46e99d6edcf71e45218e46b6","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-03-19T03:00:34Z","title_canon_sha256":"41fa79d3cc727e4b464268826eeb15e92ac0937a97468db83f5df5828eed29bf"},"schema_version":"1.0","source":{"id":"0903.3267","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.3267","created_at":"2026-05-18T00:23:41Z"},{"alias_kind":"arxiv_version","alias_value":"0903.3267v1","created_at":"2026-05-18T00:23:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.3267","created_at":"2026-05-18T00:23:41Z"},{"alias_kind":"pith_short_12","alias_value":"BQYQ7YPZRHTE","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"BQYQ7YPZRHTEIHTO","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"BQYQ7YPZ","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:da100e1aa86af856245e8a7f0b876544af02834800b2c023ac3eb1e31f8cccb6","target":"graph","created_at":"2026-05-18T00:23:41Z","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":"We offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet algorithms; even spectral theory for fractal measures. In each case, there is an associated class of harmonic functions which we study. And in addition, we study three questions in depth:\n  In specific applications, and for a specific stochastic process, how do we realize the transfer operator $T$ as an operator in a suitable Hilbert space? And how to spect","authors_text":"Myung-Sin Song, Palle E. T. Jorgensen","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-03-19T03:00:34Z","title":"Spectral Theory of Discrete Processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.3267","kind":"arxiv","version":1},"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":"sha256:023b83f17710273c79e29e8f24a9e8629a62c3d694ea00a5d6948d49624df5cc","target":"record","created_at":"2026-05-18T00:23:41Z","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":"83ebc696e506b6b0e24c51729e8d9be010f3465f46e99d6edcf71e45218e46b6","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-03-19T03:00:34Z","title_canon_sha256":"41fa79d3cc727e4b464268826eeb15e92ac0937a97468db83f5df5828eed29bf"},"schema_version":"1.0","source":{"id":"0903.3267","kind":"arxiv","version":1}},"canonical_sha256":"0c310fe1f989e6441e6ea4eefdfd994f20395c475c5fb7260fe6f716fecb7ded","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0c310fe1f989e6441e6ea4eefdfd994f20395c475c5fb7260fe6f716fecb7ded","first_computed_at":"2026-05-18T00:23:41.261847Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:23:41.261847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wOrBARQBsiMx5/UmPjCla4bBUVVJIrntdJkaIcuTJCg2+fAB/xs3HilucfwFlZnNYyq0zUPA3WnYtdn0gX7zAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:23:41.262437Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.3267","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:023b83f17710273c79e29e8f24a9e8629a62c3d694ea00a5d6948d49624df5cc","sha256:da100e1aa86af856245e8a7f0b876544af02834800b2c023ac3eb1e31f8cccb6"],"state_sha256":"ffa3748c483fe3bb6e610659ebc72e9682bf0b705c57e06c3508860193720e15"}