{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:F7VZIYTAVXQACKHAGXEU4GCRRT","short_pith_number":"pith:F7VZIYTA","canonical_record":{"source":{"id":"1110.4962","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-22T09:39:12Z","cross_cats_sorted":[],"title_canon_sha256":"76bda6035f7e2781f3b9b7a242bb77ea07f066e5a45d21044daf5d12e3450986","abstract_canon_sha256":"fd92661ea9f9382ce9a98926242a1862a168e8d7b5d2552df775784bdb4d8ab6"},"schema_version":"1.0"},"canonical_sha256":"2feb946260ade00128e035c94e18518cce6af4d447c25fdbcfd59fab7189cefe","source":{"kind":"arxiv","id":"1110.4962","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.4962","created_at":"2026-05-18T03:21:21Z"},{"alias_kind":"arxiv_version","alias_value":"1110.4962v1","created_at":"2026-05-18T03:21:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.4962","created_at":"2026-05-18T03:21:21Z"},{"alias_kind":"pith_short_12","alias_value":"F7VZIYTAVXQA","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"F7VZIYTAVXQACKHA","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"F7VZIYTA","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:F7VZIYTAVXQACKHAGXEU4GCRRT","target":"record","payload":{"canonical_record":{"source":{"id":"1110.4962","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-22T09:39:12Z","cross_cats_sorted":[],"title_canon_sha256":"76bda6035f7e2781f3b9b7a242bb77ea07f066e5a45d21044daf5d12e3450986","abstract_canon_sha256":"fd92661ea9f9382ce9a98926242a1862a168e8d7b5d2552df775784bdb4d8ab6"},"schema_version":"1.0"},"canonical_sha256":"2feb946260ade00128e035c94e18518cce6af4d447c25fdbcfd59fab7189cefe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:21.741509Z","signature_b64":"yoGEPKjOIstugWJkU5rvp+SjB1O5kcgUQePBuoNDPcOfUNNdvSeVx3HKC99Nobbq1uoVQZwlo/DLg+V8OhLIAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2feb946260ade00128e035c94e18518cce6af4d447c25fdbcfd59fab7189cefe","last_reissued_at":"2026-05-18T03:21:21.740875Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:21.740875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.4962","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:21:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4uXzFnYgZppliYduYZwh4dXm/oL6IHsJbNQeGTlgHeuJaAnns2cbUz4kB0fxp+qTVNjPm5Jr8i/hFnZr7nE5AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T17:58:26.363699Z"},"content_sha256":"8b3eeb9d27f3456d2ea9341a43c74324c53275263cdeecaf3032d0e6c7169d8f","schema_version":"1.0","event_id":"sha256:8b3eeb9d27f3456d2ea9341a43c74324c53275263cdeecaf3032d0e6c7169d8f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:F7VZIYTAVXQACKHAGXEU4GCRRT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Convex conjugates of analytic functions of logarithmically convex functionals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Krzysztof Zajkowski","submitted_at":"2011-10-22T09:39:12Z","abstract_excerpt":"Let $f_{\\bf c}(r)=\\sum_{n=0}^\\infty e^{c_n}r^n$ be an analytic function; ${\\bf c}=(c_n)\\in l_\\infty$. We assume that $r$ is some logarithmically convex and lower semicontinuous functional on a locally convex topological space $L$. In this paper we derive a formula on the Legendre-Fenchel transform of a functional $\\hat{\\lambda}({\\bf c},\\phi)=\\ln f_{\\bf c}(e^{\\lambda(\\phi)})$, where $\\lambda(\\phi)=\\ln r(\\phi)$ ($\\phi\\in L$). In this manner we generalize to the infinite case Theorem 3.1 from \\cite{OZ1}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.4962","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:21:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IAPR1tvSereOcTYDhVOQGqBrsFhXaVac/CT4Koft504hhNFyuq2yH03c7jpxn8WwXI13X+hGz1S/2hAOS317Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T17:58:26.364042Z"},"content_sha256":"68d98db7e621f902d8c7492f55bfd89db417309cf2c3caf652473f60fc811721","schema_version":"1.0","event_id":"sha256:68d98db7e621f902d8c7492f55bfd89db417309cf2c3caf652473f60fc811721"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F7VZIYTAVXQACKHAGXEU4GCRRT/bundle.json","state_url":"https://pith.science/pith/F7VZIYTAVXQACKHAGXEU4GCRRT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F7VZIYTAVXQACKHAGXEU4GCRRT/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T17:58:26Z","links":{"resolver":"https://pith.science/pith/F7VZIYTAVXQACKHAGXEU4GCRRT","bundle":"https://pith.science/pith/F7VZIYTAVXQACKHAGXEU4GCRRT/bundle.json","state":"https://pith.science/pith/F7VZIYTAVXQACKHAGXEU4GCRRT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F7VZIYTAVXQACKHAGXEU4GCRRT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:F7VZIYTAVXQACKHAGXEU4GCRRT","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":"fd92661ea9f9382ce9a98926242a1862a168e8d7b5d2552df775784bdb4d8ab6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-22T09:39:12Z","title_canon_sha256":"76bda6035f7e2781f3b9b7a242bb77ea07f066e5a45d21044daf5d12e3450986"},"schema_version":"1.0","source":{"id":"1110.4962","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.4962","created_at":"2026-05-18T03:21:21Z"},{"alias_kind":"arxiv_version","alias_value":"1110.4962v1","created_at":"2026-05-18T03:21:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.4962","created_at":"2026-05-18T03:21:21Z"},{"alias_kind":"pith_short_12","alias_value":"F7VZIYTAVXQA","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"F7VZIYTAVXQACKHA","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"F7VZIYTA","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:68d98db7e621f902d8c7492f55bfd89db417309cf2c3caf652473f60fc811721","target":"graph","created_at":"2026-05-18T03:21:21Z","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":"Let $f_{\\bf c}(r)=\\sum_{n=0}^\\infty e^{c_n}r^n$ be an analytic function; ${\\bf c}=(c_n)\\in l_\\infty$. We assume that $r$ is some logarithmically convex and lower semicontinuous functional on a locally convex topological space $L$. In this paper we derive a formula on the Legendre-Fenchel transform of a functional $\\hat{\\lambda}({\\bf c},\\phi)=\\ln f_{\\bf c}(e^{\\lambda(\\phi)})$, where $\\lambda(\\phi)=\\ln r(\\phi)$ ($\\phi\\in L$). In this manner we generalize to the infinite case Theorem 3.1 from \\cite{OZ1}.","authors_text":"Krzysztof Zajkowski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-22T09:39:12Z","title":"Convex conjugates of analytic functions of logarithmically convex functionals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.4962","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":"sha256:8b3eeb9d27f3456d2ea9341a43c74324c53275263cdeecaf3032d0e6c7169d8f","target":"record","created_at":"2026-05-18T03:21:21Z","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":"fd92661ea9f9382ce9a98926242a1862a168e8d7b5d2552df775784bdb4d8ab6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-22T09:39:12Z","title_canon_sha256":"76bda6035f7e2781f3b9b7a242bb77ea07f066e5a45d21044daf5d12e3450986"},"schema_version":"1.0","source":{"id":"1110.4962","kind":"arxiv","version":1}},"canonical_sha256":"2feb946260ade00128e035c94e18518cce6af4d447c25fdbcfd59fab7189cefe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2feb946260ade00128e035c94e18518cce6af4d447c25fdbcfd59fab7189cefe","first_computed_at":"2026-05-18T03:21:21.740875Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:21.740875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yoGEPKjOIstugWJkU5rvp+SjB1O5kcgUQePBuoNDPcOfUNNdvSeVx3HKC99Nobbq1uoVQZwlo/DLg+V8OhLIAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:21.741509Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.4962","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8b3eeb9d27f3456d2ea9341a43c74324c53275263cdeecaf3032d0e6c7169d8f","sha256:68d98db7e621f902d8c7492f55bfd89db417309cf2c3caf652473f60fc811721"],"state_sha256":"a184213f45a49f2ad2e9f96a232798c796d5ea06660b0d5065d59209a5f46d1c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+Ktkkxq5lokjUjt3HQYcApwMVbINGBHBtSOD6bVpTiygUFOXysk9NQiJjx0Vz013teUBetaWHfpnCqHLUNzGBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T17:58:26.366044Z","bundle_sha256":"88b33d0a7195f133e2d37534b701dddc8e4072835053e56cab16a07a4aa09eaf"}}