{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1996:VCK5JKTN3YVYCMZ7HZNQBN26HP","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":"32022c952f92de2332510765fa2ce48c1e6806349d5346feb7e7ec9deeb06cf8","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1996-10-01T00:00:00Z","title_canon_sha256":"1081b18be28478c7d36db8d21ef0639da7d053b47c2958d4e77bce4f3b190fda"},"schema_version":"1.0","source":{"id":"math/9610208","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9610208","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"arxiv_version","alias_value":"math/9610208v1","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9610208","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"pith_short_12","alias_value":"VCK5JKTN3YVY","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_16","alias_value":"VCK5JKTN3YVYCMZ7","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_8","alias_value":"VCK5JKTN","created_at":"2026-05-18T12:25:48Z"}],"graph_snapshots":[{"event_id":"sha256:a33d1898617aed47b72a85fcf22262ee047abef61ea92b8d58f9c98f1d59705c","target":"graph","created_at":"2026-05-18T01:05:47Z","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 define embedding of an $n$-dimensional normed space into $L_{-p},\\ 0<p<n$ by extending analytically with respect to $p$ the corresponding property of the classical $L_p$-spaces. The well-known connection between embeddings into $L_p$ and positive definite functions is extended to the case of negative $p$ by showing that a normed space embeds in $L_{-p}$ if and only if $\\|x\\|^{-p}$ is a positive definite distribution. Using this criterion, we generalize the recent solutions to the 1938 Schoenberg's problems by proving that the spaces $\\ell_q^n,\\ 2<q\\le \\infty$ embed in $L_{-p}$ if and only i","authors_text":"Alexander Koldobsky","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"1996-10-01T00:00:00Z","title":"Positive definite distributions and subspaces of $L_{-p}$ with applications to stable processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9610208","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":"sha256:3ebe29a2acd05b74f83097d1f55deaa105d98beb6de5ea1fe83b333091a779ae","target":"record","created_at":"2026-05-18T01:05:47Z","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":"32022c952f92de2332510765fa2ce48c1e6806349d5346feb7e7ec9deeb06cf8","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1996-10-01T00:00:00Z","title_canon_sha256":"1081b18be28478c7d36db8d21ef0639da7d053b47c2958d4e77bce4f3b190fda"},"schema_version":"1.0","source":{"id":"math/9610208","kind":"arxiv","version":1}},"canonical_sha256":"a895d4aa6dde2b81333f3e5b00b75e3bf9b2d80478f9a8d2f67a13303ae7ddde","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a895d4aa6dde2b81333f3e5b00b75e3bf9b2d80478f9a8d2f67a13303ae7ddde","first_computed_at":"2026-05-18T01:05:47.009037Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:47.009037Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Cr9IiHKd9+lDLa9RTSVuJqPYI6+M4yAGaB9Bni27q3iRxhuia/Nwmiyi6spaQbW/uIOsChDONBTxqDd76IEpBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:47.009570Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9610208","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3ebe29a2acd05b74f83097d1f55deaa105d98beb6de5ea1fe83b333091a779ae","sha256:a33d1898617aed47b72a85fcf22262ee047abef61ea92b8d58f9c98f1d59705c"],"state_sha256":"54102bffef731b1fd71ad64f6226e8131119f08c5ba86839966bea31e047de9e"}