{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:NS5RVFFE2ATJYHJUOJXQEEP6MU","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":"2e6c05644b9f1fe4253b565f6eeea4b22c808e6fe4cc2dcc917ee7349fff79dd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-07-20T14:40:06Z","title_canon_sha256":"e7c7ffc3ccb4bed360dda93eab267438ea7354ac2decfb46cb8add4a17bdd2bb"},"schema_version":"1.0","source":{"id":"1607.05978","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.05978","created_at":"2026-05-18T01:10:43Z"},{"alias_kind":"arxiv_version","alias_value":"1607.05978v1","created_at":"2026-05-18T01:10:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.05978","created_at":"2026-05-18T01:10:43Z"},{"alias_kind":"pith_short_12","alias_value":"NS5RVFFE2ATJ","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"NS5RVFFE2ATJYHJU","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"NS5RVFFE","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:14bd768ae7044a631cd8e6bbe5f6547cf1c39163d386a87ad4cebce9d28bbf4b","target":"graph","created_at":"2026-05-18T01:10:43Z","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 present an approach to defining Hilbert spaces of functions depending on infinitely many variables or parameters, with emphasis on a weighted tensor product construction based on stable space splittings, The construction has been used in an exemplary way for guiding dimension- and scale-adaptive algorithms in application areas such as statistical learning theory, reduced order modeling, and information-based complexity. We prove results on compact embeddings, norm equivalences, and the estimation of $epsilon$-dimensions. A new condition for the equivalence of weighted ANOVA and anchored nor","authors_text":"Michael Griebel, Peter Oswald","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-07-20T14:40:06Z","title":"Hilbert function space splittings on domains with infinitely many variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.05978","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":"sha256:5f2e82e4ac88c3b9b6e292d638ff5570be936d6ff58872dca0842da2c349086e","target":"record","created_at":"2026-05-18T01:10:43Z","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":"2e6c05644b9f1fe4253b565f6eeea4b22c808e6fe4cc2dcc917ee7349fff79dd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-07-20T14:40:06Z","title_canon_sha256":"e7c7ffc3ccb4bed360dda93eab267438ea7354ac2decfb46cb8add4a17bdd2bb"},"schema_version":"1.0","source":{"id":"1607.05978","kind":"arxiv","version":1}},"canonical_sha256":"6cbb1a94a4d0269c1d34726f0211fe651d0d2fe7ff204425ce0985f6c76c3e41","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6cbb1a94a4d0269c1d34726f0211fe651d0d2fe7ff204425ce0985f6c76c3e41","first_computed_at":"2026-05-18T01:10:43.528509Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:43.528509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/GN7elDHz5lryHjL6Z+REntBKzFIDr+XPgIc7aG9ZUVyx7BFUKWSWwYlpEb6JdQOoUipt09NwE7MlxxXlpl7BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:43.528897Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.05978","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5f2e82e4ac88c3b9b6e292d638ff5570be936d6ff58872dca0842da2c349086e","sha256:14bd768ae7044a631cd8e6bbe5f6547cf1c39163d386a87ad4cebce9d28bbf4b"],"state_sha256":"f7c3bcbaac3b09ccdba7fb08e6fcc0bfcc24139a7e8ea012f9be4782ad55c6ab"}