{"as_of":"2026-08-21T20:42:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:702239ed57895f144734dbae3e99ec2b48e772b7daadbde1b27d77bc7e27ab5f","coverage":[{"denominator":57,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":57,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-04T13:43:13.379692Z","state":"measured"},{"denominator":57,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":57,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-21T06:32:19.484+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2509.26371/citation-record","integrity":"/paper/2509.26371/integrity","json":"/paper/2509.26371/citation-record.json","paper":"/paper/2509.26371"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.206593Z","title":"Neural operator: Graph kernel network for partial differential equations","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.206593Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:23713af7799a856adc8e372b03bf5179b0510ef690726849735ed21dd8d42648","observation_id":"490748b9-975e-4d5f-97bd-c8686e2b6c9e","resolution":{"observed_at":"2026-08-04T13:43:13.206593Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.210738Z","title":"Theory of reproducing kernels.Transactions of the American mathematical society, 68(3):337–404, 1950","venue":null,"work_id":null,"year":1950},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.210738Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:3d5ddd1847b207bb6e8aa3219f2e45aa5ea2182fde743ae0267b9a111e65d7e7","observation_id":"dbe541b3-f162-4a86-b128-b2d0ade28676","resolution":{"observed_at":"2026-08-04T13:43:13.210738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.213893Z","title":"Structured sparsity-inducing norms through submodular functions.Advances in Neural Information Processing Systems, 23, 2010","venue":null,"work_id":null,"year":2010},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.213893Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:140a1098ca92b08d52745fd9965b50ddbf8f4205b8dfc3cd2f521d5f7692e0a2","observation_id":"405e0581-31ff-42fb-94c5-55c551c75210","resolution":{"observed_at":"2026-08-04T13:43:13.213893Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.217193Z","title":"Breaking the curse of dimensionality with convex neural networks.Journal of Machine Learning Research, 18(19):1–53, 2017","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.217193Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:f6f93ff105a6dbcca9e9220ab975f0cdcc7c964faa4ca6cd44143047c6abe5db","observation_id":"e9554622-aa60-470e-9056-dc4064808528","resolution":{"observed_at":"2026-08-04T13:43:13.217193Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.220410Z","title":"Iglesias, Yury Korolev, Emanuele Naldi, and Stefano Vi- gogna","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.220410Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:80dc3ca3b7af5ce275191c5cebdc5eb26dbe4cb29dd80f869ccb45179145a37c","observation_id":"bdb71335-5e3a-4b4a-a821-0fc8ec4f8499","resolution":{"observed_at":"2026-08-04T13:43:13.220410Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.223605Z","title":"Understanding neural networks with reproducing kernel Banach spaces.Applied and Computational Harmonic Analysis, 62:194– 236, 2023","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.223605Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:056b46e470c8eeeab83df839fe693c910541c51aa4398eb6e31b7c0382ee9e7f","observation_id":"259920d8-3506-474f-ac7e-796fe38a4b6e","resolution":{"observed_at":"2026-08-04T13:43:13.223605Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2403.08750","last_updated":"2025-08-21T08:54:47Z","snapshot_observed_at":"2026-08-21T04:06:05.666028Z","submitted_at":"2024-03-13T17:51:02Z","title":"Neural reproducing kernel Banach spaces and representer theorems for deep networks","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2403.08750","snapshot_observed_at":"2026-08-04T13:43:13.227017Z","title":"Neural reproducing kernel Banach spaces and representer theorems for deep networks.arXiv preprint arXiv:2403.08750, 2024","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.227017Z"},"links":{"cited_paper":"/paper/2403.08750","citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:47cb9fc1ff268b86a2a027b56148d69c3b88784520a04accbe84b03b0d73268a","observation_id":"33b99903-f95d-4cf0-91e7-459dc04961ee","resolution":{"observed_at":"2026-08-04T13:43:13.227017Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.230178Z","title":"Kernel methods are competitive for operator learning.Journal of Computational Physics, 496:112549, 2024","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.230178Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:eaca973b45ed7d65d2072cc62445be1626ee3e2e6517380b0c0bb5daa04ea5ce","observation_id":"f0e43723-46da-4635-a76f-fd2d83e8adde","resolution":{"observed_at":"2026-08-04T13:43:13.230178Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.233077Z","title":"Kovachki, and Andrew M","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.233077Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:a80bce63d44aff971319e4d0ba14e65a786c9c89820460bf24da8fdf8b5f655e","observation_id":"0f8a2b07-2c10-4f89-bea7-8eb69834b6ef","resolution":{"observed_at":"2026-08-04T13:43:13.233077Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.235812Z","title":"Springer, 2007","venue":null,"work_id":null,"year":2007},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.235812Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:01ef29461d1531221b953dd963eead64ecc3683b565492bdb4423a1c619fe063","observation_id":"fee5375e-ed27-4dfb-846f-86a78ff63b1b","resolution":{"observed_at":"2026-08-04T13:43:13.235812Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.238380Z","title":"A mathematical guide to operator learning","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.238380Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:c13589c46f61a99aa64c101910b29003463ad185fed24b25a14c2cad402ddef0","observation_id":"ad41cc79-abeb-4d61-8f1d-9f0706b55f15","resolution":{"observed_at":"2026-08-04T13:43:13.238380Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.241303Z","title":"Sparsity of solutions for variational inverse problems with finite- dimensional data.Calculus of Variations and Partial Differential Equations, 59(1):14, 2020","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.241303Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:c384b34384c8fdbea59a15b7b5f88bde1cc4b7becda17588ae406b3b56597e6a","observation_id":"c9931484-d4bc-4afc-bc6d-3e73805a60db","resolution":{"observed_at":"2026-08-04T13:43:13.241303Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.244382Z","title":"Iglesias, and Daniel Walter","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.244382Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:f3b88faf39963aac7fa870c272830c287b706c330b2263d00770b76886b715f0","observation_id":"e637a37d-b0c6-4978-84db-30d4eee4e772","resolution":{"observed_at":"2026-08-04T13:43:13.244382Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"0910.1013","last_updated":"2009-10-06T14:19:08Z","snapshot_observed_at":"2026-08-15T05:37:23.470462Z","submitted_at":"2009-10-06T14:19:08Z","title":"Functional learning through kernels","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"0910.1013","snapshot_observed_at":"2026-08-04T13:43:13.247191Z","title":"Functional learning through kernels.arXiv preprint arXiv:0910.1013, 2009","venue":null,"work_id":null,"year":2009},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.247191Z"},"links":{"cited_paper":"/paper/0910.1013","citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:3ae897552542ce4d667a097e61368869f56d265c67dbb7273b2bce36b3e3e5c3","observation_id":"5584dd90-16e1-4648-9d41-566ee3fb3a12","resolution":{"observed_at":"2026-08-04T13:43:13.247191Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.251297Z","title":"Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem.Analysis and Applications, 4(04):377–408, 2006","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.251297Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:5d1784d4af88c399fc5920dadf0f2bc1b7ffa4b1d6df0dcb0007513af2a36c8c","observation_id":"27d47cde-42cf-44dc-9e0c-8782bd483984","resolution":{"observed_at":"2026-08-04T13:43:13.251297Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.254417Z","title":"Vector valued reproducing kernel Hilbert spaces and universality.Analysis and Applications, 8(01):19–61, 2010","venue":null,"work_id":null,"year":2010},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.254417Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:8946d8517e655fef553747c375ff7bbdecd549478edf5293d49ff93010938e47","observation_id":"a5d3221e-2e2e-4035-aeb5-6a8b57495058","resolution":{"observed_at":"2026-08-04T13:43:13.254417Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1903.00819","last_updated":"2025-08-04T09:01:57Z","snapshot_observed_at":"2026-08-15T00:45:31.103333Z","submitted_at":"2019-03-03T03:54:17Z","title":"Vector-valued Reproducing Kernel Banach Spaces with Group Lasso Norms","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1903.00819","snapshot_observed_at":"2026-08-04T13:43:13.257238Z","title":"Vector-valued reproducing kernel Banach spaces with group lasso norms.arXiv preprint arXiv:1903.00819, 2019","venue":null,"work_id":null,"year":1903},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.257238Z"},"links":{"cited_paper":"/paper/1903.00819","citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:4a5abdd4fd0e093ac1ab5465f2a02868d2d55f14319528d2f615416c4bd0f23a","observation_id":"0c038192-9009-4b4d-a55e-8b3f5dadd40b","resolution":{"observed_at":"2026-08-04T13:43:13.257238Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.260465Z","title":"Combettes, Saverio Salzo, and Silvia Villa","venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.260465Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:5c811f660e9d4e86d94becdc0466d639e55f89c40fc649a40e2a260804c88d29","observation_id":"2fe7c0b6-d163-4896-a4d2-20314ab4209c","resolution":{"observed_at":"2026-08-04T13:43:13.260465Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.263535Z","title":"Verduyn Lunel","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.263535Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:7e2eb82882b63ba51f0d96bcc9c2e59f1dd50d2525818c211221e9eaff79fd70","observation_id":"5c3bedcf-0c16-4445-ae12-c39508c0a7b9","resolution":{"observed_at":"2026-08-04T13:43:13.263535Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.266418Z","title":"RDA-INR: Riemannian diffeomorphic autoen- coding via implicit neural representations.SIAM Journal on Imaging Sciences, 17(4):2302–2330, 2024","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.266418Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:cdf29b7afa1abf31f09f00f6c5d496f0fa83530a9e49a3e8b6746c322d7928e5","observation_id":"7fc78416-cbda-4345-8646-972c36264757","resolution":{"observed_at":"2026-08-04T13:43:13.266418Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.269568Z","title":"RONOM: Reduced-order neural operator modeling","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.269568Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:ab7a4da1a74a82b0c22bc38036577f734fbe32168c5fde79f1b837cc8bbc20ea","observation_id":"9b791f79-9e5e-406c-8a1d-9b0ce398fc1a","resolution":{"observed_at":"2026-08-04T13:43:13.269568Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.272616Z","title":"Springer Science & Business Media, 2011","venue":null,"work_id":null,"year":2011},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.272616Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:6e7c2c7cd96a808ba7d3a470e7bf9ae75ab611a8320fd48a4b8dbf2c70ed5d8f","observation_id":"bf65fd6f-5f6e-4cdd-a88c-c1a7ff29963f","resolution":{"observed_at":"2026-08-04T13:43:13.272616Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.275643Z","title":"Georgiev, Luis S´ anchez-Gonz´ alez, and Panos M","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.275643Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:e939eb05e130b09c7bca53d918325e580690ba44ba0088d2ea44329b8bc48439","observation_id":"bb5f5774-428a-425b-bbf4-c3886b66d98f","resolution":{"observed_at":"2026-08-04T13:43:13.275643Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2501.03697","last_updated":"2025-01-07T11:01:24Z","snapshot_observed_at":"2026-08-16T12:59:32.178517Z","submitted_at":"2025-01-07T11:01:24Z","title":"Deep Networks are Reproducing Kernel Chains","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2501.03697","snapshot_observed_at":"2026-08-04T13:43:13.278402Z","title":"Deep networks are reproducing kernel chains.arXiv preprint arXiv:2501.03697, 2025","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.278402Z"},"links":{"cited_paper":"/paper/2501.03697","citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:4b63b3b0d00a9da6b1a2ef5bef23552cfb7e9ddc3557a5f4f5e9e91004b7d4d8","observation_id":"209d2af4-5aa6-4bb7-b5d7-5cbdb26fc5fb","resolution":{"observed_at":"2026-08-04T13:43:13.278402Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.281698Z","title":"Koopman and Perron–Frobenius operators on reproducing kernel Banach spaces.Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(12), 2022","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.281698Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:004cb0629cbf9ce7c7b54ef8a144cbda51f5641edd85feb14ca1d1569db75e6b","observation_id":"1bb06496-9e3c-4e3e-be2f-0699b7962f8d","resolution":{"observed_at":"2026-08-04T13:43:13.281698Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2304.07599","last_updated":"2023-04-15T17:13:09Z","snapshot_observed_at":"2026-08-16T15:40:13.016283Z","submitted_at":"2023-04-15T17:13:09Z","title":"Learning in latent spaces improves the predictive accuracy of deep neural operators","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2304.07599","snapshot_observed_at":"2026-08-04T13:43:13.284569Z","title":"Learning in latent spaces improves the predictive accuracy of deep neural operators.arXiv preprint arXiv:2304.07599, 2023","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.284569Z"},"links":{"cited_paper":"/paper/2304.07599","citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:bd974040360f064771fa8e914e0c718b961af176adbac09cb10fdf095c5feb82","observation_id":"3022987c-8eaf-4e87-b15a-cfd4161f6832","resolution":{"observed_at":"2026-08-04T13:43:13.284569Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.287989Z","title":"Two-layer neural networks with values in a Banach space.SIAM Journal on Mathematical Analysis, 54(6):6358–6389, 2022","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.287989Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:559145f442c68b3f4af7ffe589ab227d9c6a2359e44b07468b9c50647de4c050","observation_id":"ffdbe8a5-8123-438a-9fef-553bec89abba","resolution":{"observed_at":"2026-08-04T13:43:13.287989Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.291089Z","title":"On universal approximation and error bounds for fourier neural operators.Journal of Machine Learning Research, 22(290):1–76, 2021","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.291089Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:4fc78c3922ef1404d43e025f0051eeafc6e893f2da002e0e19bc5512c62cbfdb","observation_id":"1e9a0c87-b658-4dc3-b498-4e88cd8ee347","resolution":{"observed_at":"2026-08-04T13:43:13.291089Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.294334Z","title":"Fourier neural operator for parametric partial differential equations","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.294334Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:5089b25967a6a89cc9f6105bc847b2030b9493c2e6bd0732b29000f69b271974","observation_id":"25530467-1fe8-4033-a80e-0656230faea4","resolution":{"observed_at":"2026-08-04T13:43:13.294334Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.297867Z","title":"On reproducing kernel Banach spaces: Generic definitions and unified framework of constructions.Acta Mathematica Sinica, English Series, 38(8):1459– 1483, 2022","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.297867Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:6b13903f4223b44f86b40b2ac19d87298ab0d958cda2232fd563d9d53b60462d","observation_id":"97799abc-9100-40d5-a8c7-19059f9e5557","resolution":{"observed_at":"2026-08-04T13:43:13.297867Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.300864Z","title":"Multi-task learning in vector-valued reproducing kernel Banach spaces with theℓ 1 norm.Journal of Complexity, 63:101514, 2021","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.300864Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:17fc20f4a2d0c5f07c34ea7ebd4430112106bb9a1676ff4efcdbec1211929f66","observation_id":"6369bfd1-1516-4722-b377-301f5884e960","resolution":{"observed_at":"2026-08-04T13:43:13.300864Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.303837Z","title":"Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.Nature machine intelligence, 3(3):218–229, 2021","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.303837Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:56fe1b4069ef1d766222b614070dc00f47b0bc1cdd5a1c26f5786c5f067daf9b","observation_id":"340dcf25-ccf7-4039-b257-1f174c09658c","resolution":{"observed_at":"2026-08-04T13:43:13.303837Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.307107Z","title":"A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data.Computer Methods in Applied Mechanics and Engineering, 393:114778, 2022","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.307107Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:cec6d120af1a6712fd0d0ed574b0c19a4015e26ec3b9b4c923ac07e10cc81dbb","observation_id":"ea4fa666-9c5b-4754-a0fd-dcaac7062050","resolution":{"observed_at":"2026-08-04T13:43:13.307107Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2009.10713","last_updated":"2020-12-07T23:36:20Z","snapshot_observed_at":"2026-08-12T11:27:30.709881Z","submitted_at":"2020-09-22T17:55:47Z","title":"Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2009.10713","snapshot_observed_at":"2026-08-04T13:43:13.310394Z","title":"Towards a mathematical understanding of neural network- based machine learning: what we know and what we don’t.arXiv preprint arXiv:2009.10713, 2020","venue":null,"work_id":null,"year":2009},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.310394Z"},"links":{"cited_paper":"/paper/2009.10713","citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:34623f0e4be36f9ff64608e2627755eb2a1954671c57ea00127217001a877a56","observation_id":"8c85d4a9-0ee4-4138-a93f-515a9e3b99ed","resolution":{"observed_at":"2026-08-04T13:43:13.310394Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.313809Z","title":"A priori estimates of the population risk for two-layer neural networks.Commu- nications in Mathematical Sciences, 17(5):1407–1425, 2019","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.313809Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:8612be37e143061db52ade1254580bdd0326a0258aa17b969409fdea6ef4fb17","observation_id":"b47c4ec9-9564-4d04-9eb9-f06cb18031ef","resolution":{"observed_at":"2026-08-04T13:43:13.313809Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.317350Z","title":"The Barron space and the flow-induced function spaces for neural network models","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.317350Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:8ae778ec7f09e428171785fd4adcd36a75078fd4e39b3212559f1f2be82462a3","observation_id":"07e754e8-28d5-4069-b77e-8b71636d0b6f","resolution":{"observed_at":"2026-08-04T13:43:13.317350Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.320448Z","title":"On the dual spaceC ∗ 0 (S, X).Acta Math","venue":null,"work_id":null,"year":2009},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.320448Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:c85a5d04d85e275ff5e5bf86efbafdecc8a6604a89c44df00441cea9b006a4ab","observation_id":"c1bdc914-c64f-4113-905c-d7d498b36a11","resolution":{"observed_at":"2026-08-04T13:43:13.320448Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.323104Z","title":null,"venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.323104Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:53376c9dfba061bae217b4108c0c9ee4490d8fe86f8c1ffc9308e8a1a65e5b41","observation_id":"ddf3da54-2452-4b9d-8f9f-ab03617781f8","resolution":{"observed_at":"2026-08-04T13:43:13.323104Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.326029Z","title":null,"venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.326029Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:2f97604b50b631da641c12a0dfda5166678ad70f2af373c2ff8c734b7ecfee3b","observation_id":"e242a1f8-a63c-49f9-a448-bebe65c785fe","resolution":{"observed_at":"2026-08-04T13:43:13.326029Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.329031Z","title":"A measure-theoretic approach to kernel conditional mean embed- dings.Advances in Neural Information Processing Systems, 33:21247–21259, 2020","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.329031Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:35e79bbc05c6ded1487d9b916f1b195a802766666646f384b8220498175e414e","observation_id":"98ba6f88-0407-45fd-a8ba-0cc6e7503055","resolution":{"observed_at":"2026-08-04T13:43:13.329031Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.331676Z","title":"Paulsen and Mrinal Raghupathi.An introduction to the theory of reproducing kernel Hilbert spaces, volume 152 ofCambridge Studies in Advanced Mathematics","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.331676Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:2ea60bce783ecfc45138d6cf874dddbe917e373678a27fe55826bfc0345d3c94","observation_id":"9c666c50-79a4-4eeb-aee9-ba5085b4aa4c","resolution":{"observed_at":"2026-08-04T13:43:13.331676Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.334378Z","title":"Con- volutional neural operators","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.334378Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:0a44a588326085dcbc23642d0660b5d91a45956fe63b4efa189d2067d7df8bdc","observation_id":"c72ceec7-d7b2-4b1b-9b1d-6804fdace818","resolution":{"observed_at":"2026-08-04T13:43:13.334378Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.337256Z","title":null,"venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.337256Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:c48646d136598b911d29e92df122e5bbf4a69f7296cc4122392a5a9dbe3eec30","observation_id":"fd67546b-1ecb-46de-b7f9-7552572f4021","resolution":{"observed_at":"2026-08-04T13:43:13.337256Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.339945Z","title":null,"venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.339945Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:94909e9b0f4f27f125f9dd064a2e175029295fb08940cbe5b688640bdc527541","observation_id":"c48dc22b-7d14-45b1-b02d-07ab74d8d7e4","resolution":{"observed_at":"2026-08-04T13:43:13.339945Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.343076Z","title":"Linear functionals on the space of continuous mappings of a compact Hausdorff space into a Banach space.Rev","venue":null,"work_id":null,"year":1957},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.343076Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:feb213e375102aa5d248d5007ee24f0304fbe6f279d4cbc3b680ba61e4d849f4","observation_id":"e200e514-e5c5-40de-843d-4e17bc309741","resolution":{"observed_at":"2026-08-04T13:43:13.343076Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.346097Z","title":"Implicit neural representations with periodic activation functions.Advances in Neural Information Processing Systems, 33:7462–7473, 2020","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.346097Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:6e0fad8a8e0325d8c35c312313dffb028003a2a0b30b115370b8445483d71874","observation_id":"edbcd896-b072-4c8e-bfb0-3699a3b3ea7c","resolution":{"observed_at":"2026-08-04T13:43:13.346097Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.349174Z","title":"Duality for neural networks through reproducing kernel Banach spaces.Applied and Computational Harmonic Analysis, 78:101765, 2025","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.349174Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:71147b1007d829b0375e9733f9e5ac9387f4082b39547379eee3acbc1376d853","observation_id":"8ab85ac6-360f-4e17-b6b6-c0e3821ed89f","resolution":{"observed_at":"2026-08-04T13:43:13.349174Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.352083Z","title":"Reproducing kernel Hilbert spaces cannot contain all continuous functions on a compact metric space.Archiv der Mathematik, 122(5):553–557, 2024","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.352083Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:37a71be8b870dc5e68bafa51febda3dfd4972c9536088d8a6e3e8b9f3c5e78e4","observation_id":"e7cc7f6d-d3ef-4c7a-9631-e502ac021e96","resolution":{"observed_at":"2026-08-04T13:43:13.352083Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.355088Z","title":"American Mathematical Society, 2011","venue":null,"work_id":null,"year":2011},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.355088Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:e3d7551cfa60ac72cc6cc40b8fbd65dd925eda687800544298179dfd8c90ed55","observation_id":"869a0015-1de1-4277-83c5-72d918398cf2","resolution":{"observed_at":"2026-08-04T13:43:13.355088Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.357865Z","title":"Sparse representer theorems for learning in reproducing kernel Banach spaces.Journal of Machine Learning Research, 25(93):1–45, 2024","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":50,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.357865Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:08b0e97e9a3daa64a6136d1b0307f51eb71a2a770f19caec12331f724a42e289","observation_id":"b01eabd9-29ab-4511-8079-623079dd782f","resolution":{"observed_at":"2026-08-04T13:43:13.357865Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.361010Z","title":"Hypothesis spaces for deep learning.Neural Networks, 193:107995, 2026","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":51,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.361010Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:578d46dc8342617e2e74050aba2f3689cf80d1b332703489ad567cb987e8788b","observation_id":"12f9e6ea-9133-41c0-8ee5-58fa457a48a4","resolution":{"observed_at":"2026-08-04T13:43:13.361010Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.363896Z","title":"Extreme points in spaces of operators and vector-valued measures.Rend","venue":null,"work_id":null,"year":1984},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":52,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.363896Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:e2031fa6ff9e56d0926690a1f1ee16e52d5a99e6c97378300d5ae5c5020f748b","observation_id":"57c8d926-e364-48dd-9412-2f2993a18579","resolution":{"observed_at":"2026-08-04T13:43:13.363896Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.367225Z","title":null,"venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":53,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.367225Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:5953c7e1b780c982ffa3352f0cb393d4d82199b2def593e224af6c980d965294","observation_id":"c43fe657-2f98-4ae0-87a3-72987f7f7aa1","resolution":{"observed_at":"2026-08-04T13:43:13.367225Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.370220Z","title":"Representation formulas and pointwise properties for Barron functions.Cal- culus of Variations and Partial Differential Equations, 61(2):1–37, 2022","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":54,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.370220Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:09900f8b0d000c63df720d870a68564ce4eaca31a8a7b60ea0fff4fa72c24ae8","observation_id":"28262cdc-114a-4dbb-bd54-4d81786bae61","resolution":{"observed_at":"2026-08-04T13:43:13.370220Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.373037Z","title":"Sparse machine learning in Banach spaces.Applied Numerical Mathematics, 187:138–157, 2023","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":55,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.373037Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:1d9cc57dc4db60151f29cbc4f845dec8653b16dc9a2eb57e5e441a537eb46a5e","observation_id":"fc9a65b3-ee89-45d1-9c26-e31aacc936b5","resolution":{"observed_at":"2026-08-04T13:43:13.373037Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.376600Z","title":"Reproducing kernel Banach spaces for machine learning","venue":null,"work_id":null,"year":2009},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":56,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.376600Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:58119ad932aba4ebdfdc7d432f81ac58562218f7f1e0239e8913f74b3b6431d6","observation_id":"1a2d27c9-ff72-4fda-a9cb-d13265cd1797","resolution":{"observed_at":"2026-08-04T13:43:13.376600Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T13:43:13.379692Z","title":"Vector-valued reproducing kernel Banach spaces with applications to multi-task learning.Journal of Complexity, 29(2):195–215, 2013","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","version":3},"reference_index":57,"source":"pdf_text","source_observed_at":"2026-08-04T13:43:13.379692Z"},"links":{"citing_paper":"/paper/2509.26371"},"observation_digest":"sha256:34dbff67ee02b7f7f3c2ce06b98e298430152d24fedbb619e22bce169d1c5b9e","observation_id":"c8c35158-1ca9-471d-beca-a5f1c0a354be","resolution":{"observed_at":"2026-08-04T13:43:13.379692Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2509.26371","last_updated":"2026-07-24T13:26:28Z","latest_version":3,"primary_category":"math.FA","snapshot_observed_at":"2026-08-14T01:15:31.574561Z","submitted_at":"2025-09-30T15:06:24Z","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators"},"reference_resolution":{"displayed":57,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":57,"verified_exact":0,"verified_fuzzy":0},"total_outbound_references":57},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-21T06:32:19.484+00:00","source":"crossref"},{"observed_at":"2026-08-21T06:32:16.066871+00:00","source":"retraction_watch"}],"thesis":"As of 21 August 2026, this Paper Citation Record lists 57 of 57 outbound references and 0 inbound Pith citation observations for arXiv:2509.26371."}