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Also for any given positive integer $m$, let ${\\cal A}(m,K)$ denote the space of all $f$ such that $f$ is holomorphic in a neighborhood of $K$ and $f(0)=f^{\\prime}(0)=...=f^{(m)}(0)=0$. Then ${\\cal A}(m,K)$ is dense in ${\\cal A}(K)$ under the supremum norm on $K$ provided that there exists "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0312123","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"2003-12-05T10:55:48Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"dbfc3e9f05e9948216e260e92f0e3782d8ba63595a941d5f335118da5eada966","abstract_canon_sha256":"dbba6793364444ec430ab0cf4d1daeb47664c3b52608a0dc9701a8b5707ceaa6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:26.691936Z","signature_b64":"xGDuflqcdVZr1jnBnWVQjZp1ZAfzqnOEN8D3XsqRl+wKEfbRAtXGGEK+yfFTEwgrwgEL++UMYEHHAc0qNLBGDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0c9badd38804e8783ec59f4fb99d95a2afae1065ce4345575835f31ea95ca1b","last_reissued_at":"2026-05-18T01:05:26.691216Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:26.691216Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some approximation theorems","license":"","headline":"","cross_cats":["math.CV"],"primary_cat":"math.FA","authors_text":"N. V. Rao","submitted_at":"2003-12-05T10:55:48Z","abstract_excerpt":"The general theme of this note is illustrated by the following theorem:\n Theorem 1. Suppose $K$ is a compact set in the complex plane and 0 belongs to the boundary $\\partial K$. Let ${\\cal A}(K)$ denote the space of all functions $f$ on $K$ such that $f$ is holomorphic in a neighborhood of $K$ and $f(0)=0$. Also for any given positive integer $m$, let ${\\cal A}(m,K)$ denote the space of all $f$ such that $f$ is holomorphic in a neighborhood of $K$ and $f(0)=f^{\\prime}(0)=...=f^{(m)}(0)=0$. Then ${\\cal A}(m,K)$ is dense in ${\\cal A}(K)$ under the supremum norm on $K$ provided that there exists "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0312123","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0312123","created_at":"2026-05-18T01:05:26.691308+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0312123v1","created_at":"2026-05-18T01:05:26.691308+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0312123","created_at":"2026-05-18T01:05:26.691308+00:00"},{"alias_kind":"pith_short_12","alias_value":"6DE3VXJYQBHI","created_at":"2026-05-18T12:25:51.375804+00:00"},{"alias_kind":"pith_short_16","alias_value":"6DE3VXJYQBHIPA7M","created_at":"2026-05-18T12:25:51.375804+00:00"},{"alias_kind":"pith_short_8","alias_value":"6DE3VXJY","created_at":"2026-05-18T12:25:51.375804+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6DE3VXJYQBHIPA7MLH2PXGOZLI","json":"https://pith.science/pith/6DE3VXJYQBHIPA7MLH2PXGOZLI.json","graph_json":"https://pith.science/api/pith-number/6DE3VXJYQBHIPA7MLH2PXGOZLI/graph.json","events_json":"https://pith.science/api/pith-number/6DE3VXJYQBHIPA7MLH2PXGOZLI/events.json","paper":"https://pith.science/paper/6DE3VXJY"},"agent_actions":{"view_html":"https://pith.science/pith/6DE3VXJYQBHIPA7MLH2PXGOZLI","download_json":"https://pith.science/pith/6DE3VXJYQBHIPA7MLH2PXGOZLI.json","view_paper":"https://pith.science/paper/6DE3VXJY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0312123&json=true","fetch_graph":"https://pith.science/api/pith-number/6DE3VXJYQBHIPA7MLH2PXGOZLI/graph.json","fetch_events":"https://pith.science/api/pith-number/6DE3VXJYQBHIPA7MLH2PXGOZLI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6DE3VXJYQBHIPA7MLH2PXGOZLI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6DE3VXJYQBHIPA7MLH2PXGOZLI/action/storage_attestation","attest_author":"https://pith.science/pith/6DE3VXJYQBHIPA7MLH2PXGOZLI/action/author_attestation","sign_citation":"https://pith.science/pith/6DE3VXJYQBHIPA7MLH2PXGOZLI/action/citation_signature","submit_replication":"https://pith.science/pith/6DE3VXJYQBHIPA7MLH2PXGOZLI/action/replication_record"}},"created_at":"2026-05-18T01:05:26.691308+00:00","updated_at":"2026-05-18T01:05:26.691308+00:00"}