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Let $A$ and $B$ be $d\\times d$ and $s\\times s$ \\thematrix matrices with $d, s\\geq 1$ respectively and let $\\varphi_A$ be a scaling function associated with matrix $A$ and generated by a finite solution. There always exists a scaling function $\\varphi_B$ associated with matrix $B$ such that \\begin{equation*}\n  \\varphi_B \\simeq \\varphi_A. \\end{equation*} An example shows that t"},"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":"1904.07139","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-04-15T15:45:41Z","cross_cats_sorted":[],"title_canon_sha256":"fb762f03f3faf90ec68ec6c2cfe38a04f97c79a05162296c55b389e9ce700484","abstract_canon_sha256":"ec1ed900bf346c2284675da1c21e2e49beba0bef4efce7bdab1e65f16558bb74"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:35.160644Z","signature_b64":"s8vw6L1wDIhZzKxdCCZqeaBw1VA4EStr5tdOElsEoubPE9FywLXaSNxxWr0xw4o9KSQBWoPZfyFRUh1oHpbiDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65e72493085c80a0c40f776607b1e5ccea584e5e57a6cfa4fe425b2d5154085b","last_reissued_at":"2026-05-17T23:48:35.160106Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:35.160106Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Isomorphism in Wavelets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Wei Huang, Xingde Dai","submitted_at":"2019-04-15T15:45:41Z","abstract_excerpt":"Two scaling functions $\\varphi_A$ and $\\varphi_B$ for Parseval frame wavelets are algebraically isomorphic, $\\varphi_A \\simeq \\varphi_B$, if they have matching solutions to their (reduced) isomorphic systems of equations. Let $A$ and $B$ be $d\\times d$ and $s\\times s$ \\thematrix matrices with $d, s\\geq 1$ respectively and let $\\varphi_A$ be a scaling function associated with matrix $A$ and generated by a finite solution. 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