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The first construction is obtained by choosing functions $\\phi_i$, $i=1,...,r$, with compactly supported Fourier transforms $\\hat{\\phi}_i$, $i=1,...,r$. The second construction, with compactly supported $\\phi_i,"},"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":"1109.3285","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-09-15T08:17:23Z","cross_cats_sorted":[],"title_canon_sha256":"30f5dfa3747cdef233f2c69eed66c1a19744c5d6b51775528102abb807c81c6e","abstract_canon_sha256":"12d35a9faa322c0e2d9c6685f72ba8751f42d45a597fe6ac94ff845b72ec11aa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:23.433542Z","signature_b64":"RKGbshUgZYwjdeQcZ0AK8VZ7t/oluRbiXRrsFp4ulsCNXz2NYEWaBjZmoFMFUq9Zg0pKg1LYY5cDUuORJpSdCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cab365181f69f1431de8ea2208f3d8bd09a184a311c7955418f66a1d5cf56f29","last_reissued_at":"2026-05-18T03:48:23.432878Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:23.432878Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Construction of frames for shift-invariant spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Stevan Pilipovic, Suzana Simic","submitted_at":"2011-09-15T08:17:23Z","abstract_excerpt":"We construct a sequence ${\\phi_i(\\cdot-j)\\mid j\\in{\\ZZ}, i=1,...,r}$ which constitutes a $p$-frame for the weighted shift-invariant space [V^p_{\\mu}(\\Phi)=\\Big{\\sum\\limits_{i=1}^r\\sum\\limits_{j\\in{\\mathbb{Z}}}c_i(j)\\phi_i(\\cdot-j)  \\Big| {c_i(j)}_{j\\in{\\mathbb{Z}}}\\in\\ell^p_{\\mu}, i=1,...,r\\Big}, p\\in[1,\\infty],] and generates a closed shift-invariant subspace of $L^p_\\mu(\\mathbb{R})$. 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