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It is known that the sequence $(x_n)$, defined recursively by $x_0=x$ and $x_{n+1}=P_N\\cdots P_1x_n$ for $n\\ge0$, converges in norm to $P_Mx$ as $n\\to\\infty$ for all $x\\in X$, where $P_M$ denotes the orthogonal projection onto $M=M_1\\cap\\dotsc\\cap M_N$. Moreover, the rate of convergence is either exponentially fast for all $x\\in X$ or as slow as one likes for appropriately chosen initial vectors $x\\in X$. We give a new estimate in terms of natural geome"},"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":"1510.04560","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-10-15T14:50:15Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"40926cf8b4be80eede666fdf808ce02cb63a7568a6f379bf0ca622bfa021b9db","abstract_canon_sha256":"55b03eb3ff44bb0d07acf725a51a48f227fb7d3f0fd8ddfc9ec88a6d85c77389"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:07.813306Z","signature_b64":"/2nnBIsnANM+dyIudgMYaZrEaloaRGUpDtJbOL9V4fbUz75zdIak3O5Shl5QWRiHgP0S1/k54F6q15zIIM0kDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0372513da96322fcd8c17a819723a46cd603d766e29bce2c4248e60fa80f746","last_reissued_at":"2026-05-17T23:54:07.812751Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:07.812751Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ritt operators and convergence in the method of alternating projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.FA","authors_text":"Catalin Badea, David Seifert","submitted_at":"2015-10-15T14:50:15Z","abstract_excerpt":"Given $N\\ge2$ closed subspaces $M_1,\\dotsc, M_N$ of a Hilbert space $X$, let $P_k$ denote the orthogonal projection onto $M_k$, $1\\le k\\le N$. It is known that the sequence $(x_n)$, defined recursively by $x_0=x$ and $x_{n+1}=P_N\\cdots P_1x_n$ for $n\\ge0$, converges in norm to $P_Mx$ as $n\\to\\infty$ for all $x\\in X$, where $P_M$ denotes the orthogonal projection onto $M=M_1\\cap\\dotsc\\cap M_N$. Moreover, the rate of convergence is either exponentially fast for all $x\\in X$ or as slow as one likes for appropriately chosen initial vectors $x\\in X$. 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