Ritt operators and convergence in the method of alternating projections
Abstract. Given N ≥ 2 closed subspaces M1, . . . , MN of a Hilbert space X, let Pk denote the orthogonal projection onto Mk, 1 ≤ k ≤ N. It is known that the sequence (xn), defined recursively by x0 = x and xn+1 = PN · · · P1xn for n ≥ 0, converges in norm to PMx as n → ∞ for all x ∈ X, where PM deno...
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| Формат: | Journal article |
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Elsevier
2016
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| _version_ | 1826272310932275200 |
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| author | Badea, C Seifert, D |
| author_facet | Badea, C Seifert, D |
| author_sort | Badea, C |
| collection | OXFORD |
| description | Abstract. Given N ≥ 2 closed subspaces M1, . . . , MN of a Hilbert space X, let Pk denote the orthogonal projection onto Mk, 1 ≤ k ≤ N. It is known that the sequence (xn), defined recursively by x0 = x and xn+1 = PN · · · P1xn for n ≥ 0, converges in norm to PMx as n → ∞ for all x ∈ X, where PM denotes the orthogonal projection onto M = M1∩. . .∩MN . Moreover, the rate of convergence is either exponentially fast for all x ∈ X or as slow as one likes for appropriately chosen initial vectors x ∈ X. We give a new estimate in terms of natural geometric quantities on the rate of convergence in the case when it is known to be exponentially fast. More importantly, we then show that even when the rate of convergence is arbitrarily slow there exists, for each real number α > 0, a dense subset Xα of X such that kxn − PMxk = o(n −α ) as n → ∞ for all x ∈ Xα. Furthermore, there exists another dense subset X∞ of X such that, if x ∈ X∞, then kxn−PMxk = o(n −α ) as n → ∞ for all α > 0. These latter results are obtained as consequences of general properties of Ritt operators. As a by-product, we also strengthen the unquantified convergence result by showing that PMx is in fact the limit of a series which converges unconditionally |
| first_indexed | 2024-03-06T22:10:32Z |
| format | Journal article |
| id | oxford-uuid:51a5d018-a154-4a09-aeeb-9b48cfbcd407 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T22:10:32Z |
| publishDate | 2016 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:51a5d018-a154-4a09-aeeb-9b48cfbcd4072022-03-26T16:20:48ZRitt operators and convergence in the method of alternating projectionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:51a5d018-a154-4a09-aeeb-9b48cfbcd407Symplectic Elements at OxfordElsevier2016Badea, CSeifert, DAbstract. Given N ≥ 2 closed subspaces M1, . . . , MN of a Hilbert space X, let Pk denote the orthogonal projection onto Mk, 1 ≤ k ≤ N. It is known that the sequence (xn), defined recursively by x0 = x and xn+1 = PN · · · P1xn for n ≥ 0, converges in norm to PMx as n → ∞ for all x ∈ X, where PM denotes the orthogonal projection onto M = M1∩. . .∩MN . Moreover, the rate of convergence is either exponentially fast for all x ∈ X or as slow as one likes for appropriately chosen initial vectors x ∈ X. We give a new estimate in terms of natural geometric quantities on the rate of convergence in the case when it is known to be exponentially fast. More importantly, we then show that even when the rate of convergence is arbitrarily slow there exists, for each real number α > 0, a dense subset Xα of X such that kxn − PMxk = o(n −α ) as n → ∞ for all x ∈ Xα. Furthermore, there exists another dense subset X∞ of X such that, if x ∈ X∞, then kxn−PMxk = o(n −α ) as n → ∞ for all α > 0. These latter results are obtained as consequences of general properties of Ritt operators. As a by-product, we also strengthen the unquantified convergence result by showing that PMx is in fact the limit of a series which converges unconditionally |
| spellingShingle | Badea, C Seifert, D Ritt operators and convergence in the method of alternating projections |
| title | Ritt operators and convergence in the method of alternating projections |
| title_full | Ritt operators and convergence in the method of alternating projections |
| title_fullStr | Ritt operators and convergence in the method of alternating projections |
| title_full_unstemmed | Ritt operators and convergence in the method of alternating projections |
| title_short | Ritt operators and convergence in the method of alternating projections |
| title_sort | ritt operators and convergence in the method of alternating projections |
| work_keys_str_mv | AT badeac rittoperatorsandconvergenceinthemethodofalternatingprojections AT seifertd rittoperatorsandconvergenceinthemethodofalternatingprojections |