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A note on alternating projections in Hilbert space

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Abstract

We provide a direct proof of a result regarding the asymptotic behavior of alternating nearest point projections onto two closed and convex sets in a Hilbert space. Our arguments are based on nonexpansive mapping theory.

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References

  1. Aronszajn N (1950) Theory of reproducing kernels. Trans. Amer. Math. Soc. 68: 337–404

    Article  MATH  MathSciNet  Google Scholar 

  2. J.-B. Baillon, R. E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4 (1978), 1–9.

    Google Scholar 

  3. H. H. Bauschke, The composition of projections onto closed convex sets in Hilbert space is asymptotically regular. Proc. Amer. Math. Soc. 131 (2003), 141–146.

    Google Scholar 

  4. H. H. Bauschke and J. M. Borwein, On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1 (1993), 185–212.

    Article  MATH  MathSciNet  Google Scholar 

  5. H. H. Bauschke and J. M. Borwein, Dykstra’s alternating projection algorithm for two sets. J. Approx. Theory 79 (1994), 418–443.

    Google Scholar 

  6. H. H. Bauschke, V. Martín Márquez, S. M. Moffat and X. Wang, Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular. Fixed Point Theory Appl. 2012 (2012), 1–11.

  7. H. H. Bauschke, E. Matoušková and S. Reich, Projection and proximal point methods: Convergence results and counterexamples. Nonlinear Anal. 56 (2004), 715–738.

    Google Scholar 

  8. Bregman L.M (1965) Finding the common point of convex sets by the method of successive projection. Soviet Math. Dokl. 6: 688–692

    MATH  Google Scholar 

  9. R. E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3 (1977), 459–470.

    Google Scholar 

  10. W. Cheney and A. A. Goldstein, Proximity maps for convex sets. Proc. Amer. Math. Soc. 10 (1959), 448–450.

  11. F. Deutsch, Best Approximation in Inner Product Spaces. Springer, New York, 2001.

  12. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York, 1984.

    MATH  Google Scholar 

  13. H. S. Hundal, An alternating projection that does not converge in norm. Nonlinear Anal. 57 (2004), 35–61.

    Google Scholar 

  14. E. Kopecká, Spokes, mirrors and alternating projections. Nonlinear Anal. 68 (2008), 1759–1764.

    Google Scholar 

  15. E. Kopecká and S. Reich, A note on the von Neumann alternating projections algorithm. J. Nonlinear Convex Anal. 5 (2004), 379–386.

    Google Scholar 

  16. E. Kopecká and S. Reich, Another note on the von Neumann alternating projections algorithm. J. Nonlinear Convex Anal. 11 (2010), 455–460.

    MATH  MathSciNet  Google Scholar 

  17. E. Kopecká and S. Reich, Alternating projections and orthogonal decompositions. J. Nonlinear Convex Anal. 12 (2011), 155–159.

  18. E. Matoušková and S. Reich, The Hundal example revisited. J. Nonlinear Convex Anal. 4 (2003), 411–427.

  19. H. Nakano, Spectral Theory in Hilbert Space. Japan Soc. Promotion Sci., Tokyo, 1953.

  20. J. von Neumann, On rings of operators. Reduction theory. Ann. of Math. (2) 50 (1949), 401–485.

    Article  MATH  Google Scholar 

  21. N. Wiener, On the factorization of matrices. Comment. Math. Helv. 29 (1955), 97–111.

    Google Scholar 

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Correspondence to Simeon Reich.

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Kopecká, E., Reich, S. A note on alternating projections in Hilbert space. J. Fixed Point Theory Appl. 12, 41–47 (2012). https://doi.org/10.1007/s11784-013-0097-4

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