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Iterative signal recovery projection algorithms have also been implemented optically without sampling the continuous waveforms (e.g., Ref. 57). In such instances, the underlying signal space is ℒ itself.
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Let A be a set of real numbers. If R ≠ Ø, then inf(R) stands for the infimum of R, i.e., the largest number in [-∞, +x] that is smaller than or equal to all elements of R. By convention, inf(Ø) = +∞.
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60
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note
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For theoretical reasons, the sets (and functions) that we deal with must be "measurable" - this is not the same as being "physically measurable" or "observable"! For our purposes, measurable sets and functions constitute a sufficiently large class to work with; thus all closed and open subsets (and all continuous functions) are measurable, as well as various combinations of those.
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Mathematically, this set is assumed to have nonzero measure.
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The complex Hubert space ℒ from the phase retrieval problem is also a real Hubert space, provided that we use the real part of the inner product as the new inner product.
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64
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Recall the notation from Remark 3.4
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Recall the notation from Remark 3.4.
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N: A(t) ∩ Z ≠ Ø} is measurable for every closed (or, equivalently, open) set Z in C; see Section 8.1 in Ref. 59 and Section 14.A in Ref. 67.
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76
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84894002082
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note
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Alternative descriptions of these algorithms have been proposed; see, for instance, Ref. 77.
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77
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0025430092
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Image recovery using iterative data refinement with relaxation
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84894012228
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note
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14
-
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79
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84893997058
-
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note
-
n). In practical experiments for problem (5), however, this ambiguity has hardly an impact, as the sets γ and C-D almost always coincide.
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80
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0032620199
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Reconstruction of an object from its noisy Fourier modulus: Ideal estimate of the object to be constructed and a method that attempts to find that object
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H. Takajo, T. Shizuma, T. Takahashi, and S. Takahata, "Reconstruction of an object from its noisy Fourier modulus: ideal estimate of the object to be constructed and a method that attempts to find that object," Appl. Opt. 38, 5568-5576 (1999).
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81
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0000635375
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Further study on the convergence property of the hybrid input-output algorithm used for phase retrieval
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H. Takajo, T. Takahashi, and T. Shizuma, "Further study on the convergence property of the hybrid input-output algorithm used for phase retrieval," J. Opt. Soc. Am. A 16, 2163-2168 (1999).
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82
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Study on the convergence property of the hybrid input-output algorithm used for phase retrieval
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H. Takajo, T. Takahashi, R. Ueda, and M. Taninaka, "Study on the convergence property of the hybrid input-output algorithm used for phase retrieval," J. Opt. Soc. Am. A 15, 2849-2861 (1998).
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Takajo, H.1
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83
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84893999618
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note
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The corresponding mask is certainly much easier to implement.
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84
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84975571733
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Phase retrieval stagnation problems and solutions
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J. R. Fienup and C. C. Wackerman, "Phase retrieval stagnation problems and solutions," J. Opt. Soc. Am. A 3, 1897-1907 (1986).
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Fienup, J.R.1
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85
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84893992567
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note
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The algorithms discussed here for solving problem (25) can be viewed in the broader context of finding a zero of the sum of two maximal monotone operators. Good starting points are Refs. 86-88.
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86
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0043100459
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Fejér-monotonicity in convex optimization
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C. A. Floudas and P. M. Pardalos, eds. (Kluwer, New York)
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P. L. Combettes, "Fejér-monotonicity in convex optimization," in Encyclopedia of Optimization, C. A. Floudas and P. M. Pardalos, eds. (Kluwer, New York, 2001), Vol. 2, pp. 106-114.
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Combettes, P.L.1
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87
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0004177997
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Ph.D. thesis (Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Mass.), available as Rep. LIDS-TH-1877 (Laboratory for Information and Decision Sciences, MIT)
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J. Eckstein, "Splitting methods for monotone operators with applications to parallel optimization," Ph.D. thesis (Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Mass., 1989), available as Rep. LIDS-TH-1877 (Laboratory for Information and Decision Sciences, MIT).
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Splitting Methods for Monotone Operators with Applications to Parallel Optimization
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Eckstein, J.1
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88
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On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators
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J. Eckstein and D. P. Bertsekas, "On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators," Math. Program. (Ser. A) 55, 293-318 (1992).
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Eckstein, J.1
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89
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84894009949
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note
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90
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90
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0000256894
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Nonexpansive projections and resolvents of accretive operators in Banach spaces
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R. B. Bruck and S. Reich, "Nonexpansive projections and resolvents of accretive operators in Banach spaces," Houston J. Math. 3, 459-470 (1977).
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Bruck, R.B.1
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91
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0030246542
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On projection algorithms for solving convex feasibility problems
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H. H. Bauschke and J. M. Borwein, "On projection algorithms for solving convex feasibility problems," SIAM Rev. 38, 367-426 (1996).
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Bauschke, H.H.1
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D. Butnariu, Y. Censor, and S. Reich, eds. (Elsevier, Amsterdam)
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H. H. Bauschke, "Projection algorithms: results and open problems," in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Studies in Computational Mathematics, D. Butnariu, Y. Censor, and S. Reich, eds. (Elsevier, Amsterdam, 2001), Vol. 8, pp. 11-22.
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Bauschke, H.H.1
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93
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77956655101
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Quasi-Fejérian analysis of some optimization algorithms
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D. Butnariu, Y. Censor, and S. Reich, eds. (Elsevier, Amsterdam)
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P. L. Combettes, "Quasi-Fejérian analysis of some optimization algorithms," in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Studies in Computational Mathematics, D. Butnariu, Y. Censor, and S. Reich, eds. (Elsevier, Amsterdam, 2001), Vol. 8, pp. 115-152.
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Combettes, P.L.1
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94
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84948499836
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An algorithm for restricted least squares regression
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R. L. Dykstra, "An algorithm for restricted least squares regression," J. Am. Stat. Assoc. 78, 837-842 (1983).
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A method for finding projections onto the intersection of convex sets in Hubert spaces
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Springer, Berlin
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J. P. Boyle and R. L. Dykstra, "A method for finding projections onto the intersection of convex sets in Hubert spaces," in Advances in Order Restricted Statistical Inference (Springer, Berlin, 1986), pp. 28-47.
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Boyle, J.P.1
Dykstra, R.L.2
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96
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84894010481
-
-
note
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85 Dykstra's algorithm can be interpreted as a tight version of the Peaceman-Rachford algorithm. See page 77 in Ref. 87 for further information. Let us also note that in the standard linear case, the Peaceman-Rachford and Douglas-Rachford algorithms can be viewed from a unifying standpoint (see Section 7.4 in Ref. 97).
-
-
-
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98
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0037976260
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Dykstra's alternating projection algorithm for two sets
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H. H. Bauschke and J. M. Borwein, "Dykstra's alternating projection algorithm for two sets," J. Approx. Theory 79, 418-443 (1994).
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Bauschke, H.H.1
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99
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0042391294
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Dykstra's algorithm with Bregman projections: A convergence proof
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H. H. Bauschke and A. S. Lewis, "Dykstra's algorithm with Bregman projections: a convergence proof," Optimization 48, 409-427 (2000).
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Bauschke, H.H.1
Lewis, A.S.2
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100
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0001886656
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A cyclic projection algorithm via duality
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N. Gaffke and R. Mathar, "A cyclic projection algorithm via duality," Metrika 36, 29-54 (1989).
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Gaffke, N.1
Mathar, R.2
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101
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0023860607
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A successive projection method
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S.-P. Han, "A successive projection method," Math. Program. (Ser. A) 40, 1-14 (1988).
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Han, S.-P.1
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102
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Two generalizations of Dykstra's cyclic projections algorithm
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H. Hundal and F. Deutsch, "Two generalizations of Dykstra's cyclic projections algorithm," Math. Program. (Ser. A) 77, 335-355 (1997).
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Hundal, H.1
Deutsch, F.2
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103
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0027579732
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Signal recovery by best feasible approximation
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P. L. Combettes, "Signal recovery by best feasible approximation," IEEE Trans. Image Process. 2, 269-271 (1993).
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IEEE Trans. Image Process.
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Combettes, P.L.1
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104
-
-
84893987214
-
-
note
-
The Douglas-Rachford algorithm was originally developed as a linear implicit iterative method to solve partial differential equations in Ref. 105 (see also Chap. 7 in Ref. 97). It was extended to an operator splitting method for finding a zero of the sum of two maximal monotone operators by Lions and Mercier in Ref. 106. When it is applied to the normal cone maps of the constraint sets, one obtains a method for solving problem (25). See Refs. 86-88 and 106 for further information.
-
-
-
-
105
-
-
84967782959
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On the numerical solution of heat conduction problems in two or three space variables
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J. Douglas and H. H. Rachford, "On the numerical solution of heat conduction problems in two or three space variables," Trans. Am. Math. Soc. 82, 421-439 (1956).
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Trans. Am. Math. Soc.
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Douglas, J.1
Rachford, H.H.2
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106
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0000345334
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Splitting algorithms for the sum of two nonlinear operators
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SIAM (Soc. Ind. Appl. Math.)
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P.-L. Lions and B. Mercier, "Splitting algorithms for the sum of two nonlinear operators," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 16, 964-979 (1979).
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, pp. 964-979
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Lions, P.-L.1
Mercier, B.2
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107
-
-
84893994753
-
-
note
-
kn ⇀ u.
-
-
-
-
108
-
-
84894005553
-
-
note
-
If we had used the literal update rule for the HIO algorithm, the present observation would change only in one respect: the set A would be replaced with S(n) (see Remark 4.1 and Ref. 79) and hence vary with n.
-
-
-
-
109
-
-
0002130882
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Projections on convex sets in Hubert space and spectral theory
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E. H. Zarantonello, ed. (Academic, New York)
-
E. H. Zarantonello, "Projections on convex sets in Hubert space and spectral theory," in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, ed. (Academic, New York, 1971), pp. 237-424.
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Zarantonello, E.H.1
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110
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0003712747
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Academic, San Diego, Calif.
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L. Debnath and P. Mikusiński, Introduction to Hilbert Spaces with Applications, 2nd ed. (Academic, San Diego, Calif., 1999).
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Debnath, L.1
Mikusiński, P.2
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111
-
-
84893991498
-
-
note
-
n) of points in M converges weakly to a point x, then x may not be in M.
-
-
-
-
112
-
-
84894009913
-
-
note
-
Bx), as is the case when dim ℋ < +∞ (or when B is a closed affine subspace). Note, however, that the projector onto a closed convex set may fail to be weakly continuous. An example is on page 245 in Ref. 109.
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