Titlebook: Best Approximation in Inner Product Spaces; Frank Deutsch Textbook 2001 Springer-Verlag New York 2001 Convexity.Hilbert space.algorithms.c

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Qiang Xu,Changjun Li,Ming Ronnier LuoWe will describe the general problem of best approximation in an inner product space. A uniqueness theorem for best approximations from convex sets is also provided. The five problems posed in Chapter 1 are all shown to be special cases of best approximation from a convex subset of an appropriate inner product space.

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Antagonism

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Constrained Interpolation from a Convex Set,In many problems that arise in applications, one is given certain function values or “data” along with some reliable evidence that the unknown function that generated the data has a certain shape. For example, the function may be nonnegative or nondecreasing or convex. The problem is to recover the unknown function from this information.

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Lecture Notes in Electrical Engineering” approximation. While these problems seem to be quite different on the surface, we will later see that the first three (respectively the fourth and fifth) are special cases of the general problem of . (respectively .. In this latter formulation, the problem has a rather simple geometric interpretat

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https://doi.org/10.1007/978-981-19-9024-3ce theorems of interest. In particular, the two most useful existence and uniqueness theorems can be deduced from it. They are: (1) Every finite-dimensional subspace is Chebyshev, and (2) every closed convex subset of a Hilbert space is Chebyshev.

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https://doi.org/10.1007/978-981-19-9024-3deed, it will be the basis for . characterization theorem that we give. The notion of a dual cone plays an essential role in this characterization. In the particular case where the convex set is a subspace, we obtain the familiar orthogonality condition, which for finite-dimensional subspaces reduce

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Qiang Xu,Changjun Li,Ming Ronnier Luoand, if . is a subspace, even linear. There are a substantial number of useful properties that . possesses when . is a subspace or a convex cone. For example, every inner product space is the direct sum of any Chebyshev subspace and its orthogonal complement. More generally, a useful duality relatio