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Construction of best Bregman approximations in reflexive Banach spaces. (English) Zbl 1040.41016

Let \(X\) be a reflexive Banach space and let \(f: X \to (-\infty,\infty]\) be a lower semicontinuous convex function which is Gâteaux differentiable on \(\operatorname{int} \operatorname{dom}f \neq \emptyset\). Assume further that \(\partial f\) is both locally bounded and single-valued on its domain and \((\partial f)^{-1}\) is locally bounded on its domain, \(f\) is strictly convex on every convex subset of \(\operatorname{dom} \partial f\). The Bregman distance associated with \(f\) is the function \(D: X \times X \to [0,\infty]\) defined by \(D(x,y) = f(x)-f(y)-(x-y,\nabla f(y))\) if \(y \in \operatorname{int} \operatorname{dom}f\) and as \(\infty\) otherwise. Let \(x_0 \in \operatorname{int} \operatorname{dom}f\), \((\operatorname{int} \operatorname{dom} f)\cap \bigcap S_i \neq \emptyset,\) and \(S=\overline{\operatorname{dom}}f \cap \bigcap S_i\) where \(\{S_i\}_{i \in I}\) is a countable family of closed and convex subsets of \(X\).
In this paper the authors present a method for finding the best Bregman approximation to \(x_0\) from \(S\). A renorming result that the authors attribute to Professor J. D. Vanderwerff, saying that any reflexive space can be equivalently renormed so that in the new norm the space is strictly convex, Gâteaux smooth but fails to have the Kadec-Klee property will also be of interest to a general reader.

MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A29 Approximation with constraints
41A50 Best approximation, Chebyshev systems
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