### A best approximation property of the generalized spline functions.

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Let V ⊂ Z be two subspaces of a Banach space X. We define the set of generalized projections by ${}_{V}(X,Z):={P\in (X,Z):P|}_{V}=id$. Now let X = c₀ or ${l}_{\infty}^{m}$, Z:= kerf for some f ∈ X* and $V:=Z\cap l{\u207f}_{\infty}$ (n < m). The main goal of this paper is to discuss existence, uniqueness and strong uniqueness of a minimal generalized projection in this case. Also formulas for the relative generalized projection constant and the strong uniqueness constant will be given (cf. J. Blatter and E. W. Cheney [Ann. Mat. Pura Appl. 101 (1974), 215-227] and G. Lewicki...

Let $D$ be a nonempty compact subset of a Banach space $X$ and denote by $S\left(X\right)$ the family of all nonempty bounded closed convex subsets of $X$. We endow $S\left(X\right)$ with the Hausdorff metric and show that there exists a set $\mathcal{F}\subset S\left(X\right)$ such that its complement $S\left(X\right)\setminus \mathcal{F}$ is $\sigma $-porous and such that for each $A\in \mathcal{F}$ and each $\tilde{x}\in D$, the set of solutions of the best approximation problem $\parallel \tilde{x}-z\parallel \to min$, $z\in A$, is nonempty and compact, and each minimizing sequence has a convergent subsequence.

Let $Y\subset {l}_{\infty}^{\left(n\right)}$ be a hyperplane and let $A\in \mathcal{L}\left(Y\right)$ be given. Denote $$\begin{array}{ccc}\hfill \mathcal{A}=& \{L\in \mathcal{L}({l}_{\infty}^{\left(n\right)},Y):L\mid Y=A\}\phantom{\rule{4.0pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}\hfill & \hfill {\lambda}_{A}=inf\{\parallel L\parallel :L\in \mathcal{A}\}.\end{array}$$ In this paper the problem of calculating of the constant ${\lambda}_{A}$ is studied. We present a complete characterization of those $A\in \mathcal{L}\left(Y\right)$ for which ${\lambda}_{A}=\parallel A\parallel $. Next we consider the case ${\lambda}_{A}>\parallel A\parallel $. Finally some computer examples will be presented.

As a counterpart to best approximation, the concept of best coapproximation was introduced in normed linear spaces by C. Franchetti and M. Furi in 1972. Subsequently, this study was taken up by many researchers. In this paper, we discuss some results on the existence and uniqueness of best approximation and best coapproximation when the underlying spaces are metric linear spaces

In this paper we deal with a best approximation of a vector with respect to a closed semi-algebraic set C in the space ℝⁿ endowed with a semi-algebraic norm ν. Under additional assumptions on ν we prove semi-algebraicity of the set of points of unique approximation and other sets associated with the distance to C. For C irreducible algebraic we study the critical point correspondence and introduce the ν-distance degree, generalizing the notion developed by other authors for the Euclidean norm. We...