On best approximation in \(L_ p\) spaces
Let \(L_ p:=L_ p(S,\Sigma,\mu)\), \(1\leq p<\infty\), be the Banach space of all \(\mu\)-measurable extended real valued functions (equivalence classes) y on S such that \(\| y\|:=[\int | y(s)|^ p\mu (ds)]^{1/p}<\infty,\) where (S,\(\Sigma\),\(\mu)\) is a positive measure space. If X is a convex closed nonempty subset of \(L_ p\), then an element z in X is called a best approximation to an element y in \(L_ p\) if \(\| y-z\| \leq \| y-x\|\), for all x in S. The author proved [ibid. 47, 184-194 (1986; Zbl 0615.41027) and J. Math. Anal. Appl. 115, 155-172 (1986; Zbl 0593.49004)] that there exists a positive constant \(c_ p\leq 1\) independent of the element y in \(L_ p\), \(2\leq p<\infty\), such that the strong unicity inequality (*) \(\| y-z\|^ p\leq \| y-x\|^ p-c_ p\| z-x\|^ p\) holds for all x in X. The largest constant \(c_ p\) in this inequality is \(c_ p=(1+t_ 0^{p- 1})(1+t_ 0)^{1-p}=(p-1)(1+t_ 0)^{2-p},\) where \(t_ 0=t_ 0(p)\) denotes the unique zero of the function \(g(t)=-t^{p-1}+(p-1)t+p-2\) in the interval (1,\(\infty)\) for \(p>2\), and \(t_ 0(2)=1.\) In this paper the author establishes a counterpart to (*) for a subspace of \(L_ p\) where \(1\leq p<2\). More precisely, let X be a subspace of \(L_ p(S,\Sigma,\mu)\), \(1\leq p<2\), and let \(z\in X\) be a best approximation to an element y in \(L_ p\). Then \(\| y-z\|^ p\leq \| y-x\|^ p\leq \| y-z\|^ p+c_ p\| z-x\|^ p,\) for all x in X, where it is additionally assumed that \(\mu\) (\(\{\) \(s\in S:\) \(y(s)=z(s)\})=0\) when \(p=2\).
scientific article; zbMATH DE number 4011130
scientific article