The equivalence of the moduli of continuity of the best approximation operator and of strong unicity in \(L^ 1\)
Let X be a normed linear space and \(M_ n\) a finite dimensional subspace of X. For \(x\in X\) denote the distance from x to \(M_ n\) by \(\rho (x,M_ n)=\inf \{\| x-q\|:q\in M_ n\}\) and define \(P:X\to 2^ Mn\) by \(P(x)=\{q\in M_ n:\| x-q\| =\rho (x,M_ n)\}\). P is known as the best approximation operator. The local modulus of continuity of P at \(x\in X\) is \(\Omega (M_ n,x,\delta)=\sup \{\rho (q,P(x)):q\in P(x_ 1),x_ 1\in X,\| x-x_ 1\| \leq \delta \}\) and the local modulus of strong unicity of P at \(x\in X\) is \(\Omega *(M_ n,x,\delta)=\sup \{\rho (q,P(x)):q\in M_ n,\| x-q\| -\rho (x,M_ n)\leq \delta \}\). It is easily seen that \(\Omega (M_ n,x,\delta)\leq \Omega *(M_ n,x,2\delta)\). The question raised in this paper is whether these moduli can be of the same order and so we say that X satisfies the E-property if for any \(M_ n\subset X\) and \(x\in X\) there is a constant \(\gamma >0\) depending only on x and \(M_ n\) such that \(\Omega *(M_ n,x,\delta)\leq \Omega (M_ n,x,\gamma \delta)\). It is noted that all of the \(L^ p\)- spaces, \(1<p<\infty\), fail to have the E-property. It is proven that \(X=L^ 1=L^ 1(T,\Sigma,\mu)\), (T,\(\Sigma\),\(\mu)\) a complete, \(\sigma\)- finite, positive measure space, does satisfy the E-property and if (T,\(\Sigma\),\(\mu)\) is non-atomic we may take \(\gamma =1\). Derived from this is the equivalence of the Lipschitz continuity of P and strong unicity of best approximations in \(L^ 1\) as well as the equivalence of the Lipschitz and strong unicity constants for \(L^ 1\) approximation.
scientific article; zbMATH DE number 3939920
scientific article