Local Lipschitz constants (Q801497)

Let X be a closed subset of \(I=[-1,1]\). For \(f\in C[X]\), the local Lipschitz constant is defined to be \(\lambda_{n\delta}(f)=\sup \{\| B_ n(f)-B_ n(g)\| /\| f-g\|:0<\| f-g\| \leq \delta \}\), where \(B_ n(g)\) is the best approximation in the sup norm to g on X from the set of polynomials of degree at most n. It is shown that under certain assumptions the norm of the derivative of the best approximation operator at f is equal to the limit as \(\delta\) \(\to 0\) of the local Lipschitz constant of f, and an explicit expression is given for this common value. Then possibly very different, characterizations of local and global Lipschitz constants are also considered.