On Stability of the Metric Projection Operator
András Kroó, Allan M. Pinkus · SIAM Journal on Mathematical Analysis · 2013
Let $M$ be a closed linear subspace of a normed linear space $X$. For a given $f\in X$ denote by $P_M f$ the set of best approximations to $f$ from $M$. The operator $P_M$ is termed the metric projection onto $M$. In this paper we are interested in the stability of the metric projection $P_M$ relative to perturbations of the subspace $M$. We mainly consider the case where $X=L^p$, $p\in [1,\infty]$. We consider a measure of distance $d(M,N)$ between subspaces $M$ and $N$ and estimate $\|P_Mf-P_Nf\|$ in terms of $d(M,N)$ and $\|f\|$. Typically such an estimate will be of order $d(M,N)^\beta$ with some $\beta$ which, in general, depends on the geometry of the space $X$.