APPROXIMATION NUMBERS OF SOBOLEV EMBEDDING OPERATORS ON AN INTERVAL

Abstract

Consider the Sobolev embedding operator from the space of functions in $W^{1,p}(I)$ with average zero into $L^p$, where $I$ is a finite interval and $p\,{>}\,1$. This operator plays an important role in recent work. The operator norm and its approximation numbers in closed form are calculated. The closed form of the norm and approximation numbers of several similar Sobolev embedding operators on a finite interval have recently been found. It is proved in the paper that most of these operator norms and approximation numbers on a finite interval are the same.