Approximation Numbers of Sobolev Embedding Operators on an Interval

Consider the Sobolev embedding operator from the space of functions in W 1,p(I) with average zero into Lp , 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. 1. Preliminaries Let I be an open, finite interval and let 1 p ∞. We will need to consider both real and complex spaces, and therefore by L K (I) we denote the complex L space over I for K = C and the corresponding real space for K = R. Similarly, W 1,p K (I) denotes the first-order L Sobolev space, real or complex according to whether K = R or K = C. We also denote the subspace of W 1,p K (I) consisting of functions vanishing at the endpoints of I by W 1,p 0,K(I). For f ∈ L p K (I), the average f̃ of f is defined by f̃ = 1 |I| ∫