Bounds for orthogonal polynomials for exponential weights

Orthogonal polynomials pn(W2,x) for exponential weights W2 = e−2Q on a finite or infinite interval I, have been intensively studied in recent years. We discuss efforts of the authors to extend and unify some of the theory; our deepest result is the bound ¦pn(W2,x)¦W(x)¦(x−a−n)(x−an)¦14 ⩽ C, x∈I with C independent of n and x. Here a±n are the Mhaskar-Rahmanov-Saff numbers for Q and Q must satisfy some smoothness conditions on I.