Limit theorems of polynomial approximation with exponential weights

We develop the limit relations between the errors of polynomial approximation in weighted metrics and apply them to various problems in approximation theory such as asymptotically best constants, convergence of polynomials, approximation of individual functions, and multidimensional limit theorems of polynomial approximation. In addition we establish new inequalities for polynomials in complex domains and new asymptotics and estimates for orthogonal polynomials with exponential weights. More detailed information on approximation properties of functions is obtained for the canonical weights W(x) = exp(-vertical bar x vertical bar(alpha)), 0 < alpha < infinity, and some other weights.