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DE LA CALLE YSERN in both sides of the inequality. This fact is denoted in Table 1 by adding a prime on those conditions. Overall it can be noticed how, over the years, the approximants become more and more general, the conditions get weaker and weaker, and the results become more and more precise. 3. 1. Relative asymptotics and varying measures. We will see now how varying measures are useful not only to prove results on rational approximation but to obtain theorems about polynomials orthogonal with respect to a fixed measure as well.

25) ΘT = δξ . deg T ξ: T (ξ)=0 The sum is taken over all the zeros of T and δξ denotes the Dirac measure concentrated at ξ. Any subsequence of the probability measures {ΘA2n } has a weak star limit. If we are interested in describing the exact rate of convergence of the approximants πnA,B it is natural to require that limit to be made explicit. Thus, it is said that the sequence of polynomials {A2n }n∈N has the measure α as its asymptotic ∗ zero distribution if ΘA2n −→ α, n → ∞. By λΣ,α we will denote the equilibrium measure of the set Σ in the presence of the external field given by (minus) the logarithmic potential of α.

4 below). 1. Meromorphic Stieltjes functions. In the classical theorems mentioned above we find the simplest possible situation. Let us describe a much more sophisticated one whose solution was given by L´opez in [L89b]. We have opted to begin with that work because it is possible to find in it the most common ingredients and difficulties in Pad´e approximation and shows how varying measures come into the game and help to solve it. Let σ be a measure supported on Σ = [0, +∞) such that all its moments cm are finite numbers.

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