By Amir

Each mathematician operating in Banaeh spaee geometry or Approximation concept is familiar with, from his personal experienee, that the majority "natural" geometrie houses could faH to carry in a generalnormed spaee except the spaee is an internal produet spaee. To reeall the weIl recognized definitions, this suggests IIx eleven = *, the place is an internal (or: scalar) product on E, Le. a functionality from ExE to the underlying (real or eomplex) box pleasing: (i) O for x o. (ii) is linear in x. (iii) = (intherealease, thisisjust =

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3 in [BeMi]. Summing up, it is possible to obtain convergence results if either the speed of approach of the interpolation points to the singularities of the function is not very quick or the measure does not have much weight where we are interpolating. 17) or an analog of it. 10? The way of proceeding is similar in all cases. The problem is ﬁrstly solved for polynomials orthogonal with respect to a (varying) measure supported on the unit circle. 8. For each n ∈ N, let Wn (z) = z n + . . be a monic polynomial whose zeros wn,1 , .

Xn of the Pad´e approximant πn belong to ∆. 4) πn (z) = i=1 λn,i , z − xn,i where λn,i , i = 1, . . , n, are the Christoﬀel numbers of the Gauss-Jacobi quadrature corresponding to the measure µ. Hence, λn,i > 0, i = 1, . . , n, and the family {πn }n∈N is normal in C \ ∆ whence the proof follows. Let g K stand for the maximum absolute value of the continuous function g on the compact set K. 5) lim sup µ − πn n→∞ 1/2n K ≤ e−τ , τ = inf gC\Σ (z, ∞), z ∈ K , where K is any compact subset of C\ ∆ and gΩ (z, ∞) stands for the Green function of the region Ω.

Then lim µ(z) − πnA,B (z) n→∞ 1/2n = e−τ (z) , τ (z) = F gC\Σ (z, ζ) dα(ζ), uniformly on compact subsets of C \ (F ∪ ∆), and lim sup µ(z) − πnA,B (z) 1/2n = e−τ (z) , n→∞ uniformly on compact subsets of C \ ∆ of positive capacity. 5) when a2n,i = ∞ for all i = 1, . . , n, and n ∈ N. The work [L98c] essentially deals with the same problem. e. on [−1, 1]) while another choice of the degrees of the polynomials pn and qn makes the result to be displayed in a slightly diﬀerent fashion. Also, the rate k(n)/n must have a limit.