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**Sample text**

Let W := diag(w0 , w1 , . . , wN ) be the diagonal matrix of the polynomial weights to x0 , x1 , . . , xN . 1 shows that one has the bijective map u = Wq; the kernel equation Au = 0 may thus be written AWq = 0. 3. 2) with n ≤ m exists, then the vector q of its denominator values belongs to the kernel of the N × (N + 1)-matrix AW. The elements of AW are given in [Zhu-Zhu]. 2. Reduced complexity barycentric rational interpolation Besides its many advantages, the barycentric representation of r ∈ Rm,n also has some disadvantages in comparison with the canonical r(x) = (am xm + · · · + a1 x + a0 )/(bn xn +· · ·+b1 x+b0 ).

Nodes like Chebyshev’s, distributed like 1 important since they lead to exponential convergence for holomorphic functions (see [Ber-Tre] for citations). , faster than polynomial. N. 3) is the formula of choice if one wishes to evaluate p as stably as possible (even when p is a very bad approximant of f ). 20 years ago already, W. Werner [WeW] has given an algorithm for updating the weights wj in O(N ) operations when a point is added. Werner’s algorithm Recent Developments in Barycentric Rational Interpolation 29 contains an unstable expression for the weight corresponding to the new point.

N 2n 4n2 We are now in the position to state the following result. 1. For every f ∈ C[0, 1], lim Ln f = f n→∞ uniformly on [0, 1]. Moreover, for every x ∈ [0, 1] and n ≥ M0 , a+b+2 |v(x)| x(1 − x) + M ω2 f, 1 − , 2n 2n n where ω1 and ω2 denote the ordinary first and second moduli of smoothness and M is a suitable constant independent of f, n and x. |Ln f (x) − f (x)| ≤ ω1 f, Proof. The first statement follows from formulae (5)–(7) and the Korovkin theorem. 43) of [4] into account, we get |Ln f (x) − f (x)| v v(x) v(x) (x) − f x + | + |f x + − f (x)| 2n 2n 2n v |v(x)| x(1 − x) f ◦ e1 + , + ω1 f, 2n n 2n |v(x)| a+b+2 x(1 − x) + ω1 f, .