By J Szabados, Peter Vertesi

This publication offers a scientific survey at the most important result of interpolation thought within the final 40 years. It offers with Lagrange interpolation together with decrease estimates, wonderful and tough thought, interpolatory proofs of Jackson and Teliakovski-Gopengauz theorems, Lebesgue functionality, Lebesgue consistent of Lagrange interpolation, Bernstein and Erdös conjecture at the optimum nodes, the just about far and wide divergence of Lagrange interpolation for arbitrary process of nodes, Hermite-Fejer variety and lacunary interpolation and different comparable themes.

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13) and n\J{\ = A J W ) * ( £ + j)A, »= 0 , . . , [ ^ ] - l , ^ m a x ^ . n ^ ) ) . /< is > A( M+12' , k = 1,... e, at least A e ( ± - ±) > 3 more than AW). We shall construct the 17*'s in two phases. In the first phase, in each Ji where at least one 0* occurs, replace B/t's by these 0/fc's, and leave the remaining 6*'s unchanged. According to the previous argument, there is at least one such unchanged 6 * in each J t (call them free node(s)). This system fulfils so far only (a). We would like to ensure (b).

IS E ^ g J & ^ . 1 vA-4»-i n ^Vi-4»-i x ^[ k* - x n , n _i | n^ 1 - sjfc,n-i n-1 = X>,„-i(T,l)| k=l = 0(logn), H > - . 59). 59) is that qn can be chosen such that An(V(q)) = Jin{X) ~ n6. 58) implies Urn fA„(V(g)) = oo. , Qn—►<» Qn-+00 pends on qn continuously, there exists a qn with the said property. Then define Y^ =V{q) with these values of qn. 69) fc=m where 6 is a positive number independent of n. 60) where 6 is a positive number independent of n. n — 3*,n-l + " Xk,n-1 )■ — ^n,n-l ' 0

23) m = [£] + 1. 24) Now Em(f) is the best approximation of f(z) on \z\ < 1 by complex poly nomials of degree at most m. In particular, if zk = and we have 2nki n c deg pn < 6n, l, k = 1 , . . , n, then dn = 2a, m = 3n + 1 max \f(z) - pn(z)\ < cE$n+x(f) Proof. Let hm(z) G Pm be such that m™\f(z)-hm(z)\ = Em(f). Some Convergent Interpolator)/ Processes 55 We shall construct polynomials qn(z) G Pm such that