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9): namely, we show that pn(fi*) can be chosen as a linear operator^ and we also give the order of approximation instead of the mere convergence. 11) are called Erdos conditions. The proof of this theorem is long and complicated, therefore we break it into lemmas. 4. 11), for any e > 0 there exists a system of nodes (not necessarily in decreasing order) yk=co8nk, 2k-l T]k = r)kM =—M + dkx + l—2' M : = [ n ( l + e)]; = "" ' n>n0 such that (a) the Xi 's are among the yk }s (b) lim n(nk+i - nk) > 0, k = 1 , .

Consider the nodes VkM = cos QkM , QkM = 2k — 1 0]Lr * , A; = 1 , . . , M, Some Convergent Interpolator]! Processes 53 +1 with -s - Now for each QkM i if there is a 0jn in the interval (©jkA/ ~ 2M» ®*A* + 2Af ) then delete this &HM and replace it by 0 Jfi . , there is at most one 9jn in it (cf. 22)). If there is no 0jn in the interval considered, then leave &kM unchanged. Denote by z^Mi k = 1 , . . , M the system of nodes thus obtained. By construction, the Xfcn's are among the ZfcA/'s. 1l) hold for the ZkM1* (with n replaced by M).

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*,n-i(r 2) i)| k=m _ rn-i(g») p £-*> V V fc=m n > ^ f ^ n X *' * \q *•" \ g n„ - Xkfc,n,-1 n _i Zfc, Xfc,n-1 ~ XI nn,,n»«-i 1/' n -1 - J£I*=±_, n>6.