Interpolation of Individual Functions

1. I N T R O D U C T I O N A well-known Faber theorem asserts that given any sequence Pn of projections from C ([0, 1]) onto the space q3n of polynomials of degree n 1, there exists a continuous function f such that Pn f does not converge to f in the uniform norm. In particular, let A n : 0 < t~ n) < t~ n) < . . . < t (n) < 1 be a sequence of partitions on the interval [0, 1]. Let L (An) : C ([0, 1]) -~ q3n be the usual Lagrange interpolation operators. Then there exists f • C([0, 1]) such that L(An) f 7 4 f. On the other hand, if the function f • C ([0, 1]) is given, then one can always find a sequence An C [0, 1] such that L (An) f --* f . The familiar proof of it goes as follows: Let bn (f) be a polynomial of best approximation to f . Then f bn (f) alternates sign at least n + 1 times on [0, 1]; hence (by the intermediate value theorem), there exists at least n points t~n), . . . , t(n ~) E [0, 1] such tha t bn(f)(t~ n))''" = f(t~n').'" Pick these points to be An. Then L(An) f = bn(f)and N / L(An) f --+ f. What happens if at least one of the arguments in this proof does not apply? As far as I know, nothing is known. Here is a partial list of questions: 1. Given two functions f ,g E C([0,1]), does there exist a sequence An C [0,1] such that L(An) f --* f and L(An)g --* g? 2. Let X C [0, 1] be a compact subset. Let f • C (X). Does there exist An C X such that L (An) f ~ f in C ( X ) ? 3. Let Vn be n-dimensional subspaces in C [0, 1] which are dense in C [0, 1]. Given f • C [0, 1], does there exist a sequence An C [0, 1] such that P ( A n ) f --~ f , where P (An) is an interpolation projection from C ([0, 1]) onto Vn? The set of questions goes on (cf. [1]). The set of answers remains empty (cf. 0). In this paper, I describe some partial results relating to questions 1, 2, and 3 above. I also hope to stimulate an interest to these problems.