ABSTRACT

Let f : [−1, 1] → R be an arbitrary continuous function, let pn(f ; ·) ∈ Pn be such that

( −1 + 2ν

n

) = f

( −1 + 2ν

n

) (ν = 0, 1, . . . , n) .

It was observed by Runge [34] that the sequence {pn(f ;x)} may not converge uniformly to f(x) as n → ∞. Subsequently, Bernstein (see [31, pp. 30-35] for details and pertinent remarks) noted that the sequence {pn(|x|; ·)} converges to |x| at no point of [−1, 1] other than −1, 0, 1. Does there exist a universal infinite triangular matrix, whose n-th row consists of points x1,n, . . . , xn,n belonging to [−1, 1], such that for any continuous function f : [−1, 1] → R the sequence {pn−1(f ; ·)} of polynomials pn−1 ∈ Pn−1, satisfying pn−1(xν,n) = f(xν,n) for ν = 1, · · · , n, converges uniformly to f ? The famous theorem of Faber (see [13] ; also see [6]) says that the answer to this question is “no.” Similarly, there does not exist a universal infinite triangular matrix, whose n-th row consists of points θ0,n, θ1,n, . . . , θn,2n belonging to [−1, 1], such that for any continuous 2π-periodic function f : [0, 2π] → R the sequence {Tn(f ; θ)} of trigonometric polynomials Tn(f ; .) of degree at most n, satisfying Tn(θν,2n) = f(θν,2n) for ν = 0, 1, · · · , 2n, converges uniformly to f(θ) (see [38 , Chapter VIII]).