On the unbounded divergence in the best approximation on equidistant nodes

In this work, we point out a superdense (meaning residual, dense and uncountable) set X0 in the Banach space of all functions f:[−1,1]→R possessing rth continuous derivatives (r∈N) such that for each function in X0 the discrete best approximation polynomials associated with the equidistant nodes in [−1,1] unboundedly diverge on a superdense set in [−1,1] of full measure.