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 X 0 in the Banach space of all functions f : [ − 1 , 1 ] → R possessing r th continuous derivatives ( r ∈ N ) such that for each function in X 0 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.