ON DENSITY OF INTERPOLATION POINTS, A KADEC-TYPE THEOREM, AND SAFF PRINCIPLE OF CONTAMINATION IN LP-APPROXIMATION

Let f is-an-element-of L(p)(I) and denote by B(n,p)(f) the polynomial of best L(p-) approximation to f of degree n (1 < p < infinity, I = [-1, 1], the weighted L(p)-norm with an arbitrary positive weight). Extending a result proved by Saff and Shekhtman for p = 2 we show that for every 1 < p < infinity and f is-an-element-of L(p)(I) (not a polynomial) points of sign change of the error function f - B(n,p)(f) are dense in I as n --> infinity.