Dolomites Research Notes on Approximation

Telescoping polynomial series with specified restrictions on the zeros of the polynomial components turn out to be entire functions. Applied to polynomial Lp-approximation, 1 < p ≤ ∞, on a compact set E, we obtain a converse theorem based only on the location of the zeros of the difference of consecutive polynomials and the asymptotic behavior of the zeros of the polynomials. In contrast to the Bernstein- Walsh theorem, no information about the asymptotic behavior of the error of approximation is needed.