Weighted approximation for Erdðs weights

Abstract

We investigate Mean Convergence of Lagrange Interpolation and Rates of Approximation for Erd5's Weights on the Real line. An Erdg's Weight is of the form, W : • expI-Q]' where typically Q is even, continuous and is of faster than polynomial growth at infinity. Concerning Lagrange Interpolation, we obtain necessaryand sufficient conditions for convergence in Lp (1::; p < 00) and in particular, sharp results for p > 4 and 1 <p < 4. On Rates of Approximation, we first investigate the problem of formulating and proving the correct Jackson Theorems for Erdifs Weights. This is accomplished in Lp(O < p < 00) with endpoint effects in [-an, anI, the Mhaskar-Rahmanov-Saff interval. We next obtain a net ural Realisation Functional for our class of weights and prove its fundamental equivalence to our.modulus of continuity. Finally, we prove the correct converse or Bernstein Theorems in Lp (0 < p :5 00) and deduce a Marchaud Inequality for our modulus.