Damelin, Steven Benjamin2017-03-012017-03-011995http://hdl.handle.net/10539/22133We 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.enThesis