FORWARD AND CONVERSE THEOREMS OF POLYNOMIAL-APPROXIMATION FOR EXPONENTIAL WEIGHTS ON [-1, 1] .2.

We consider exponential weights of the form w := e(-Q) on [-1, 1] wher e Q(x) is even and grows faster than (1 - x(2))(-delta) near +/- 1, so me delta > 0. For example, we can take Q(x): = exp(k)((1-x(2))(-alpha) ), k greater than or equal to 0, alpha > 0, where exp(h) denotes the k th iterated exponential and exp(0)(x) = x. We prove theorems of polyno mial approximation in weighted L-p spaces with norm \\fw\\(Lp[-1, 1]) for all 0 < p less than or equal to x, to match the forward theorems p roved in part 1 of this paper. (C) 1997 Academic Press.