Let W β (x):=exp(-1 2|x| β ), x∈ℝ, β>1. Given f:ℝ→ℝ, let L n [f](x) denote the Lagrange interpolation polynomial to f at the zeros of the orthonormal polynomial of degree n for the weight W β 2 . Let 1<p<∞, Δ∈ℝ, α>0, and α ^:=min{1,α}. Moreover, let τ=τ(p):=1/p-α ^+0,p≤4,(β/6)(1-4/p),p>4· It is shown that for lim n→∞ ∥(f(x)-L n [f](x))W(x)(1+|x|) -Δ ∥ L p (ℝ) =0,(1) to hold for every continuous
... [Show full abstract] function f:ℝ→ℝ satisfying lim |x|→∞ |f(x)|W(x)(1+|x|) α =0,(2) it is necessary and sufficient that Δ>τif1<p≤4;Δ>τifp>4andα=1;Δ≥τifp>4andα≠1· Moreover, it is shown that (1) holds for every 1<p<∞ and every continuous function f satisfying (2) if and only if Δ≥-α ^+max{1,β/6}. These are special cases of results for more general Freud weights.