- #1

rainwyz0706

- 36

- 0

x − n, x ≥ n,

Is kn(x) uniformly convergent on R?

I can show that it is uniformly convergent on any closed bounded interval [a,b], but I don't think it is on R. Could anyone please give me some hints how to prove it?

2.Fix 0 < η < 1. Suppose now that h : [0, 1] → R is continuous. Prove that the series

t(x) = ∑ x^n h(x^n ) is uniformly convergent on [0, η]. Deduce that t(x) is continuous.

I'm not sure how to treat h(x^n) here, since it's not bounded. Could anyone help me figure it out?

Any help is appreciated!