Math205b Winter Term 2003

Assignment 13

1. Let f ∈ C([0, 1], ℝ) and define the Bernstein polynomials by

p_{n}(f, x) = ∑_{k=0}^{n} \binom{n}{k}f(\frac{k}{n})x^{k}(1 − x)^{n−k} , x ∈ [0, 1] , n ∈ ℕ .

Show that

∥p_{n}(f, ·) − f∥_{∞} ⟶_{n→∞} 0 .

[Hint: Use the identity

∑_{k=0}^{n}(k − nx)^{2}\binom{n}{k}x^{k}(1 − x)^{n−k} = nx(1 − x)

and split the sum into two parts according to whether |x − \frac{k}{n}| ≤ δ or |x − \frac{k}{n}| > δ > 0 .]

2. Determine the radius of convergence of the following power series

∑_{n=0}^{∞} \frac{\sqrt{n 2^{n}}}{(n + 1)^{5}} x^{n} , ∑_{n=0}^{∞}(−1)^{n} \frac{n!}{n^{n}} x^{n} .

3. Assume that the power series

∑_{n=0}^{∞} a_{n}x^{n} and ∑_{n=0}^{∞} b_{n}x^{n}

have positive radii of convergence. Suppose that there exists a sequence (y_{j})_{j∈ℕ} with y_{j} → 0 (j → ∞) and y_{j} ≠ 0 such that

∑_{n=0}^{∞} a_{n}y_{j}^{n} = ∑_{n=0}^{∞} b_{n}y_{j}^{n} .

Prove that a_{n} = b_{n} , n ∈ ℕ.

4. Assume D ⊂^{o} ℝ and let f : D → ℝ be analytic. Show that, for every x_{0} ∈ D, constants M, r, δ > 0 can be found such that

|f^{(k)}(x)| ≤ Mk! r^{k} , x ∈ (x_{0} − δ, x_{0} + δ) .

5. You ask a question.

The Homework is due on Friday, February 21.