Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 9 / 4 / c

Table of contents
- Solution c

Solution

The functions are nonnegative and normalized to have integral one. On the disk

∣ x ∣ \leq (2 n)^{- 1/4}

, Bernoulli inequality gives

(1 - ∣ x ∣^{4})^{n} \geq 1 - n ∣ x ∣^{4} \geq 1/2

. Consequently

c_{n}^{- 1} \geq \frac{1}{2} π (2 n)^{- 1/2}, c_{n} \leq \frac{2 2 n}{π} .

For any fixed

0 < a < 1

, the mass outside radius

a

is at most

\int_{∣ x ∣ \geq a} g_{n} (x) d x \leq π c_{n} (1 - a^{4})^{n} \leq 2 2 n (1 - a^{4})^{n} ⟶ 0.

This is an approximate identity. For any test function,

\int g_{n} φ - φ (0) \leq sup_{∣ x ∣ \leq a} ∣ φ (x) - φ (0) ∣ + 2∥ φ ∥_{\infty} \int_{∣ x ∣ \geq a} g_{n} .

Let

n \to \infty

and then

a ↓ 0

to conclude

g_{n} \to δ_{0} in D^{'} (R^{2})

.

Polar coordinates also give the exact normalization through the beta function:

c_{n}^{- 1} = \frac{π}{2} B (1/2, n + 1), c_{n} = \frac{2Γ ( n + 3/2 )}{π ^{3/2} Γ ( n + 1 )} .

Figure 1.
Normalized radial densities and cumulative mass concentrating at the origin
. Details.

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