Real Analysis II.

Instructor: Dmitry Ryabogin

Assignment IV.

1. Problem 1. Make a proper change of variables to compute the double integral

∫_{A} (y^{2} − x^{2})dxdy, A := {(x, y) ∈ R^{2} : 0 < x < y, xy < 1, x^{2} + y^{2} < 4}.

2. Problem 2.

a) Let K = [−1, 1]^{k} be the cube in R^{k}. Compute the volume of K. What is the length of the main diagonal (the segment joining the points (−1, −1, ..., −1) and (1, 1, ..., 1))?

b) Prove that the volume of the cube can be written as

vol_{k}(K) = v_{k} ∫_{S^{k−1}} r(u)^{k}dσ(u),

where r(u) is the ”radius” of K in direction u,

r(u) := max{t > 0 : tu ∈ K}.

Conclude that the ”radius” of K satisfies

∫_{S^{k−1}} r(u)^{k}dσ(u) = \frac{2^{k}}{v_{k}} ≈ (\frac{2k}{πe})^{k},

(use 3 g) of the previous assignment), and the ”average radius” of K is about \sqrt{2k/πe}. This indicates that the volume of the cube tends to lie in its corners, where the radius close to \sqrt{k}, not in the middle of its facets, where the radius is close to 1.

3. Problem 2. ”How is the mass of the ball distributed” ?

a) Let B be the ball of volume 1 in R^{k}. What is its radius?

b) The central slice A_{e_{1}}(0) := {y ∈ B : y_{1} = 0} of B is an (k − 1)-dimensional ball of the same radius. Prove that the volume of the slice vol_{k−1}(A_{e_{1}}(0)) = v_{k−1}v_{k}^{−(k−1)/k}.

c) Use

Γ(\frac{k}{2} + 1) ≈ \sqrt{2π}e^{−k/2}(\frac{k}{2})^{(k+1)/2}.

to find that vol_{k−1}(A_{e_{1}}(0)) ≈ e for large k.

d) Prove that a parallel slice having a distance t from the origin

A_{e_{1}}(t) = {y ∈ B : y_{1} = t}

has a volume

vol_{k−1}(A_{e_{1}}(t)) ≈ \sqrt{e}(1 − \frac{t^{2}}{r^{2}} )^{(k−1)/2},

where r is the radius of B.

e) Since r ≈ \sqrt{k/(2πe)}, prove that

vol_{k−1}(A_{e_{1}}(t)) ≈ \sqrt{e} e^{−πet^{2}}.

f) Draw the graph of f(t) := vol_{k−1}(A_{e_{1}}(t)) and observe that it does not depend on the dimension. Conclude that the volume of the ball concentrates close to any subspace of dimension k − 1.

g) The part f) suggests that the volume concentrates near the center of the ball, where the subspaces all meet. On the other hand, prove that, for k large, most of the volume of B lies near its surface. How do you explain this phenomena?

4. Problem 3.

a) Let

h_{λ}(x) = \frac{2}{π} \frac{λ}{λ^{2} + x^{2}} , λ > 0.

Prove that

∫_{R} h_{λ}(x)dx = 1 ∀λ > 0.

b) Let g be a bounded function. Prove that

lim_{λ→0} g ∗ h_{λ}(x) = g(x),

provided g is continuous at x.

c) Let g be integrable on R. Prove that

lim_{λ→0} ∫_{R} |g ∗ h_{λ}(x) − g(x)|dx = 0.