MATH 519 Homework
Fall 2013
36. If (X, h·, ·i) is an inner product space show that
hx, yi =
1
||x + y||2 − ||x − y||2 + i||x + iy||2 − i||x − iy||2
4
Thus, in any normed linear space, there can exist at most one inner product giving rise
to the norm.
37. Let Ω ⊂ Rn , ρ be a measurable function on Ω,R and ρ(x) > 0 a.e. on Ω. Let X
denote the set of measurable functions u for which Ω |u(x)|2 ρ(x) dx is finite. We can
then define the weighted inner product
Z
hu, viρ =
u(x)v(x)ρ(x) dx
Ω
p
and corresponding norm ||u||ρ = hu, uiρ on X. The resulting inner product space is
complete, often denoted L2ρ (Ω). (As in the case of ρ(x) = 1 we regard any two functions
which agree a.e. as being the same element of X, so X is again really a set of equivalence
classes.)
a) Verify that all of the inner product axioms are satisfied.
b) Suppose that there exist constants C1 , C2 such that 0 < C1 ≤ ρ(x) ≤ C2 a.e.
Show that un → u in L2ρ (Ω) if and only if un → u in L2 (Ω).
P
inx
for the function f (x) = x on (−π, π). Use
38. Find the Fourier series ∞
n=−∞ cn e
some sort of computer graphics to plot a few of the partial sums of this series on the
interval [−3π, 3π].
39. Use the Fourier series in problem 38 to find the exact value of the series
∞
X
1
n2
n=1
∞
X
n=1
1
(2n − 1)2
40. (Exercise 7.2a in text) If f ∈ C(T) and SN denotes the N ’th partial sum of its
Fourier series, show that SN = DN ∗ f where DN is the so-called Dirichlet kernel
DN (x) =
1 sin [(N + 1/2)x]
2π
sin (x/2)
Produce a sketch of DN , either by hand or by computer, for some reasonably large value
of N .