Homework N. 4
Real Analysis - Academic Year 2013-2014
Proff. K. Payne and G. Molteni
(1) Verify the following formulas for Gaussian integrals, where h·, ·i is the standard inner product
on Rn .
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(a) If f (x) = e−|x| /(2σ) with x ∈ Rn and σ > 0, then
Z
σ n/2
1
2
−ihx,ξi
fˆ(ξ) :=
e
f
(x)
dx
=
e−σ|ξ| /2 .
n
(2π) Rn
2π
(b) If A is a positive definite symmetric n × n matrix with real entries, then
Z
e−πhAx,xi dx = (det A)−1/2 .
Rn
Hint: For part (b), consider diagonalizing the matrix A with a suitable change of basis.
(2) Let S(Rn ) be the Schwartz space of functions of rapid decay; that is, functions f ∈ C ∞ (Rn )
such that for every multi-index α, β ∈ Nn0 one has
sup xβ Dxα f (x) < +∞.
x∈Rn
(a) Prove the following version of Heisenberg’s uncertainty principle: for every f ∈ S(Rn )
with ||f ||L2 (Rn ) = 1
Z
Z
n2
2
2
2 ˆ
2
|x| |f (x)| dx
|ξ| |f (ξ)| dξ ≥
.
4(2π)n
Rn
Rn
(b) Show that the constant cannot be improved.
Hint: For part (b), think again about Gaussians.
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(3) Let g : R → R be the Gaussian function g(x) = e−x and Pk the set of polynomials of degree
k (with complex coefficients).
(a) Show that Fourier transform F : Hk → Hk where
Hk := {h(x) = Pk (x)e−x
2 /2
: Pk ∈ Pk }, k ∈ N0 .
(b) Consider the Hermite functions defined by
k
d
k x2 /2
hk (x) := (−1) e
g(x).
dx
Show that hk ∈ Hk and that {hk }k∈N0 is complete in the sense that: if f ∈ S(R) and
Z
f (x)hk (x) dx = 0 for all k ≥ 0
R
then f = 0.
(c) Show that the functions hk are eigenfunctions of F.
P
ikx with N ∈ N and x ∈ Q = [−π, π), prove the
(4) For the Dirichlet kernel DN (x) = N
k=−N e
following statements that we have made in class (Lemma 4.3.1):
sin ((N + 1/2)x)
(a) DN is real valued, even and DN (x) =
sin (x/2)
Z 0
Z π
1
1
(b)
DN (x) dx =
DN (x) dx = 1.
π −π
π 0
Hint: For part (a), think about the formula for the partial sum of a geometric series.
(5) Let Q = [−π, π) and Sf (x) the Fourier series expansion in terms of the real trigonometric
system. Following the ideas used in class for f (x) = x:
(a) Expand f (x) = |x| in a real Fourier series for each x ∈ Q and deduce that
+∞
+∞
π4 X
1
=
96
(2n + 1)4
and
n=0
π4 X 1
.
=
90
n4
n=1
(b) Expand f (x) = x(π − |x|) in a real Fourier series for each x ∈ Q and deduce that
+∞
+∞
X
π6
1
=
960
(2n + 1)6
and
n=0
X 1
π6
=
.
945
n6
n=1
(6) For Q = [−π, π) and s ≥ 0, define the Sobolev space
(
)
X
H s (Q) := f ∈ L2 (Q) :
(1 + k 2 )s |fˆ(k)|2 < +∞
k∈Z
(a) Prove that
H s (Q)
is a Hilbert space with respect to the inner product
X
(f, g)s :=
(1 + k 2 )s fˆ(k)ĝ(k).
k∈Z
(b) Prove that the span of the trigonometric polynomials is dense in each H s (Q).
(c) Prove that if 0 ≤ s < t then the inclusion map I : H t (Q) → H s (Q) is well defined, linear,
bounded and compact.
Hints: For the density in part (b), show that ||SN f − f ||s → 0 as N → +∞ where as always
SN f (x) =
N
X
fˆ(k)eikx .
k=−N
For the compactness in part (c), exploit that SN : H t (Q) → H s (Q) is of finite rank.
(7) Several integral inequalities involving Fourier series.
(a) Let f : [0, 1] → C, f ∈ C 1 ([0, 1]), with f (0) = f (1). Prove that
Z 1
Z 1
Z 1
2
1
2
|f 0 (x)|2 dx
|f (x)| dx − f (x)dx ≤ 2
4π 0
0
0
and that the equality holds if and only if f (x) ∈ span{e2πix , 1, e2πix }.
(b) Let f : [0, 1] → C, f ∈ C 1 ([0, 1]). Prove that
Z 1
Z 1
Z 1
2
1
2
|f (x)| dx − f (x)dx ≤ 2
|f 0 (x)|2 dx
π 0
0
0
and that the equality holds if and only if f (x) ∈ span{1, cos(πx)}.
(c) Let f : [0, 1] → C, f ∈ C 2 ([0, 1]), with f (0) = f (1) = 0. Prove that
Z 1
Z 1
1
0
2
|f (x)| dx ≤ 2
|f 00 (x)|2 dx
π
0
0
and that the equality holds if and only if f (x) ∈ span{sin(πx)}.
(d) (Wirtinger’s inequality) Let f : [0, 1] → C, f ∈ C 1 ([0, 1]), with f (0) = f (1) = 0. Prove
that
Z 1
Z 1
1
2
|f 0 (x)|2 dx
|f (x)| dx ≤ 2
π 0
0
and that the equality holds if and only if f (x) ∈ span{sin(πx)}.
Hints: For part (a), notice that {e2πikx }k∈Z is an o.n.b. for L2 ([0, 1]) and use the Bessel
inequality. For part (b), extend f to [−1, 1] as an even function and adapt the argument of
part (a) to the interval [−1, 1]. For part (d), extend f to [−1, 1] as an odd function.
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