Ph.D. Qualifying Exam, Real Analysis Spring 2015, part I Do all five

Ph.D. Qualifying Exam, Real Analysis
Spring 2015,
part I
Do all five problems. Write your solution for each problem in a separate blue book.
1
Suppose A, B ⊂ R/Z are measurable of positive Lebesgue measure: m(A), m(B) > 0.
a. Show that there exists y ∈ R/Z such that m((A + y) ∩ B) > 0.
b. Show that in fact there exists y ∈ R/Z such that m((A + y) ∩ B) ≥ m(A)m(B).
2
Let X, Y be Banach spaces.
a. Show that if Tn ∈ L(X, Y ) are compact, and Tn → T ∈ L(X, Y ) in norm, then T is compact.
b. Show that if X, Y are separable Hilbert spaces then every compact operator T ∈ L(X, Y ) is the
norm limit of finite rank operators.
3
Let S(R) be the set of Schwartz functions on R, i.e. the set of C ∞ functions φ on R with xα ∂xβ φ
bounded for all α, β ∈ N.
R
a. With the Fourier transform given by (Fφ)(ξ) = e−ixξ φ(x) dx, show the Poisson summation
formula for φ ∈ S(R):
X
X
2π
φ(x + 2nπ) =
(Fφ)(n)einx ,
x ∈ R.
n∈Z
b.
n∈Z
Show that for any t > 0 we have
X
1 X
2
(−1)k e−k /(2t) .
exp(−t(2πn + π)2 /2) = √
2πt k∈Z
n∈Z
4
Let S(Rn ) denote set of Schwartz functions, and S 0 (Rn ) the dual space of tempered distributions. For
u ∈ S 0 (Rn ) let Dj u denote the distributional derivative of u in the jth coordinate. Let H = {u ∈
L2 (R2 ) : D2 u ∈ L2 (R2 )}, equipped with the norm kuk2H = kuk2L2 + kD2 uk2L2 .
a. Show that H is a Hilbert space (with the norm being induced by the inner product), and S(R2 ) is
dense in H.
b. Show that the restriction map R : S(R2 ) → S(R), (Rφ)(x1 ) = φ(x1 , 0), to x2 = 0, has a unique
continuous extension to a map H → L2 (R).
5
Suppose that X is a Banach space, and let B = {x ∈ X : kxkX ≤ 1} be the unit ball in X.
a. Suppose Z is a finite dimensional subspace of X. Show that there exists a closed subspace W of
X such that X = Z ⊕ W (direct sum).
b. If X is infinite dimensional, show that B is not compact in the norm topology.
c. Suppose that X, Y are Banach spaces, X ⊂ Y with the inclusion map ι : X → Y continuous and
compact. Let T ∈ L(X, Y ), and suppose that for all x ∈ X, kxkX ≤ C(kT xkY + kxkY ). Show that
KerT is finite dimensional, RanT is closed, and the induced map X/KerT → RanT is invertible as a
bounded linear map.
Ph.D. Qualifying Exam, Real Analysis
Spring 2015,
part II
Do all five problems. Write your solution for each problem in a separate blue book.
1
Suppose that f ∈ L1 ([0, 1]). Prove that there are nondecreasing sequences of continuous functions,
∞
{ϕk }∞
k=1 and {ψk }k=1 , on [0, 1] such that for a.e. x ∈ [0, 1] (with respect to Lebesgue measure), both
ϕk (x) and ψk (x) are bounded sequences, and moreover,
f (x) = lim ϕk (x) − lim ψk (x).
k→∞
k→∞
2
Let X be a vector space over C, F a vector space of linear maps X → C, and equip X with the weakest
topology in which all members of F are continuous. Show that the only continuous linear maps X → C
are those in F.
3
P
inx of the Fourier series of continuous functions
Consider the partial sums (SN f )(x) =
|n|≤N cn e
f on T = R/(2πZ), where cn are the Fourier coefficients of f , and recall that SN f is given by the
convolution of the Dirichlet kernel DN with f .
a.
Show that kDN kL1 → ∞ as N → ∞.
b. Show that there exists a continuous function f on T = R/(2πZ) such that the Fourier series of f
does not converge uniformly to f , i.e. SN f does not converge uniformly to f as N → ∞.
4
Suppose H is a Hilbert space. Recall that U ∈ L(H) is unitary if U U ∗ = I = U ∗ U .
Show that if U is unitary then Ran(I − U ) ⊕ Ker(I − U ) = H (orthogonal direct sum).
Pn−1 j
U . Show that Sn → P in
b. Let P be orthogonal projection to Ker(I − U ). Let Sn = n1 j=0
the strong operator topology (i.e. Sn f → P f in H for all f ∈ H). (This is the von Neumann, or mean
ergodic theorem.)
a.
c.
5
Give an example of a unitary operator U on `2 such that Sn does not converge to P in norm.
R
Let T = R/(2πZ) be the unit circle, and consider the integral I(r) = T eir cos θ ϕ(θ) dθ, ϕ ∈ C ∞ (T),
where dθ is the Lebesgue measure on T. Show that there exists C > 0 such that |I(r)| ≤ Cr−1/2 , r ≥ 1.
Hint: Show that if ϕ is supported away from [0], [π] ∈ R/(2πZ), then I(r) is rapidly decreasing as
r → R∞; then assume ϕ is supported near [0] or [π], and change variables to obtain an integral of the
2
form e±irs ϕ(s)
˜ ds (times a prefactor). (Note: I(r) is essentially the Fourier transform, evaluated at
(r, 0), of a delta distribution on the unit circle in R2 .)