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 .)