Practice Final

This exam is closed book, no books, papers or recording devices permitted. You may use theorems from class, or the book, provided you can recall them correctly.

Problem 1

Suppose $ f\in L^1([0,1])$ and $ \int_{[0,1]} f\chi dx=0$ for all simple measurable functions $ \chi$ on $ [0,1].$ Show thatt $ f=0$ almost everywhere with respect to Lebesgue measure.

Ans. Take $ \chi$ to be the characteristic function of the measureable set $ \{x\in[0,1];f(x)>0\}.$ Then $ \int_{[0,1]}f\chi dx=\int_{[0,1]}f^+dx=0$ implies that $ f^+=0$ a.e. - the same argument works for $ f^-$ so $ f=0$ a.e.

Problem 2

Suppose $ A$ is a compact operator on a Hilbert space $ H,$ and that $ A^*A$ has no positive eigenvalues, show that $ A=0.$

Ans. Since $ A^*A$ is compact and selfadjoint there is a complete orthonormal basis of its eigenvectors. All the eigenvectors must be non-negative, since $ A^*Au=\lambda u$ imples $ \Vert Au\}^2=\lambda \Vert u\Vert^2.$ So, if there are no positive eigenvalues they must all be zero. Thus $ A^*Au=0$ for all $ u$ implies $ \Vert Au\Vert=0$ for all $ u,$ so $ A=0.$

Problem 3

Give an example of a function $ u\in L^2(\mathbb{R})$ which is continuous but is such that its Fourier transform $ \hat u\notin L^1(\mathbb{R}).$

Ans. For any $ N$ it is easy to find a non-negative continuous function, $ g,$ with maximum $ N$ supported in $ [0,N^{-4}].$ The integral of its square is then less than $ 1/N^2.$ Consider the function

$\displaystyle f(x)=\sum\limits_{N\ge 1}g(x-N).

All the terms have disjoint supports and $ f$ is square integrable. If its Fourier transform was in $ L^1$ the function $ f$ would have to be continuous, which it is, and would also have to vanish infinity, which it does not. [This one is a bit tricky.]

Problem 4

Suppose $ u\in L^2([-\pi,\pi])$ and there exists $ v\in L^2([-\pi,\pi])$ such that

$\displaystyle \int_{[-\pi,\pi]} u(x) \frac{d}{dx}\phi(x) dx=\int_{[-\pi,pi]} v(x)\phi(x)dx$    

for all smooth $ 2\pi$-periodic functions, $ \phi,$ on the real line. Show that $ u$ has a continuous representative in $ L^2([-\pi,\pi]).$

Ans. Plug $ e^{ikx}$ into the identity for each $ k\in\mathbb{N}$ and you find that the Fourier coefficients $ c_k$ of $ u$ satisfy $ ikc_k=d_k$ where $ d_k$ are the Fourier coefficients of $ v\in L^2([-\pi,\pi]).$ Thus

$\displaystyle \sum\limits_{k}k^2\vert c_k\vert^2<\infty.

This is enough to imply that the Fourier series for $ u$ converges uniformly so $ f$ `is' continuous - has a representative which is continuous.

Problem 5

Suppose $ f\in\mathcal{S}(\mathbb{R})$ has Fourier transform satisfying $ \hat f(\xi)=0$ in $ \vert\xi\vert<1.$ Show that there exists $ g\in\mathcal{S}(\mathbb{R})$ such that $ f(x)=\frac{d^2}{dx^2}g(x)$ for all $ x\in\mathbb{R}.$

Ans. The function $ h(\xi)=-\hat f(\xi)/\xi^2$ in $ \vert\xi\vert>1/2,$ $ h(\xi)=0$ in $ \vert\xi\vert\le 1/2$ is in $ \mathcal{S}(\mathbb{R})$ and satisfies $ (-\xi)^2h(\xi)=\hat f(\xi).$ Thus if $ g$ is the inverse Fourier transform of $ h$ it is in $ \mathcal{S}$ and satisfies $ \frac{d^2}{dx^2}g=f.$

Problem 6

Show that there is no element of $ L^1([-\pi,\pi])$ satisfying

$\displaystyle \int_{[-\pi,\pi]} f(x)e^{ik^3x}=1\ \forall\ k\in\mathbb{N}.$    

Ans. This stops the Fourier coefficietns of $ f$ from vanishing at $ \infty.$

Problem 7

Suppose $ f\in L^2([-\pi,\pi]))$ had Fourier coefficients $ c_j,$ $ j\in\mathbb{Z}$ satisfying

$\displaystyle \sum\limits{k\in\mathbb{Z}}k\vert c_k\vert<\infty.$    

Show that there exists a function $ g\in L^2([-\pi,\pi])$ such that

$\displaystyle \int_{[-\pi,pi]} g(x)\phi(x)dx=\int_{[-\pi,pi]} f(x)\left(\frac{d}{dx}\phi(x)+\cos(x)\phi(x)\right)dx$    

for all smooth $ 2\pi$-periodic functions $ \phi$ on the real line.

Ans. The given condition implies he uniform convergence of the (formal) Fourier series for $ df/dx.$ So $ f$ has a continuous first derivative. Integration by parts is then justified in the given identity so we can take $ g=-\frac{df(x)}{dx}+f(x)\cos x.$

Problem 8

If $ f\in\mathcal{C}([0,1]),$ show that

$\displaystyle (Au)(x)=\int_x^1 f(t)u(t)dt,\ x\in[0,1],$ (45)

defines a compact operator $ A:L^2([0,1])\longrightarrow L^2([0,1]).$

Ans. Change variables to shift the integral to $ [-\pi,\pi]$ (or use Fourier series on $ [0,1])$ by setting $ y=2\pi x-\pi.$ The

$\displaystyle Au'(y)=\frac1{2\pi}\int_y^{\pi} f(s/2\pi-1)u'(s)ds.

Anyway, we get the same sort of integral with a different continuous function. Now, integrating agains $ e^{ikx}$ we again find that the Fourier coefficients $ a_k$ of $ Au'$ satisfy $ \sum\limits_{k}k^2\vert a_k\vert^2<C$ whenever $ \Vert u'\Vert\le 1.$ This implies that $ A$ maps the ball into a compact set.

Problem 9

Show that there is an infinite orthonormal sequence $ u_j\in L^2(\mathbb{R})$ with each element satifying

$\displaystyle \widehat{u_j}=c_ju_j,\ c_j\in\mathbb{C}.$    

Ans. The eigenfunctions of the harmonic oscillator.

Richard B. Melrose 2004-05-24