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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Ans. The eigenfunctions of the harmonic oscillator.