515 – Homework 2

This set is due Feb. 29 at the beginning of lecture. Let me know if more time is needed or anything like that. Problem 4 was incorrect as stated; I have fixed it now. Thanks to Tara Sheehan for bringing the problem to my attention.

1. We have used repeatedly that if $f:[a,b]\to{\mathbb R}$ is Riemann integrable, then it is bounded. We briefly discussed this in class, but I did not give a proof. Please prove it.

2. Recall that Cantor’s middle third set is defined as $C=\bigcap_n I_n$ where $I_0=[0,1]$ and, for each $n$, $I_{n+1}$ is the result of removing from each of the closed intervals making up $I_n$ their (open) middle third, so

$I_1=[0,1/3]\cup[2/3,1]$,

$I_2=[0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1]$,

etc. Show that $C$ is a Jordan measurable set of measure 0.

3. Solve exercise 1.1.12 from the book. To be specific, if $A$ is Jordan null and $B\subseteq A$, you need to prove that $B$ is also measurable, and its measure is 0. (In particular, every subset of the Cantor set $C$ is Jordan measurable.)

4. Modify the construction of the Cantor set as follows: Let $r$ be a constant, $0. Define $C_r$ as $\bigcap_n I_{n,r}$, where $I_{0,r}=[0,1]$ and, for each $n$, $I_{n+1,r}$ is the result of removing from each of the closed intervals making up $I_{n,r}$ their (open) middle $r$th portion, so each of these intervals ${}[a,b]$ is replaced with ${}[a,c]\cup[d,b]$ where $\displaystyle c=\frac{a+b}2-\frac{(b-a)r}2$ and $\displaystyle d=\frac{a+b}2+\frac{(b-a)r}2.$ (In particular, $C=C_{1/3}$.)

Also, for $0, define $C_r^\prime$ so that at stage $n$ we remove from each interval its middle portion of length $r^n$ (and if the interval has length less than $r^n$, then we remove it).

Check that all $C_r$ are Jordan measurable of measure 0, and find (with proof) all the values of $r$ for which $C_r^\prime$ is Jordan measurable. (If proceeding in this generality is not possible, at least prove that $C_{1/5}^\prime$ is not Jordan measurable.)

On the other hand, all the sets $C_r^\prime$ are Lebesgue measurable, as we will see in lecture.

5. Suppose that $f_n:[a,b]\to{\mathbb R}$ are continuous functions, and that $f_n\to f$ uniformly. Suppose that $g:[a,b]\to{\mathbb R}$ is continuous. Prove that

$\displaystyle \int_a^b g(x)f_n(x)dx\to\int_a^b g(x)f(x) dx.$

6. Suppose that $f:[0,1]\to{\mathbb R}$ is continuous, and that $\displaystyle \int_0^1 f(x)x^n dx=0$ for all $n=0,1,2,\dots$ Prove that $f$ is the constant zero function. This is a theorem of Hausdorff.

What if we are only given that the integrals vanish for integers $n>0$?

The quantities $\displaystyle \int_0^1 f(x) x^n dx$ are called moments. They are particularly useful in probability theory and in physics. (For example, in classical mechanics, if $f(x)\ge0$ for all $x\in[0,1]$, one thinks of $f$ as a mass density, and of the zero-th moment $M=\displaystyle \int_0^1 f(x)dx$ as its associated mass; the quantity $\displaystyle \frac1M\int_0^1 f(x)x dx$ is the center of gravity, and $\int_0^1 f(x) x^2 dx$ is the moment of inertia about 0.)

7. Hausdorff’s moment problem asks whether the moments of a continuous $f$ uniquely determine $f$. If $f$ is defined on a bounded interval, the answer is yes, by the result from the previous problem. But this is no longer the case if we consider functions $f$ defined on unbounded intervals (where, of course, the integrals are now interpreted as improper integrals). For a specific example, define $f:[0,\infty)\to{\mathbb R}$ by

$\displaystyle f(x)=e^{-x^{1/4}}\sin(x^{1/4})$.

Show that $\displaystyle \int_0^\infty f(x)x^n dx=0$ for all $n=0,1,\dots$

Hint: Let $\zeta=e^{i\pi/4}$. Show that

$\displaystyle \int_0^\infty t^{n+1} e^{-\zeta t}dt =\frac {n+1}{\zeta}\int_0^\infty t^n e^{-\zeta t}dt$.

Conclude that $\displaystyle \int_0^\infty t^{4n+3}e^{-\zeta t}dt$ is real. Consider its imaginary part and, by an appropriate change of variables, deduce that all the moments

$\displaystyle \int_0^\infty e^{-x^{1/4}}\sin(x^{1/4}) x^n dx$

are 0.

8. A trigonometric series is a series $\displaystyle \sum_{n\in{\mathbb Z}} c_n e^{inx}$, where the series is understood as

$\displaystyle \lim_{N\to\infty}\sum_{n=-N}^N c_n e^{inx}$.

Such a series is a Fourier series iff there is a function $f$ periodic of period $2\pi$ and integrable on the interval ${}[-\pi,\pi]$ such that $\displaystyle c_n=\hat f(n)=\frac1{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}dx$ for all integers $n$. Not all trigonometric series are Fourier series; for example, $\displaystyle \sum_{n=2}^\infty\frac{\sin nx}{\ln n}$ is not a Fourier series. (You do not need to prove this.)

Suppose that $f$ is continuous and periodic of period $2\pi$. Show that $\hat f(n)\to0$ as $|n|\to\infty$. This is a particular case of the Riemann-Lebesgue theorem.

(Cantor used this result in 1870 to prove that if $\displaystyle \sum_{n\in{\mathbb Z}} c_n e^{inx}=0$ for all $x$, then $c_n=0$ for all $n$. Set theory was born of his research in generalizations of this result.)

Extra Credit Problem.

Suppose that $S(x)=\sum_n c_n e^{inx}$ is a trigonometric series, and that the $c_n$ are bounded. The result of formally integrating $S$ twice is the Riemann function $F_S$ of $S$, defined by

$\displaystyle F_S(x)=\frac{c_0x^2}2-{\sum_{n\in{\mathbb Z}}}'\frac1{n^2}c_ne^{inx},$

where the $\sum'$ means that $n=0$ is excluded.  Show that $F_S$ is continuous (though in general, it is no longer periodic). One would expect that $F_S''$ exists and equals $S$, but this is in general not true. However, one can prove a variant of this:

Define the second symmetric derivative of a function $F$ by

$\displaystyle D^2 F(x)=\lim_{h\to 0}\frac{\Delta^2 F(x,h)}{h^2},$

where $\Delta^2F(x,h)=F(x+h)-2F(x)+F(x-h)$.

Prove that if $F''(x)$ exists, then so does $D^2 F(x)$ and $F''(x)=D^2 F(x)$. Give an example where $D^2 F(x)$ exists but $F''(x)$ does not.

Prove Riemann’s first lemma: If $S(x)$ is a trigonometric series with bounded coefficients (that converges at the point $x_0$) then $D^2 F_S(x_0)$ exists and equals $S(x_0)$.

For much more on this topic, see for example

• Alexander Kechris, Set theory and uniqueness for trigonometric series, 1997,

available at Kechris’s webpage.