## Neugebauer’s theorem

December 30, 2014

The problem of characterizing when a given function is a derivative has been studied at least since 1911 when Young published a paper recognizing its relevance:

W. H. Young. A note on the property of being a differential coefficient, Proc. Lond. Math. Soc., 9 (2), (1911), 360–368.

The paper opens with the following comment (Young refers to derivatives as “differential coefficients”):

Recent research has provided us with a set of necessary and sufficient conditions that a function may be an indefinite integral, in the generalised sense, of another function, and the way has thus been opened to important developments. The corresponding, much more difficult, problem of determining necessary and sufficient conditions that a function may be a differential coefficient, has barely been mooted; indeed, though we know a number of necessary conditions, no set even of sufficient conditions has to my knowledge ever been formulated, except that involved in the obvious statement that a continuous function is a differential coefficient.

It is more or less understood nowadays that no completely satisfactory characterization is possible. We know that derivatives are Darboux continuous (that is, they satisfy the intermediate value property), and are Baire one functions (that is, they are the pointwise limit of a sequence of continuous functions). But this is not enough: For instance, any function $f$ such that $f(x)=\sin(1/x)$ for $x\ne 0$, and $f(0)\in[-1,1]$ is Darboux continuous and Baire one, but only the function with $f(0)=0$ is a derivative.

Andrew Bruckner has written excellent surveys on derivatives and the characterization problem. See for instance:

Andrew M. Bruckner and John L. Leonard. Derivatives. Amer. Math. Monthly, 73 (4, part II), (1966), 24–56. MR0197632 (33 #5797).

Andrew M. Bruckner. Differentiation of real functions. Second edition. CRM Monograph Series, 5. American Mathematical Society, Providence, RI, 1994. xii+195 pp. ISBN: 0-8218-6990-6. MR1274044 (94m:26001).

Andrew M. Bruckner. The problem of characterizing derivatives revisited. Real Anal. Exchange, 21 (1),  (1995/96), 112–133. MR1377522 (97g:26004).

Here I want to discuss briefly a characterization obtained by Neugebauer, see

Christoph Johannes Neugebauer. Darboux functions of Baire class one and derivatives. Proc. Amer. Math. Soc., 13 (6), (1962), 838–843. MR0143850 (26 #1400).

For concreteness, I will restrict discussion to functions $f:[0,1]\to\mathbb R$, although this makes no real difference. Whenever an interval is mentioned, it is understood to be nondegenerate. For any closed subinterval $I\subseteq [0,1]$, we write $I^\circ$ for its interior, and $\mathrm{lh}(I)$ for its length. Given a point $x\in[0,1]$, we write $I\to x$ to indicate that $\mathrm{lh}(I)\to 0$ and $x\in I$.

Theorem (Neugebauer). A function $f:[0,1]\to\mathbb R$ is a derivative iff to each closed subinterval $I$ of ${}[0,1]$ we can associate a point $x_I\in I^\circ$ in such a way that the following hold:

1. For all $x\in[0,1]$, if $I\to x$, then $f(x_I)\to f(x)$, and

2. For all  closed subintervals $I,I_1,I_2$ of ${}[0,1]$, if $I=I_1\cup I_2$ and ${I_1}^\circ\cap {I_2}^\circ=\emptyset$, then $\mathrm{lh}(I)f(x_I)=\mathrm{lh}(I_1)f(x_{I_1})+\mathrm{lh}(I_2)f(x_{I_2})$.

## Analysis – HW 3 – Strong measure zero

October 17, 2013

This set is due Wednesday, October 30, at the beginning of lecture.

[Edit, Oct. 30: The original version of the problem set had some mistakes, and has been replaced accordingly.]

Recall that a set $A\subseteq \mathbb R$ is measure zero iff for all $\epsilon>0$ there is a sequence $(I_n\mid n\in\mathbb N)$ of open intervals such that $\displaystyle \sum_{n\in\mathbb N}\mathrm{lh}(I_n)<\epsilon$ and $\bigcup_n I_n\supseteq A$.

Similarly, $X\subseteq\mathbb R$ is strong measure zero iff for any sequence $(\epsilon_n\mid n\in\mathbb N)$ of positive reals, there is a sequence $(I_n\mid n\in\mathbb N)$ of open intervals such that $\mathrm{lh}(I_n)\le\epsilon_n$ for all $n$, and $\bigcup_n I_n\supseteq X$. The notion is due to Borel, in 1919.

In lecture we showed that the continuous image of a measure zero set does not need to be a set of measure zero, and that the sum of two measure zero sets does not need to be a measure zero set.

As mentioned in lecture, Borel conjectured that the strong measure zero sets are precisely the countable sets. This statement turned out to be independent of the usual axioms of set theory: If the continuum hypothesis is true, the conjecture is false. On the other hand, Laver showed in 1976 that the conjecture is true in some models of set theory.

## Random series

March 2, 2013

A little while ago, a question was posted on MathOverflow on why lacunary series are “badly behaved”, in the sense that they their circle of convergence is their natural boundary. As the answers indicate, it is actually the opposite: This behavior is typical, in the sense that a random power series will have this property. I posted a short note pointing this out as an answer to a related question on Math.StackExchange. Here it is, with very minor edits:

Consider $\sum_n X_n z^n$, where the $X_n=X_n(\omega)$ are independent random (complex) variables and $z$ is complex.

First of all, the radius of convergence of the series (at a given $\omega$ in the underlying measure space) is

$r(\omega)=(\limsup_{n\to\infty}|X_n(\omega)|^{1/n})^{-1}.$

Note that $r$ is a measurable function of $\omega$, and its value does not depend on the values of a finite number of the $X_n$. Kolmogorov’s zero-one law then gives us that $r(\omega)$ is a constant, say $R$, almost surely. This $R$ lies in ${}[0,\infty]$, and both $0$ and $+\infty$ are possible values, depending on the distribution of the $X_n$, though the most interesting cases to study are perhaps when $0.

There is a nice book that presents the relevant theory:

Jean-Pierre Kahane. Some random series of functions. Second edition. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, 1985. MR0833073 (87m:60119).

(“Random Taylor series” is Chapter 4. What follows is based on Kahane’s presentation. Kahane’s book includes proofs of all the results below.)

Consideration of random series seems to have been first suggested by Borel, in

Emile Borel. Sur les séries de Taylor. C. r. hebd. Séanc. Acad. Sci., Paris 123, (1896), 1051-2.

(The journal is available here.)

Part of the problem was that at the time the concepts of probability theory were not quite formalized yet, so going from Borel’s remarks to actual theorems took some time.  Borel wrote

Si les coefficients sont quelconques, le cercle de convergence est une coupure.

What Borel is saying is that if the coefficients of a series $\sum_n X_n z^n$ are “arbitrary”, then the circle of convergence is a natural boundary for the function. What this means is that there is no way to extend $F(z)=\sum_n X_n z^n$ analytically beyond the circle of convergence (because the singular points are dense on the boundary).

The first actual result in this regard is due to Steinhaus in 1929: If the $r_n$ are positive, and $0<\limsup_n r_n^{1/n}<\infty$, and the $\omega_n$ are independent random variables equidistributed on ${}[0,1]$, then $\sum_n r_ne^{2\pi i\omega_n}z^n$ has the circle of convergence as natural boundary, almost surely. A different formalization was found later by Paley and Zygmund, in 1932, in terms of Rademacher sequences.

On the other hand, Borel’s statement cannot quite be translated as “the coefficients are independent random variables”. Kahane’s example is the series

$\sum_n (2^n\pm1)z^n,$

which has radius of convergence $1/2$, and $1/2$ is the only singular point on the circle of convergence.

Kahane mentions a conjecture of Blackwell that the general situation should be that one of the two scenarios above applies: Either

1. $F(z)=\sum_n X_n z^n$ has the circle of convergence as natural boundary, or
2. There is a series $\sum_n c_n z^n$ (the $c_n$ being constants, not random variables; Kahane calls it a sure series) that added to $F$ results on a (random) Taylor series with a strictly larger circle of convergence which is its natural boundary.

The conjecture was proved in 1953 by Ryll-Nardzewski, see

Czesław Ryll-Nardzewski. D. Blackwell’s conjecture on power series with random coefficients. Studia Math. 13, (1953). 30–36. MR0054882 (14,994e).

Kahane also wrote a nice survey of these and related matters, in

Jean-Pierre Kahane. A century of interplay between Taylor series, Fourier series and Brownian motion. Bull. London Math. Soc. 29 (3), (1997), 257–279. MR1435557 (98a:01015).