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}).

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414/514 The theorems of Riemann and Sierpiński on rearrangement of series

November 16, 2014

I.

Perhaps the first significant observation in the theory of infinite series is that there are convergent series whose terms can be rearranged to form a new series that converges to a different value.

A well known example is provided by the alternating harmonic series,

\displaystyle 1-\frac12 +\frac13-\frac14+\frac15-\frac16+\frac17-\dots

and its rearrangement

\displaystyle 1-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\dots

According to

Henry Parker Manning. Irrational numbers and their representation by sequences and series. John Wiley & Sons, 1906,

Laurent evaluated the latter by inserting parentheses (see pages 97, 98):

\displaystyle \left(1-\frac12\right)-\frac14+\left(\frac13-\frac16\right)-\frac18+\left(\frac15-\frac1{10}\right)-\dots \displaystyle=\frac12\left(1-\frac12+\frac13-\frac14+\dots\right)

A similar argument is possible with the rearrangement

\displaystyle 1+\frac13-\frac12+\frac15+\frac17-\frac14+\dots,

which can be rewritten as

\displaystyle 1+0+\frac13-\frac12+\frac15+0+\frac17+\dots \displaystyle =\left(1-\frac12+\frac13-\frac14+\frac15-\frac16+\frac17-\dots\right) \displaystyle +\left(0+\frac12+0-\frac14+0+\frac16+0-\dots\right) \displaystyle =\frac32\left(1-\frac12+\frac13-\frac14+\dots\right).

The first person to realize that rearranging the terms of a series may change its sum was Dirichlet in 1827, while working on the convergence of Fourier series. (The date is mentioned by Riemann in his Habilitationsschrift, see also page 94 of Ivor Grattan-Guinness. The Development of the Foundations of Mathematical Analysis from Euler to Riemann. MIT, 1970.)

Ten years later, he published

G. Lejeune Dirichlet. Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1837, 45-81,

where he shows that this behavior is exclusive of conditionally convergent series:

Theorem (Dirichlet). If a series converges absolutely, all its rearrangements converge to the same value.

Proof. Let u_0,u_1,\dots be the original sequence and u_{\pi(0)},u_{\pi(1)},\dots a rearrangement. Denote by U_0,U_1,\dots and V_0,V_1,\dots their partial sums, respectively. Fix \epsilon>0. We have that for any n, if m is large enough, then for all i\le n there is some j\le m with \pi(j)=i. Also, there is a k such that for all j\le m there is a i\le k with \pi(j)=i, so

|U_m-V_m|\le\sum_{i=n+1}^m|u_i|+\sum_{i=n+1}^k|u_j|.

Choosing n large enough, and using that \sum_i|u_i| converges,  we can ensure that the two displayed series add up to less than \epsilon. This gives the result. \Box

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Analysis – HW 4 – Fractals

October 29, 2013

This set is due Friday, November 15, at the beginning of lecture.

We discuss the Hausdorff metric on the collection of compact subsets of a complete metric space, and fractals obtained by contractions. (But note that not all fractals one may want to study come from this procedure.)

TeX source.

Pdf file.

[Edit, Nov. 12, 2013: Problem 4 has been changed to: Provide an example showing that, in general, if e\in\mathbb R^n and L is compact, then d_H(\{e\},L)\ne d(e,L). (There are cases where the equality holds, and it may be useful to provide some examples of this as well.)]


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.

TeX source.

Pdf file.