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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414/514 Homework 3 – Continuity and series

November 11, 2014

This set is due in three weeks, on Friday, December 5th, at the beginning of lecture.

1. Given a finite set A\subseteq\mathbb T=\{z\in\mathbb C\mid |z|=1\}, find (with proof) a power series f(x)=\sum_{n=0}^\infty a_n x^n with radius of convergence 1 and such that if z\in\mathbb T, then f(z) converges iff z\notin A.

Find (with proof) a countably infinite set  X\subseteq \mathbb T and a power series f(x)=\sum_{n=0}^\infty a_n x^n with radius of convergence 1 and such that if z\in\mathbb T, then f(z) converges iff z\notin X.

The first part is easy, the second one may be tricky. If after a few honest efforts you find yourself stuck, take a look at the literature. For instance, at either Stefan Mazurkiewicz, Sur les séries de puissances, Fundamenta Mathematicae, 3, (1922), 52–58, or Fritz Herzog, George Piranian, Sets of convergence of Taylor series. I. Duke Math. J., 16, (1949), 529–534. Both papers prove more general results, by explicit constructions. Present a detailed adaptation of their argument that solves the question. Do not simply reproduce the proof in the papers for the more general cases. Other relevant references on this topic are Fritz Herzog, George Piranian, Sets of convergence of Taylor series. II. Duke Math. J., 20, (1953), 41–54, and Thomas W. Körner, The behavior of power series on their circle of convergence. In Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981), 56–94, Lecture Notes in Math., 995, Springer, Berlin-New York, 1983.

2. Present a detailed proof of Weierstrass theorem that for any compact interval [a,b], any continuous function f:[a,b]\to\mathbb R is the uniform limit of a sequence of polynomials. The proof should be different from the one presented in lecture (based on Bernstein polynomials). Again, although many books have a proof of this result, make sure your write up is your own.


414/514 Examples of Baire class two functions

November 3, 2014

Previously, we listed some examples of Baire class one functions. Here we do the same for functions in the next class of Baire. Recall that if I is an interval, the function f:I\to\mathbb R is (in) Baire class two (\mathcal B_2) iff it is the pointwise limit of a sequence of Baire one functions.

This post comes from an answer I posted on Math.Stackexchange about a year ago.

Here are three examples:

    1. Let C be the Cantor set. For each interval (a,b) contiguous to C, define f on {}[a,b] by

      f(x)=\frac{2(x-a)}{b-a}-1,

      so f maps the interval to [-1,1]. Otherwise, let f(x)=0.

    2. Write each x\in(0,1) in binary: x=0.a_1a_2a_3\dots, not terminating in a string of 1s, and define

      \displaystyle f(x)=\limsup_{n\to\infty} \frac{a_1+\dots+a_n}n.

    3. Conway’s base 13 function.

The first two examples come from Bruckner’s book Differentiation of real functions. All three are examples of functions that are not derivatives but have the intermediate value property.

The first one is discontinuous precisely at the points of C, and it is “almost” Baire class 1, in that one can turn it into a Baire class 1 function by only modifying its values (carefully) at the endpoints of intervals contiguous to C. But if one does this, then the function no longer has the intermediate value property.

The second function has the property that the image of any subinterval of (0,1), no matter how small, is all of (0,1). The third function is in the same spirit, but it behaves even more dramatically: The image of every open interval is all of \mathbb R.

To verify that the functions are indeed in Baire class at most 2:

  1. For example 1, use that the limit of x^n on {}[0,1] is 0 for x<1, and 1 at x=1, to get for each open interval (a,b) contiguous to C a Baire class 1 function f_{[a,b]} that is zero everywhere except on {}[a,b], where it coincides with f. Now use that the sum of finitely many Baire class 1 functions is Baire class 1.
  2. For example 2, there are several ways to proceed. Here is one, which I do not think is optimal, but (I believe) is correct: Recall that a limsup is the infimum (over m) of a supremum (over all n>m), so it is enough to see that each f_m(x)= \sup_{n>m}g_n is Baire class 1, where

    \displaystyle g_n(x)=\frac{a_1+\dots+a_n}n.

    The point is that each g_n has finitely many discontinuities, all of which are jump discontinuities. Any such function is Baire class 1. This would appear to mean that f_m is Baire class 2, but we are saved by noting that f_m is the uniform limit of the g_n, n>m. (The point is that each Baire class is closed under uniform limits.)

  3. The argument for example 3 is similar. (Note that this function is unbounded.)

To see that the functions are not Baire class 1: The functions in examples 2 and 3 are discontinuous everywhere, but the set of points of continuity of a Baire class 1 function is dense. For example 1, use Baire’s extension of this result giving us that, in fact, if f is Baire class 1, then for any perfect set P, the set of points of continuity of f\upharpoonright P is comeager relative to P. In example 1 this fails (by design) when P=C. (All we need is that, for any closed set D, the restriction of a Baire one function to D has at least one continuity point on D. Baire also showed that this characterizes Baire one functions.)

Example 2 is also discussed in the van Rooij-Schikhof book (see their Exercise 9.M).

To close, let me include some examples that do not have the intermediate value property. Note first that if A\subseteq\mathbb R and \chi_A is its characteristic (or indicator) function, then \chi_A is continuous iff A=\emptyset or \mathbb R. More interestingly, \chi_A is Baire class 1 iff A is both an F_\sigma and a G_\delta set.

Recall that a set is F_\sigma iff it is the countable union of closed sets, and it is G_\delta iff it is the countable intersection of open sets. The notation F_\sigma is pronounced F-sigma. Here, the F is for fermé, “closed” in French, and the \sigma is for somme, French for “sum”, “union”. Similarly, the notation G_\delta stands for G-delta. Here, the G is for Gebiet, German for “area”, “region”— neighborhood—, and the \delta is for Durchschnitt, German for “intersection”.

Note that, in particular, open sets are both: They are clearly G_\delta, and any open interval (and therefore, any countable union of open intervals) is a countable union of closed intervals. It follows that closed sets are also both. In particular, the characteristic function of the Cantor set is Baire class 1. More generally, a function f is Baire class 1 iff the preimage f^{-1}(U) of any open set is F_\sigma.

For the more general case where A is F_\sigma or G_\delta, then \chi_A is Baire class 2. For any A which is either, but not both, \chi_A is an example of a properly Baire class 2 function. For instance, this is the case with A=\mathbb Q. In fact, \chi_A is Baire class 2 iff A is both an F_{\sigma\delta} and a G_{\delta\sigma} set (G_{\delta\sigma} sets are countable unions of G_\delta sets, that is, countable unions of countable intersections of open sets, and F_{\sigma\delta} sets are countable intersections of F_\sigma sets, that is, countable intersections of countable unions of closed sets).

More generally, f is Baire class 2 iff for any open U, the set f^{-1}(U) is G_{\delta\sigma}. For details, and a significant generalization due to Lebesgue, that characterizes each Baire class and relates it to the hierarchy of Borel sets, see section 24 in Kechris’s book Classical descriptive set theory.


414/514 References on continuous nowhere differentiable functions

October 19, 2014

Just as the last two times I have taught 414/514, I am assigning a final project on the topic of continuous nowhere differentiable functions (see here and here for the previous times).

The project requires that you choose an example of a continuous nowhere differentiable function, and to write a report describing the function, indicating who first introduced it, and presenting complete proofs of its continuity and nowhere differentiability. Additional information relevant for context is highly encouraged.

I am including links to two encyclopedic references on the subject. Feel free to follow the arguments there closely if needed, or to consult other sources, but make sure that what you turn in is your own version of the details of the argument, and that full details (rather than a sketch) are provided.

  1. Johan Thim’s Master thesis (Continuous nowhere differentiable functions), written under the supervision of Lech Maligranda.
  2. A.N. Singh’s short book on The theory and construction of non-differentiable functions. (See here for a short review.)

As I mentioned before,

Please take this project very seriously (in particular, do not copy details from books or papers, I want to see your own version of the details as you work through the arguments). Feel free to ask for feedback as you work on it; in fact, asking for feedback is a good idea. Do not wait until the last minute.

The project should be typeset and is due Wednesday, December 17 (though I strongly encourage you to turn it in earlier).

Please contact me by email as soon as you have chosen the topic you are going to cover, and I’ll list it here, to avoid repetitions.

  • Stephanie Potter: Wen’s function.
  • Jeremy Siegert: Orlicz functions.
  • Stuart Nygard: Besicovitch’s function.
  • Monica Agana: Koch’s snowflake.

414/514 Homework 2 – Monotone and Baire one functions

October 10, 2014

This set is due in three weeks, on Monday, November 3, at the beginning of lecture.

1. Let f:[a,b]\to\mathbb R be increasing. We know that f(x-) and f(x+) exist for all x\in[a,b], and that f has at most countably many points of discontinuity, say t_1,t_2,\dots For each i let I_i,J_i be the intervals (f(t_i-),f(t_i)) and (f(t_i),f(t_i+)). Some of these intervals may be empty, but for each i at least one of them is not. (Here we follow the convention that f(a-)=f(a) and f(b+)=f(b).) Let \mathrm{lh}(I) denote the length of the interval I, and say that an interval (\alpha,\beta) precedes a point t iff \beta\le t.

Verify that \sum_i(\mathrm{lh}(I_i)+\mathrm{lh}(J_i))<+\infty and, more generally, for any x,

s(x):=\sum\{\mathrm{lh}(I_i)\mid I_i precedes f(x)\} +\sum\{\mathrm{lh}(J_i)\mid J_i precedes f(x)\}<+\infty.

Define a function f_0:[a,b]\to\mathbb R by setting f_0(x)=f(x)-s(x). Show that f_0 is increasing and continuous.

Now, for each n>0, define f_n:[a,b]\to\mathbb R so that f_n\upharpoonright[a,t_n)=f_{n-1}\upharpoonright[a,t_n), f_n(t_n)=f_{n-1}(t_n)+\mathrm{lh}(I_n), and f_n(x)=f_{n-1}(x)+\mathrm{lh}(I_n)+\mathrm{lh}(J_n) for all x\in(t_n,b]. Show that each f_n is increasing, and its only discontinuity points are t_1,\dots,t_n.

Prove that f_n\to f uniformly.

Use this to provide a (new) proof that increasing functions are in Baire class one.

2. Solve exercise 3.Q in the van Rooij-Schikhof book: If f:[a,b]\to\mathbb R is such that for all x, we have that f(x-) and f(x+) exist, then f is the uniform limit of a sequence of step functions. The approach suggested in the book is the following:

Show that it suffices to argue that for every \epsilon>0 there is a step function s such that |f(x)-s(x)|<\epsilon for all x.

To do this, consider the set A=\{x\in[a,b]\mid there is a step function s on [a,x] such that |f(t)-s(t)|<\epsilon for all t\in[a,x]\}.

Show that A is non-empty. Show that if a\le y\le x and x\in A, then also y\in A. This shows that A is an interval {}[a,\alpha) or {}[a,\alpha], with \alpha\le b. Show that in fact the second possibility occurs, that is, \alpha\in A. For this, the assumption that f(\alpha-) exists is useful. Finally, show that \alpha=b. For this, use now the assumption that f(\alpha+) exists.

3. (This problem is optional.) Find a counterexample to the following statement: If f:[a,b]\to\mathbb R is the pointwise limit of a sequence of functions f_1,f_2,\dots, then there is a dense subset X\subseteq [a,b] where the convergence is in fact uniform. What if f and the functions f_n are continuous?  Can you find a (reasonable) weakening of the statement that is true?

4. (This is example 1.1 in Andrew Bruckner’s Differentiation of real functions, CRM monograph series, AMS, 1994. MR1274044 (94m:26001).) We want to define a function f:[0,1]\to\mathbb R. Let C be the Cantor set in {}[0,1]. Whenever (a,b) is one of the components of the complement of C, we define f(x)=(2(x-a)/(b-a))-1 for x\in[a,b]. For x not covered by this case, we define f(x)=0. Verify that f is a Darboux continuous function, and that it is discontinuous at every point of C.

Verify that f is not of Baire class one, but that there is a Baire class one function that coincides with f except at (some of) the endpoints of intervals [a,b] as above.

Verify that f is in Baire class two.


414/514 Simple examples of Baire class one functions

October 6, 2014

Recall that a real-valued function f defined on an interval I is (in) Baire class one (\mathcal B_1) iff it is the pointwise limit of continuous functions.

Examples are continuous functions, of course, but functions in \mathcal B_1 do not need to be continuous. An easy example is the function f:[0,1]\to\mathbb R given by f(x)=0 if x\ne 1 and f(x)=1 if x=1. This is the pointwise limit of the functions f_n(x)=x^n. By the way, an easy modification of this example shows that any function that is zero except at finitely many points is in \mathcal B_1.

Step functions are another source of examples. Suppose that a=x_0<x_1<\dots<x_{n-1}<x_n=b and that s:[a,b]\to\mathbb R is constant on each (x_i,x_{i+1}). Then s is the pointwise limit of the functions s_k, defined as follows: Fix a decreasing sequence \epsilon_k converging to 0, with \epsilon_k\le 1/k and 2\epsilon_k<x_{i+1}-x_i for all i. Now define \hat s_k as the restriction of f to

\displaystyle \{x_0,x_1,\dots,x_n\}\cup\bigcup_{i=0}^{n-1}[x_i+\epsilon_k,x_{i+1}-\epsilon_k],

and let s_k:[a,b]\to\mathbb R extend \hat s_k by joining consecutive endpoints of the components of its domain with straight segments.

An important source of additional examples is the class of derivatives. Suppose f:\mathbb R\to\mathbb R and f(x)=g'(x) for all x. This is the pointwise limit of the functions f_n(x) given by

f_n(x)=\displaystyle\frac{g\left(x+\frac1n\right)-g(x)}{\frac1n}.

This simple construction does not quite work if f is defined on a bounded interval (as x+1/n may fall outside the interval for some values of x). We can modify this easily by using straight segments as in the case of step functions: Say f:[a,b]\to\mathbb R. For n large enough so 1/n<b-a, define f_n(x) as above for x\in[a,b-1/n], and now set f_n(b)=f(b) and extend f_n linearly in the interval {}[b-1/n,b].

Additional examples can be obtained by observing, first, that \mathcal B_1 is a real vector space, and second, that it is closed under uniform limits (the latter is not quite obvious). This gives us, for instance, that all monotone functions are in \mathcal B_1, since monotone functions are the uniform limit of step functions on bounded intervals: Given an increasing f:[a,b]\to\mathbb R, let f_n(x)=\lfloor nf(x)\rfloor/n. It follows that all functions of bounded variation are in \mathcal B_1, since any such function is the difference of two increasing functions.

Another interesting source of examples is characteristic functions. Given X\subseteq\mathbb R, the function \chi_X is in \mathcal B_1 iff X is both an \mathbf F_\sigma and a \mathbf G_\delta set.

On the other hand, \chi_{\mathbb Q} is not in \mathcal B_1, since it is discontinuous everywhere while Baire class one functions are continuous on a comeager set.


414/514 Homework 1 – The reals

September 11, 2014

This set is due in two weeks, on Friday September 26, at the beginning of lecture.

1. Recall that in the construction of the reals via Dedekind cuts, a real is simply the left set of a pair (A\mid B) in a cut of \mathbb Q, that is, a “real” is a set A\subset \mathbb Q that is non-empty, bounded above, closed to the left (meaning, if x\in A, y\in\mathbb Q, and y<x, then y\in A), and such that A has no maximum. We also have a copy \mathbb Q^* of \mathbb Q inside \mathbb R, given by the identification q\mapsto q^*:=\{t\in\mathbb Q\mid t<q\}. We left a few details to be verified when we presented this construction.

Let r be a real (in the sense just described). Define carefully the real -r (meaning, describe -r as a specific set of rationals, and verify that it is a real in the sense under discussion), and verify that -r+r=0, and that -r is the only real with this property.

Define carefully the product rs of reals r and s, and verify that the distributive property holds.

Check that \mathbb R is Dedekind-complete, that is, any cut of \mathbb R is realized. (S0, ignoring the formal difference between \mathbb Q and \mathbb Q^*, this version of \mathbb R is the Dedekind-completion of \mathbb Q, and this gives us that it is also the Dedekind completion of itself. )

2. More generally, define the Dedekind completion of a dense order, and verify its existence and uniqueness (up to isomorphism). In particular, the field \mathbb R(x) of  rational functions admits a completion, call it \hat{\mathbb R}(x). Can we extend the addition operation on \mathbb R(x) so it is defined in all of \hat{\mathbb R}(x) and makes it into an abelian group? Can we extend the order so \hat{\mathbb R}(x) is in fact an ordered group? What, if any, is the problem trying to extend multiplication?

3. Recall that in the construction of the reals via Cauchy sequences, a real is an equivalence class of Cauchy sequences of rationals, under the equivalence relation that states that two Cauchy sequences q_0,q_1,\dots and r_0,r_1,\dots are equivalent iff q_0,r_0,q_1,r_1,\dots is a Cauchy sequence.

Verify that this is indeed an equivalence relation, and that, given equivalent sequences \vec q and \vec r, the resulting interleaving sequence is equivalent to both. Verify that the (pointwise) definitions of addition and multiplication make sense, and that the resulting set equipped with these operations is indeed a field. Define carefully the ordering relation, and prove that it gives us a field ordering. Finally, verify that the resulting ordered field is indeed Dedekind complete.

4. Recall the construction of the reals described in Street’s paper An efficient construction of real numbers. His short note makes many claims that are not proved there. Provide as many of the missing details as possible.

5. Given a linear order (X,<), in the order topology the open sets are (by definition) those subsets of X that are union of (bounded or unbounded) open intervals in X. Show that a linear order (X,<) is order isomorphic to \mathbb R iff the following three properties are verified:

  • X has no first or last elements.
  • X is connected, that is, we cannot write X=A\cup B where A and B are open, nonempty, and disjoint.
  • X is separable, that is, there is a countable subset of X that is dense in X.