414/514 – Faber functions

January 17, 2012

Here is Shehzad Ahmed’s project from last term, on the Faber functions, a family of examples of continuous nowhere differentiable functions. Ahmed’s project centers on one of them, but the argument can be easily adapted to all the functions in the family.

As usual, the function is given as a series F=\sum_n f_n where the functions f_n are continuous, and we can find bounds M_n with \|f_n\|\le M_n and \sum_n M_n<+\infty. By the Weierstrass M-test, F is continuous. Also, the strategy for nowhere differentiability is typical: We associate with each point x a pair of sequences (a_n)_{n\ge0} and (b_n)_{n\ge0} with a_n strictly decreasing to x and b_n strictly increasing to x. The key lemma (shown, for example, in Johan Thim’s Master thesis available here) is that, if a continuous function f is differentiable at x, then we have

\displaystyle f'(x)=\lim_{n\to\infty}\frac{f(a_n)-f(b_n)}{a_n-b_n}.

In the case of the Faber functions, the functions f_n add `peaks’ in the neighborhood of any point, and the locations of these peaks can be used as the points a_n and b_n; moreover, the slopes accumulate at these peaks, and so the limit on the right hand side of the displayed equation above does not converge and, in fact, diverges to +\infty or -\infty.

Faber’s original paper, Einfaches Beispiel einer stetigen nirgends differentiierbaren Funktion, Jahresbericht der Deutschen Mathematiker-Vereinigung, (1892) 538-540, is nice to read as well. It is available through the wonderful GDZ, the Göttinger Digitalisierungszentrum.

414/514 – Katsuura function

January 17, 2012

Here is Erron Kearns’s project from last term, on the Katsuura function, an example of a continuous nowhere differentiable function.

The presentation is nice: As usual with these functions, this one is defined as the limit of an iterative process, but the presentation makes it very clear the function is a uniform limit of continuous (piecewise linear) functions, and also provides us with a clear strategy to establish nowhere differentiability.

Actually, the function is presented in a similar spirit to many fractal constructions, where we start with a compact set K and some continuous transformations T_1,\dots,T_n. This provides us with a sequence of compact sets, where we set K_0=K and K_{m+1}=\bigcup_{i=1}^n T_i(K_m). Under reasonable conditions, there are several natural ways of making sense of the limit of this sequence, which is again a compact set, call it C, and satisfies C=\bigcup_{i=1}^n T_i(C), i.e., C is a fixed point of a natural “continuous” operation on compact sets.

This same idea is used here, to define the Katsuura function, and its fractal-like properties can then be seen as the reason why it is nowhere differentiable.