On the Extrema of Functions in the Takagi Class

Abstract

The Takagi class is a class of fractal functions on the unit interval generalizing the celebrated Takagi function. In this thesis, we study the extrema of these functions. This is a problem that goes back to J.-P. Kahane (1959). In this thesis, we state and prove the following new and original results on this long-standing problem. We characterize the set of all extrema of a given function in the Takagi class by means of a “step condition” on their binary expansions. This step condition allows us to compute the extrema and their locations for a large class of explicit examples and to deduce a number of qualitative properties of the sets of extreme points. Particularly strong results are obtained for functions in the so-called exponential Takagi class. We show that the exponential Takagi function with parameter 𝜐∈(0,1) has exactly two maximizers if 2𝜐 is not the root of a Littlewood polynomial. On the other hand, we show that there exist Littlewood polynomials such that, if 2𝜐 is a corresponding root in (0,1), the set of maximizers is a Cantor-type set with Hausdorff dimension 1/n, where n is the degree of the polynomial. Furthermore, if 𝜐 is in (-1,-0,5), the location of the maximum is a nontrivial step function with countably many jumps. Finally, we showed that, if 𝜐 is in (-1,-0.8), the minima will only attain at t = 0.2 and t = 0.8. If 𝜐 is in (-0.8,1), the only minimizer is at t = 0.5.