dc.contributor.author Han, Xiyue dc.date.accessioned 2019-01-23 16:01:09 (GMT) dc.date.available 2019-01-23 16:01:09 (GMT) dc.date.issued 2019-01-23 dc.date.submitted 2019 dc.identifier.uri http://hdl.handle.net/10012/14388 dc.description.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. en dc.language.iso en en dc.publisher University of Waterloo en dc.subject nowhere differentiable function en dc.title On the Extrema of Functions in the Takagi Class en dc.type Master Thesis en dc.pending false uws-etd.degree.department Statistics and Actuarial Science en uws-etd.degree.discipline Actuarial Science en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Master of Mathematics en uws.contributor.advisor Schied, Alexander uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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