Author
HLAVACEK, JAN

Year
1998

Advisor
BAISHANSKI, BOGDAN M.

Abstract
Given function f with an absolutely convergent Fourier series,$$f(t)=/sum/sbsp[/nu=[-]/infty][/infty]a/sb[/nu][/rm e]/sp[i/nu t],$$define the norm of f as$$/Vert f/Vert=/sum/sbsp[/nu=[-]/infty][/infty]/vert a/sb[/nu]/vert.$$We study the behavior of $/Vert f/sp[n]/Vert$ as $n/to/infty$ for f of the form e$/sp[ih(t)],$ where h is a real valued, odd, three times continuously differentiable function, $h(t+2/pi)=h(t)+2k/pi$ for some integer k, and $h/sp[/prime/prime](t)<0$ on $(0,/pi),$ with both $h/sp[/prime/prime/prime](0)$ and $h/sp[/prime/prime/prime](/pi)$ different from 0. We show that for such f the following asymptotic formula holds:$$[1/over/sqrt[n]]/Vert[/rm exp](inh)/Vert=/left([2/over/pi]/right)/sp[3/over2]/int/sbsp[0] [/pi]/sqrt[/vert h/sp[/prime/prime](t)/vert][/rm d]t+[/cal O]/left(n/sp[-[1/over10]]/right)/ [/rm as]/ n/to/infty.$$To obtain this result, we separate the Fourier coefficients of exp$(inh(t))$ into 'central' and 'noncentral' terms, and show using Van der Corput lemmas that the contribution of the noncentral terms is negligible. Each of the central terms is then approximated using the method of stationary phase. Finally, we will use certain equidistribution theorems to approximate the sum of the central terms by an integral. Our result generalizes a previous result by D. M. Girard (The behavior of the norm of an automorphism of the unit disk, Pacific J. Math., 47(2):443-456, 1973).