Extremal problems for functions with small support (Q1387328)

Given \(0< h\leq 1/2\), define the class \(K_2(h)\) of functions by the following properties: (i) \(f(x)= \sum^\infty_{n=0} a_n\cos(2\pi nx)\), (ii) \(\sum| a_n|= 1\), (iii) \(f(x)= 0\) for \(h\leq| x|\leq 1/2\). The problem the authors attack is to find an estimate of \(a_0= \int^h_{-h} f(x)dx\). Setting \[ A_2(h):= \sup_{f\in K_2(h)} \int^h_{-h} f(x)dx, \] the following theorems are proved: (1) \(\lim_{h\to 0} A_2(h)/n= L\) exists, (2) \(A_2(1/4)= 2/(\pi+ 4)\), (3) \(L> 1.16\), (4) \(L\leq{2\over \pi} \int^\pi_0 {\sin t\over t} dt= 1.179\dots\)\ .