Estimation of maximal and minimal deviation of neighboring points in Chebyshev's alternant (Q1174021)

Let \(f(x)\in C_{2\pi}\), \(T_{n-1}(x)\) be a trigonometric polynomial of the order \((n-1)\) whose deviation from \(f(x)\) in the metric of the space \(C_{2\pi}\) is the least \(E_{n-1}(f)=\max_{-\pi\leq x\leq \pi}| f(x)-T_{n-1} (x)|\). As it is known, in the interval \([-\pi,\pi)\) there exist \(2n\) points of the Chebyshev alternant \(u_ 1<u_ 2<\dots<u_{2n-1}<u_{2n}\), at which the difference \(f(x)-T_{n-1}(x)\) takes on the value \(E_{n-1}(f)\) with alternating signs. The location of the points of Chebyshev's alternant will be characterized by means of the quantities \[ D_ n(f)=\max_{1\leq i\leq 2n} | u_ i- u_{i+1}|\hbox{ and }d_ n(f)=\min_{1\leq i\leq 2n} | u_ i- u_{i+1}|, \] where \(u_{2n+1}=u_ 1+2\pi\). In this paper, bounds from below and above for \(D_ n(f)\) and \(d_ n(f)\) are obtained in terms of the best approximations \(E_{n-1}(f)\) and \(E_ n(f)\), and also quantitative estimates of the deviations of the sequences of zeros and the extremal points of a class of trigonometric polynomials are derived.