Alternating trigonometric polynomials
Set \(t_ k=h_ n+\{k\pi /(n+1)\}\), \(k=0,1,...,2n+1\), \(0\leq h_ n<\pi /(n+1)\). Let \(C_{2\pi}\) denote the class of \(2\pi\) periodic continuous functions. The author shows that for any \(f\in C_{2\pi}\) and any \(h_ n\), there exists a unique trigonometric polynomial \[ v_ n(f;h_ n;t)=\alpha_ 0/2+\sum^{n}_{m=1}(\alpha_ m\cos mt+\beta_ m\sin mt), \] the coefficients of which are defined together with the parameter \(V_ n(f;h_ n)\) from the linear system: \(f(t_ k)-v_ n(f;h_ n;t_ k)=(-1)^ kV_ n(f,h_ n),\) \(k=0,1,...,2n+1\). For corresponding results concerning algebraic polynomials case, reference may be made to \textit{G. Meinardus} [Approximation of functions: Theory and numerical methods (1967; Zbl 0154.152)]. Denoting by \(E_ n(f)\), the distance of f from the class of trigonometric polynomials of degree n, the author obtains the following lower bound for \(E_ n(f): \sqrt{2}E_ n(f)\geq \{A^ 2_ n(f)+B^ 2_ n(f)\}^{1/2},\) where \(A_ n(f)\) and \(B_ n(f)\) respectively denote the values of \(V_ n(f,h_ n)\) corresponding to \(h_ n=0\) and \(\pi /2(n+1)\). The author shows that the foregoing functionals \(A_ n(f)\) and \(B_ n(f)\) are closely related to the Fourier coefficients of f.
scientific article; zbMATH DE number 4009015
scientific article