On the best approximation and polynomials of the least quadratic deviation
Let \(C_{2\pi}\) be the space of 2\(\pi\)-periodic continuous functions f(x) with the norm \(\| f\| =\max_{x}f(x)\), and let \(E^ T_ n(f)\) be the best approximation of f by trigonometric polynomials of order n. Denote by \(L_{n,N}(f,x)\), 0\(\leq n\leq N/2\), the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system of the nodal points \(t_ n=u+2\pi k/N\), 0\(\leq k\leq N-1\), where \(N\geq 2\) is a positive integer and u a real number. If \(\{\epsilon_ n\}^{\infty}_{n=0}\) is a sequence monotonically decreasing to zero, the following class of functions \[ C(\epsilon)=\{f:\quad f\in C_{2\pi},\quad E^ T_ n(f)\leq \epsilon_ n\quad(n\in Z_+)\}, \] is defined. The author studies the problem of estimating the deviation of a function \(f\in C(\epsilon)\) from the polynomial \(L_{n,N}(f)\) in terms of the best approximation of this function. The existence of two positive absolute constants \(c_ 1\) and \(c_ 2\) such that \[ c_ 1\sum^{n}_{\nu =0}\epsilon_{n+\nu}/(\nu +1)\leq \sup_{f\in C(\epsilon)}\| f-L_{n,N}(f)\| \leq c_ 2\sum^{n}_{\nu =0}\epsilon_{n+\nu}/(\nu +1), \] is proved.
scientific article; zbMATH DE number 3853683
scientific article