A lower estimate of best approximations of continuous functions
Let \(C^*\) be the Banach space of real-valued continuous \(2\pi\)-periodic functions f with norm \(\| f\|_{C^*}=\max_{[0,2\pi)}| f|;\) let \({\mathcal T}_ n\) be the set of trigonometric polynomials of degree n; let \(Z_+\) be the set of all nonnegative integers; \(E^*_ n(f)=\inf_{T\in {\mathcal T}_ n}\| f-T\|_{C^*},\) \(f\in C^*\), \(n\in Z_+\). Lower estimates for \(E^*_ n(f)\) are usually established with the aid of the inequality (1) \(E^*_ n(f)\geq | \int^{\pi}_{-\pi}fd\mu |\), \(f\in C^*\), where \(\mu\) is a complex-valued measure on the unit circle orthogonal to elements from \({\mathcal T}_ n\) and satisfying the condition \(\mu\leq 1\). A consequence of inequality (1), in particular, is a well-known theorem of Vallée Poussin and the Newman-Rivlin inequality (2) \(E^*_ n(f)\geq \sup_{N\in Z_+}[2(N+1)]^{-1}| \sum^{N}_{k=0}(k+1)a_{k+n+1}+\sum^{N}_{k=1}(N- k+1)a_{k+n+N+1}|\), where \(f(x)=\sum^{\infty}_{k=0}a_ k \cos kx\). In the present paper, generalizing the Newman-Rivlin approach, we obtain an estimate of \(E^*_ n\) in terms of the characteristic number, largest in modulus, of a Hankel matrix formed from the Fourier coefficients of function f. As consequences, we derive an inequality for best approximations of periodic functions in an integral metric and a generalization of inequality (2) on arbitrary functions from \(C^*\). In addition, we find lower estimates for best approximations of continuous functions by means of algebraic polynomials and by integral functions of exponential type.
scientific article; zbMATH DE number 4177804