Best Approximations in the Spaces C and L

Abstract

In this chapter, we consider the best approximations of functions by trigonometric polynomials in the spaces C and L, i.e., $E_n (f)_X = \mathop {\inf }\limits_{T_{n - 1} \in \mathcal{T}_{2n - 1} } \left\| {f( \cdot ) - T_{n - 1} ( \cdot )} \right\|_X$ where X is either C or L. In these spaces, unlike the spaces L _p , 1 < p < ∞, in the general case, the approximation by Fourier sums no longer has the order of the best approximation. Moreover, as was already noted, the space C contains functions whose Fourier series diverge.