Best and near-best L1 approximations by Fourier series and Chebyshev series

AbstractApproximations Fnf and Hnf to a function f are defined, respectively, as the partial sums of order n of its expansions in Fourier series and Chebyshev series of the second kind, and they are compared, respectively, with the best trigonometric and best algebraic polynomial approximations \̊tfB and fB of degree n in ⌋1[0, 2π] and L1[−1, 1]. It is shown that the L1 norm of f − Fnf differs from that of f − \̊tfB by at most a factor of the order of log n, and that, similarly, the L1 norm of f − Hnf differs from that of f − fB by at most a factor of the order of log n. These results are discussed in the context of near-best approximations and minimal projections in Lp spaces. Also, it is shown that, if f has a certain type of lacunary series expansion, then Fnf and Hnf are identical to \̊tfB and fB, respectively