About Widths of Wiener Space in the Lq-Norm

In this paper we consider average n-widths of the Wiener space C in the Lq-norm. We study two kinds of average n-widths which describe best and best linear approximation of the Wiener space over all n-dimensional subspaces. The approximation problem on spaces with Wiener measure in the L2-norm was investigated in the book of Traub et al. (1988). Papers concerned with average n-widths in Banach spaces include Buslaev (1988), Heinrich (1990), Maiorov (1990, 1993a,b), Maiorov and Wasilkowski (1996), Mathe (1990), Ritter (1990), Pietsch (1980), and Traub et al. (1988). The approximation of the functionals on Banach space with measure can be found in Lee and Wasilkowski, (1986). The problem of computing n-widths is closely connected with the problem of complexity of approximation (Traub et al., 1988). The asymptotic n-widths dn(C, Lq , g) 3 n21/2 were calculated in Maiorov (1990) for the case 2 # q , y, and in Maiorov (1993a) for the case q 5 y. In this work we prove that the same estimate is also true for the case 1 # q , 2. Note that the estimate for n-widths coincides with the estimate for the average spline-approximation of Wiener space in the Lqnorm for all 1 # q # y (Ritter, 1990). The probabilistic n-widths of the space