Periodic Functions

then X*, is said to be optimal. A typical choice for Y{ is the image of a unit ball of some normed linear space Y (which may be different from X) under a compact linear mapping K of Y into X. Thus, Y{= {Ky: ]]y][v =< 1}. When X = Y is a Hilbert space, then it is possible to obtain an expression for d,(Y{'; X) and identify optimal subspaces. These facts originated with the example given by Kolmogorov in his paper [2] in which the concept of n-widths was introduced. In this case, d,(Y[; X) is related to the s -numbers of K. Our concern here is with computing the n-widths and determining optimal subspaces for various classes of periodic functions. In [7] and [8], C. A. Micchelli and the author were concerned with the problem of obtaining n-widths and optimal subspaces for X = L q, Y = L p (where p = ~, or q -1), and where K had the above form. The results were, in the main, restrictions to the case where the mapping K had certain total positivity properties. These properties are in a certain sense natural and give rise to particularly elegant solutions of the n-width problem. An example of the above mentioned theory provides a solution to the n-widths of the Sobolev spaces W~'~[0, 1] in Lq[0, 1] (for p = o0 or q = 1), a result which had