Spline spaces are optimal for L²\,\,n-width

IntroductionLet X (X, I1" II) be a normed linear space, A a subset of X and X. an n-dimensional linear subspace of X.The Kolmogorov n-width of : relative to X is defined by d#(A; X)= d(Ac')= inf sup inf yll.Xn x ,Yt" Y Xn X. is called an optimal subspace for : provided that d.(:) sup inf yll, Yt y Xn This concept of n-width was introduced by Kolmogorov in [8] and in his paper he finds the exact value of the n-width for W2"[0, 1] {f: f('-') abs.cont.on (0, 1), IIf(')ll <_ 1}(11 e norm on [0, 1]).Roughly speaking Kolmogorov showed that the n-width corresponds to the nth eigenvalue of a boundary value problem and an optimal subspace is spanned by the first n eigenfunctions.Kolmogorov claimed that W2"[0, 1] has a unique optimal subspace and as late as Tihomirov [13] this error was overlooked.It was first observed to be false by Karlovitz in [4] while in Ioffe and Tihomirov [2] it is conjectured that W2"[0, 1] has an optimal spline subspace.Subsequently, Karlovitz [5] explored the question of nonuniqueness of opti- mal subspaces in a general setting.The related question for min max and max min characterization of eigenvalues has been treated in Weinstein and Stenger's book [17].A main goal of this paper is to prove that W 2'' admits, for all r, optimal spline subspaces.There are in fact two; one of degree r-1 and another of degree 2r-1.