Kolmogorov width of classes of smooth functions on the sphere Sd-1
L Sd-1 := [(x(1),...,x(d)) is an element of R-d : x(1)(2)+(...)+x(d)(2) = 1} be the unit sphere of the d- dimensional Euclidean space R-d. For r > 0, we denote by B-p(r) (1 less than or equal to p less than or equal to infinity) the class of functions f on Sd-1 representable in the form f(x) = phi * F-r(x) := integral(S)(d-1) phi(y)F-r(xy)dsigma(y), parallel tophiparallel to(p) less than or equal to 1, (1.1) where dsigma(y) denotes the usual Lebesgue measure on Sd-1, Fr(xy) = Sigma(k=1)(infinity) (k(k+2lambda))-(2)/(R)(2pilambda+2)/P-Gamma(lambda)(K+lambda)(k)lambda(xy), lambda = (2)/(d-2) and P-k(lambda)(t) is the ultraspherical polynomial. For 1 less than or equal to p,q less than or equal to infinity, the Kolmogorov N-width of B-p(r) in L-q(Sd-1) is given by d(N)(B-p(r),L-q) = (XN)inf (r)(fis an element ofBp)sup (gis an element ofXN)inf parallel tof - gparallel to(q), the left-most infimum being taken over all N-dimensional subspaces X-N of L-q (Sd-1). The main result in this paper is that for rgreater than or equal to2(d - 1)(2), [GRAPHICS] where AN asymptotic to B-N means that there exists a positive constant C; independent of N, such that C(-1)A(N)less than or equal toB(N)less than or equal toCA(N). This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions. (C) 2002 Elsevier Science (USA).
kolmogorov width marcinkiewicz-zygmund inequality spherical harmonics weighted kashin-type inequality