Robert Manson Sawko, Malgorzata Zimon
SIAM/ASA JUQ
We explicitly obtain, for K(x, y) totally positive, a best choice of functions u1, ..., un and v1, ..., vn for the problem minui, vi (∝01 (∝01 |K(x, y) - ∑i = 1, n ui(x) vi(y)| dyp dx) 1 p, where ui ε{lunate} Lp[0, 1], vi ε{lunate} L1[0, 1], i = 1, ..., n, and p ε{lunate} [1, ∞]. We show that an optimal choice is determined by certain sections K(x, ξ1), ..., K(x, ξn), and K(τ1, y), ..., K(τn, y) of the kernel K. We also determine the n-widths, both in the sense of Kolmogorov and of Gel'fand, and identify optimal subspaces, for the set Kr,v = {f(x) = ∑ i=1 raiki(x) + ∫ 0 1K(x,y)h(y)dy, (a1, ..., ar)ε{lunate}Rr, {norm of matrix}h{norm of matrix}p≤1}, as a subset of Lq[0, 1], with either p = ∞ and q ε{lunate} [1, ∞], or p ε{lunate} [1, ∞] and q = 1, where {k1(x), ..., kr(x), K(x, y)} satisfy certain restrictions. A particular example is the ball Br,v = {f} in the Sobolev space. © 1978.
Robert Manson Sawko, Malgorzata Zimon
SIAM/ASA JUQ
W.F. Cody, H.M. Gladney, et al.
SPIE Medical Imaging 1994
Richard M. Karp, Raymond E. Miller
Journal of Computer and System Sciences
Ruixiong Tian, Zhe Xiang, et al.
Qinghua Daxue Xuebao/Journal of Tsinghua University