Taylor expansion of best lp-approximations about p = 1 Original Research Article

In this paper, we consider the problem of best approximation in lp (n), 1 ≤ p ≤ ∞. If hp, 1 ≤ p < ∞, denotes the best lp-approximation of the element hϵrn from a proper affine subspace K of rn, h ∉ K, then View the MathML source, where View the MathML source, is a best l1-approximation of h from K, the so-called natural best k1-approximation. We prove that, for every rϵn, the best kp-approximations have a Taylor expansion of order r of the form View the MathML source for some αlϵrn, 1 ≤ l ≤ r, and γp(r)ϵrn with |γp(r)|=o((p−1)r+1).