Characterization theorem of generalized polynomial of best approximation having bounded coefficients

Abstract

Let the set of generalized polynomials having bounded coefficients beK={p=$\mathop \Sigma \limits_{j = 1}^n$ α _jg_j.α _j≤α _j≤β _j,j=1, 2, ...,n}, whereg ₁,g ₂, ...,g _n are linearly independent continuous functions defined on the interval [a, b],α _j,β _j are extended real numbers satisfyingα _j<+∞,β _j>-∞, andα _j≤β _j. Assume thatf is a continuous function defined on a compact setX ⊂ [a, b]. This paper gives the characterization theorem forp being the best uniform approximation tof fromK, and points out that the characterization theorem can be applied in calculating the approximate solution of best approximation tof fromK.