Minimal Projections and Near-Best Approximations by Multivariate Polynomial Expansion and Interpolation

Abstract

If a multivariate function f in a space X is approximated by a function f^N in a subspace Y, then f^N is near-best within a relative distance ρ according to the definition of MASON [1] if

$$ {\text{||f - }}{{{\text{f}}}^{{\text{N}}}}{\text{||}}{\text{(1 + }}\rho {\text{)}}.{\text{||f - }}{{{\text{f}}}^{{\text{B}}}}{\text{||}} $$

(1)

where f^B is a best approximation and ∥. ∥ is a chisen norm on X. Now for any projection P of f in X to Pf in Y (i.e. a bounded, linear, idempotent mapping of X into Y), it follows (see [2]) that

$$ \parallel \text{f} - \text{Pf}\parallel \, \leqslant (1\, + \parallel \text{P}\parallel ).\parallel \text{f} - \text{f}^\text{B} \parallel $$

(2)

Hence Pf is automatically a near-best approximation, and ∥P∥ measures the relative distance from a best approximation. A projection for which ∥P∥ is smallest is termed a “minimal projection”. Note that “near-best” is not a relevant term unless ρ in (1) or ∥P∥ in (2) is suitably small.