Universal Asymptotic Optimality

A cubature formula having an optimal mode of convergence in a given Banach space certainly preserves this property under passage to an equivalent norm. A stronger assertion is often valid: a formula has an optimal mode of convergence in different nonequivalent spaces. For instance, a homogeneous error of degree M has an optimal mode of convergence in all spaces math for m ∈ (n, M/2] and p ∈ (l∞). A deeper fact was faced: the same cubature formula may be asymptotically optimal simultaneously in many spaces not necessarily equivalent to one another. We saw this by the example of a formula with the error l∞(x) acting over the periodic spaces B defined in § 1 of Chapter 4. Such results drive us to supposing that the concept of asymptotic optimality is stable under the choice of function spaces and to making a conjecture that there exist universal asymptotically optimal formulas. In our understanding, these formulas preserve the asymptotic optimality property on a wide class of spaces of integrable functions.