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Approximation Order without Quasi-Interpolants

In the study of approximation order, particularly in a multivariable setting, quasi-interpolants have played a major role. This report points out some limitations of quasi-interpolants and describes some recent results on approximation order obtained without the benefit of the quasi-interpolant idea. §1. Approximation Order In most general terms, “approximation order” is defined as follows. Definition 1.1. The indexed collection (Sh) (with h→ 0) of linear subspaces of some normed linear space X has (exact) approximation order k, in symbols: