## Approximation from shift-invariant subspaces of $$L_ 2(\mathbb{R}^ d)$$.(English)Zbl 0790.41012

Summary: A complete characterization is given of closed shift-invariant subspaces of $$L_ 2(\mathbb{R}^ d)$$ which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.

### MSC:

 41A25 Rate of convergence, degree of approximation 41A63 Multidimensional problems 41A30 Approximation by other special function classes 41A15 Spline approximation 42B99 Harmonic analysis in several variables 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: