Title: Approximation from shift-invariant subspaces of 𝐿₂(𝐑^{𝐝})
Abstract:A complete characterization is given of closed shift-invariant subspaces of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 2 left...A complete characterization is given of closed shift-invariant subspaces of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 2 left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_2}({\mathbb {R}^d})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.Read More