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• W1479895638 abstract "We prove that the exact reconstruction of a function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from its samples <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s left-parenthesis x Subscript i Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>s (x_i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on any “sufficiently dense sampling set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace x Subscript i Baseline right-brace Subscript i element-of normal upper Lamda> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{x_i}_{iin Lambda }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be obtained, as long as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is known to belong to a large class of spline-like spaces in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript p Baseline left-parenthesis script upper R Superscript n Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^p (mathcal {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittaker sampling theorem on regular sampling and the Paley-Wiener theorem on non-uniform sampling." @default.
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• W1479895638 date "1998-01-01" @default.
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• W1479895638 title "Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The 𝐿^{𝑝}-theory" @default.
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