The approximation power of moving least-squares

Abstract

A general method for near-best approximations to functionals on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <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:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, using scattered-data information is discussed. The method is actually the moving least-squares method, presented by the Backus-Gilbert approach. It is shown that the method works very well for interpolation, smoothing and derivatives’ approximations. For the interpolation problem this approach gives Mclain’s method. The method is near-best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation. The interpolation approximation in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <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:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is shown to be a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">C^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> function, and an approximation order result is proven for quasi-uniform sets of data points.