A note on generalized Floater–Hormann interpolation at arbitrary distributions of nodes

Abstract The paper is concerned with a generalization of Floater–Hormann (briefly FH) rational interpolation recently introduced by the authors. Compared with the original FH interpolants, the generalized ones depend on an additional integer parameter $$\gamma &gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , that, in the limit case $$\gamma =1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> returns the classical FH definition. Here we focus on the general case of an arbitrary distribution of nodes and, for any $$\gamma &gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , we estimate the sup norm of the error in terms of the maximum ( h ) and minimum ( $$h^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>h</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> ) distance between two consecutive nodes. In the special case of equidistant ( $$h=h^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:math> ) or quasi–equidistant ( $$h\approx h^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>≈</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:math> ) nodes, the new estimate improves previous results requiring some theoretical restrictions on $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> which are not needed as shown by the numerical tests carried out to validate the theory.