Extended interpolation on the square and a mixed interpolating sequence

Abstract This paper investigates an extended interpolation process based on the zeros of Jacobi polynomials to approximate functions on $$[-1,1]^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . By combining a classical Lagrange interpolating polynomial sequence with its extended counterpart, a new mixed polynomial sequence is introduced, which significantly reduces the number of required function evaluations. Convergence conditions in suitable weighted function spaces are rigorously analyzed and some numerical tests are presented to support the efficiency of the proposed scheme.