Uniform and $$L^p$$ Convergence of the Hermite Interpolation at Pollaczek–Laguerre Zeros

Abstract The paper deals with the weighted polynomial approximation of functions defined on $$(0,+\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , which can grow exponentially both at $$+\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> and at 0. To this aim, we introduce interpolating operators of Hermite and Hermite–Fejér-type, based at the zeros of Pollaczek–Laguerre type orthogonal polynomials. We prove that these processes converge in weighted uniform and $$L^p-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> norms and provide sharp error estimates showing that the order of convergence is the same as the best polynomial approximation, under suitable assumptions.