Rigorous root-exponential convergence theory for Laguerre approximations

Establish a rigorous proof of root-exponential convergence for generalized Laguerre polynomial approximations of the general class of functions that are analytic in a closed parabola containing the nonnegative half-line and satisfy an algebraic growth condition at infinity, including the coefficient, projection, interpolation, differentiation, and Gauss–Laguerre quadrature estimates considered by Wang.

Background

The paper analyzes claimed convergence results for generalized Laguerre polynomial approximations of functions analytic inside a parabola enclosing the nonnegative half-line. The authors argue that the proofs of those results rely on passing to a limit under an unbounded contour integral without establishing a suitable domination or uniformity condition.

Consequently, the claimed root-exponential convergence results are not fully justified for the stated general class of analytic functions. Resolving this issue would complete the rigorous convergence theory underlying the cited coefficient, approximation, interpolation, and quadrature estimates.