Asymptotic expansions for Laguerre-like orthogonal polynomials

AbstractAsymptotic expansion for the Laguerre polynomials and for their associated functions is extended to the case of a weight function which is the product of the Laguerre weight function by a polynomial, nonnegative on the interval [0,∞[

The Spherical Harmonic Spectrum of a Function with Algebraic Singularities

The asymptotic behaviour of the spectral coefficients of a function provides a useful diagnostic of its smoothness. On a spherical surface, we consider the coefficients al m of fully normalised spherical harmonics of a function that is smooth except either at a point or on a line of colatitude, at which it has an algebraic singularity taking the form θp or {pipe}θ-θ0{pipe}p respectively, where θ is the co-latitude and p>-1. It is proven that each type of singularity has a signature on the rotationally invariant energy spectrum, E(l) = √Σm(al m)2 where l and m are the spherical harmonic degree and order, of l-(p+3/2) or l-(p+1) respectively. This result is extended to any collection of finitely many point or (possibly intersecting) line singularities of arbitrary orientation: in such a case, it is shown that the overall behaviour of E(l) is controlled by the gravest singularity. Several numerical examples are presented to illustrate the results. We discuss the generalisation of singularities on lines of colatitude to those on any closed curve on a spherical surface

Polyharmonic Hardy spaces on the complexified annulus and error estimates of cubature formulas

The present paper has a twofold contribution: first, we intro- duce a new concept of Hardy spaces on a multidimensional complexified annular domain which is closely related to the annulus of the Klein-Di rac quadric important in Conformal Quantum Field Theory. Secondly, for functions in these Hardy spaces, we provide error estimate for the p oly- harmonic Gauß-Jacobi cubature formulas, which have been introduced in previous papers.AD 26/03/201