Recent Developments in Asymptotic Expansions From Numerical Analysis and Approximation Theory

Abstract

Abstract In this chapter, we discuss some recently obtained asymptotic expansions related to problems in numerical analysis and approximation theory. • We present a generalization of the Euler–Maclaurin (E–M) expansion for the trapezoidal rule approximation of finite-range integrals ∫ a b f ( x ) d x , when f ( x ) is allowed to have arbitrary algebraic–logarithmic endpoint singularities. We also discuss effective numerical quadrature formulas for so-called weakly singular, singular , and hypersingular integrals, which arise in different problems of applied mathematics and engineering. • We present a full asymptotic expansion (as the number of abscissas tends to infinity) for Gauss–Legendre quadrature for finite-range integrals ∫ a b f ( x ) d x , where f ( x ) is allowed to have arbitrary algebraic–logarithmic endpoint singularities. • We present full asymptotic expansions, as n → ∞ , (i) for Legendre polynomials P n ( x ), x ∈ (−1, 1), (ii) for the integral ∫ c d f ( x ) P n ( x ) d x , − 1 l c l d l 1, and (iii) for Legendre series coefficients e n [ f ] = ( n + 1 / 2 ) ∫ − 1 1 f ( x ) P n ( x ) d x , when f ( x ) has arbitrary algebraic–logarithmic (interior and/or endpoint) singularities in [−1, 1].