On the computation of Fourier transforms of singular functions

Efficient methods for highly oscillatory integrals with weak and Cauchy singularities

“…For the calculation of moments N k (α, β, ω), one can refer to the reference [21]. Based on the above discussion, we outline our algorithm as follows: Algorithm A c c e p t e d M a n u s c r i p t Numerical methods for oscillatory integrals 8 …”

“…Also, ClenshawCurtis method has been developed in a considerable literature of papers [2,3,5,21,24,28,30,31] to evaluate highly oscillatory integrals. In this paper, according to the ideas of numerical steepest descent method and Clenshaw-Curtis method, we introduce two efficient methods for the integral (1.1) based on the construction of Gauss-type quadrature and the fast computation of modified moments.…”

On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals

“…The Filon-like approach was also considered in Piessens [16] and Piessens and Branders [17,18], where f is replaced by its (N + 1)-term truncated Chebyshev series expansion instead of the degree N polynomial interpolant. One feature of the choosing the endpoints as collocation points is of particular relevance: it obtains an asymptotic order O(ω −2 ) on the error instead of O(ω −1 ) [11,Iserles], [12,Iserles and Nørsett].…”

Legendre-coefficients Comparison Methods for the Numerical Solution of a Class of Ordinary Differential Equations

“…Special attention has over the years been given to applications of orthogonal functions, such as Chebyshev polynomials [1], Laguerre polynomials [2], Legendre polynomials [3], Fourier series [4], block-pulse functions to mention a few. These functions and polynomial series have recieved considerable attention in dealing with various problems in Engineering and Scientific applications [5].…”