The numerical computation of some integrals on the real line

摘要

We discuss the approximation of integrals of type I(f;t) = ∫from x = R of f(x)K(x,t)e~(-x~2)|x|~αdx, α > -1, where K is the weakly singular algebraic kernel |x - t|~λ, -1 < λ < 0, for "large" value of the parameter t. Moreover, we consider strongly oscillatory kernels of type K_1(x,t) = sin(tx~2), K_2(x,t) = cos(tx~2). Weighted error estimates in uniform and L_1 norm are stated and numerical examples to confirm the efficiency of the proposed procedures are given.