The numerical computation of some integrals on the real line

We discuss the approximation of integrals of type I(f;t)=∫Rf(x)K(x,t)e−x2|x|α dx, α>−1, 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 K1(x,t)=sin(tx2), K2(x,t)=cos(tx2). Weighted error estimates in uniform and L1 norm are stated and numerical examples to confirm the efficiency of the proposed procedures are given.