Gaussian quadratures for oscillatory integrands

Abstract We consider a Gaussian type quadrature rule for some classes of integrands involving highly oscillatory functions of the form f ( x ) = f 1 ( x ) sin ζ x + f 2 ( x ) cos ζ x , where f 1 ( x ) and f 2 ( x ) are smooth, ζ ∈ R . We find weights σ ν and nodes x ν , ν = 1 , 2 , … , n , in a quadrature formula of the form ∫ − 1 1 f ( x ) d x ≈ ∑ ν = 1 n σ ν f ( x ν ) such that it is exact for all polynomials f 1 ( x ) and f 2 ( x ) from P n − 1 . We solve the existence question, partially.