Optimal quadrature formulae on the real line

Abstract

In the present paper we treat quadrature formulae on the real line with equidistant nodes involving a sequence of (not necessarily consecutive) derivatives. In Theorem 3.1 we obtain the optimal quadrature formula for Sobolev space of functions with an additional restriction on its Fourier transforms. The quadrature, which has been proved to be optimal, was introduced in [ 111 to the best of our knowledge. The quadrature formulae, considered in the present paper, were studied in [7] with respect to the optimal degree of precision. An explicit expression for the error of the optimal quadrature formula is given. A comparison of the results with those for quadrature formulae without derivatives, given in [2], under the same computational complexity is given. Other results concerning numerical integration on the real line can be found in [S, 6, 131.