Modified newton–cotes formulae for numerical quadrature of oscillatory integrals with two independent variable frequencies

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Abstract
It is shown how a function can be approximated in a unique way by an interpolation function fn (x) which is a linear combination of cos kx, sin kx, cos k′x, sin k′xand a polynomial of {n– 4)th degree, such that fn coincides with f(x) in (n + 1) equidistant points, and whereby kand k′are two arbitrary real, purely imaginary or complex conjugate parameters. Several equivalent expressions of the interpolation function fn are given, and also the error term is derived in closed form. With this type of interpolation a set of modified Newton–Cotes quadrature rules is established and the total truncation error associated with these rules is discussed. Subsequently, the five-point quadrature rule is treated in full detail and several formulae are derived which facilitate numerical computation. Finally, the proposed formulae are implemented for certain illustrative numerical examples. In particular, a technique is established for attributing values to the parameters k and k′ such that the total truncation error is minimized.

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