A new quadrature rule based on a generalized mixed interpolation formula of exponential type

A new method of approximating a function f(x) uniquely by a function fn(x) of the form fn(x)=elx(aU1(kx)+bU2(kx)+∑i=0n−2cixi), so that fn(x) interpolates f(x) at (n+1) equidistant points x0,x0+h,…,x0+nh, with h>0, is derived in a closed-form. Various equivalent forms of the interpolation formula are also derived. A closed-form expression is derived for the error involved in such an approximation. With the aid of the newly derived interpolation formula, a set of Newton Cotes quadrature rules of the closed type are also derived. The total truncation error involved in these quadrature rules are analysed and closed-form expressions for error terms are proposed as conjectures in the two cases when n is odd and when n is even, separately. A more general exponential-type interpolation formula and quadrature rules based upon the generalized mixed interpolation formula are also explained and discussed. A few numerical examples are worked out as illustrations and the results are compared with the results of some of the earlier methods.