Some extensions of Legendre quadrature

Abstract

Abstract. The m-point Gauss-Legendre formula gives an exact expression for the integral of an algebraic polynomial of maximum degree 2m -1 in terms of m ordinates. It is shown that analogous formulas can be derived for exponential and trigonometric polynomials. | The Gauss-Legendre quadrature formulas approximate the integral of a function by a weighted sum of function-values. When m function-values are used, the formula is exact for functions belonging to a specific 2m-dimensional space, namely the polynomials of degree zero through 2m -1. This finite-dimensional space is in no sense 'representative' of the infinite space over which the formula gives exact results. The latter space includes all Riemann-integrable functions which are odd with reference to an origin located at the mid-point of the domain of integration ; it also includes an infinite class of even functions, and this class contains an infinite subclass of polynomials. Similar remarks also apply to the Newton-Cotes formulas. In where qm is the mth degree polynomial belonging to a set which is orthogonal with respect to the inner product iqrix), q,ix)) = /* qrix)qsix)wOx)dx, and the x¡ are zeros of qm.Using these results as a basis, we now consider the problem of constructing an m-point quadrature formula which gives exact results whenever the integrand fOx) belongs to a 2m-dimensional space distinct from those considered in [1]. A simple example is provided by the case where fix) is a linear combination of exponentials erx, where r takes consecutive (possibly negative) integer values. In this case by writing ex = z we obtain /* fix)dx = Jfa zagOz)dz, where s is an integer and g a polynomial. The latter integral is in the form (1), so one can find an m-point formula which is valid for up to 2m terms in the linear combination. Specifically, if r takes integer values from p through p 4-2m -1, then s = p -1 ; the polynomial qm and numbers ß, are defined as in (1), (2) with wOz) = zs and integration limits ea, eh. It then follows that

Trigonometric and Gaussian Quadrature

“…In an earlier paper [1] it was noted that there exist trigonometric and exponential analogs of Gaussian quadrature formulas. We now extend those results to show several interesting features.…”

“…We will restrict our consideration to a quadrature formula of the form (1) [ fix) dx = £ ßifix,) + T,…”

“…The reduction to Gaussian form in this case is essentially given in [1], but we now incorporate the parameter oe into the approximant rather than parametrize the range of integration. Writing u = tan iwx/2)/t with t = tan (co/2) and noting that sin wx = 2r«/(l + t2u2) and cos cox = (1 -rV)/(l + t2u2), we find gix) = Q(u; co)/(l + r2«2)"-1 with Q a polynomial in u of order 2« -2.…”

“…In this case the system of equations involving the <f>¡ result by applying the method of undetermined coefficients to (9) with / E {e°x, • • • , e2N°x\, and so we know from [1] that the (b¡ are distinct and lie in the interval (0, 1). We again use Prony's method to derive a polynomial with roots <p¡, but in this case it is convenient to assume that the polynomial is <¡Y + áx<pN~l + ■ ■ ■ + <52cf> + &x-The coefficients then satisfy ; It is interesting to note that the Christoffel numbers ft-= 0(1 fa) for j Sg 1 and ßo = 2 -0(1 ¡a) with n odd, whereas the ß, = O(l) for n even.…”