Trigonometric and Gaussian Quadrature

Some relationships are established between trigonometric quadrature and various classical quadrature formulas. In particular Gauss-Legendre quadrature is shown to be a limiting case of trigonometric 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 find that Legendre and trapezoidal rule quadrature result from the trigonometric formula as a certain parameter takes special values. We also study in some detail a case in which the roots of an orthogonal polynomial tend to coalesce. The development is based on transforming the integrand into Gaussian form. The problem then reduces to finding a polynomial belonging to an orthogonal set. An algorithm is proposed for constructing the polynomial's coefficients numerically. We will restrict our consideration to a quadrature formula of the form (1) [ fix) dx = £ ßifix,) + T, J-X i-X where T is the truncation error. The formula is to be exact (i.e., T = 0) when fix) belongs to a function space with basis (2) {1, cos cox, • • • , cos (« — l)wx}. Note that we have assumed fix) is an even function in writing (2). We can append a set of linearly independent odd functions to obtain a basis set for a more general integrand, say gix). The choice of the odd basis set is arbitrary since the integration is over symmetric limits. We will discuss two choices. The parameter w, which may be complex, is to be chosen for convenience in each problem. We will only consider o> real with co E [0, ir] or imaginary with w = ia and a E [0, œ ). First suppose that n-l n-1 Six) = £ ar cos rwx + £ br sin rwx. o i The reduction to Gaussian form in this case is essentially given in [1], but we now incorporate the parameter œ into the approximant rather than parametrize the range of integration. Writing u = tan iwx/2)/t with t = tan (co/2) and noting that Received July 17, 1969, revised November 10, 1969. AMS Subject Classifications. Primary 6555; Secondary 4205.