Product integration is a well established technique for evaluating the definite integral
$$\int \begin{gathered}1 \hfill \\0 \hfill \\\end{gathered} g(t)f(t)dt$$
The function f(t) is approximated by fN(t) and the resulting integral
$$\int \begin{gathered}1 \hfill \\0 \hfill \\\end{gathered} g(t){f_N}(t)dt$$
is taken as our approximation to the integral in (1.1). fN(t) is usually constructed by piecewise-polynomial interpolation. A simple example is the product trapezoidal rule which uses piecewise-linear interpolation, namely
$${f_N}(t) = \sum\limits_{i = 0}^N {{\phi _i}(t)f(i/N)} $$
. In (1.3), the ϕi(t) I = 0,...,N are the usual “hat” functions. Then we have
$$\int \begin{gathered}1 \hfill \\0 \hfill \\ \end{gathered} g(t)f(t)dt \simeq \sum\limits_{i = 0}^N {{w_i}f(i/N)} $$
$${w_i} = \int \begin{gathered}1 \hfill \\0 \hfill \\\end{gathered} g(t){\phi _i}(t)dt$$


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© Springer Basel AG 1982

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  • K. S. Thomas

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