Abstract
This chapter presents some remarks on good, simple, and optimal quadrature formulas. The construction of a whole family of quadrature formulas of that order which are simple and whose error bound, while not quite minimal, is asymptotically equal to the minimal error bound is described. The problem of finding the optimal quadrature formula can be reduced to solving linear equations in unknowns, and solved some special cases. The chapter discusses the construction of a large class of simple schemes whose error bounds have asymptotic expressions with the same first term as that of the optimal scheme. The somewhat more general problem of tabulating the indefinite integral is considered in such a way that when a prescribed interpolation scheme is applied to the tabulated values, one obtains a good approximation in norm to this indefinite integral. These limiting values are precisely what one obtains for the optimal approximation. The corresponding formulas were discovered and called semicardinal spline quadrature formulas.
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