Summary.
In the light of the functional analysis theory we establish the optimality of the double exponential formula. The argument consists of the following three ingredients: (1) introduction of a number of spaces of functions analytic in a strip region about the real axis, each space being characterized by the decay rate of their elements (functions) in the neighborhood of the infinity; (2) proof of the (near-) optimality of the trapezoidal formula in each space introduced in (1) by showing the (near-) equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures; (3) nonexistence theorem for the spaces, the characterizing decay rate of which is more rapid than the double exponential.
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Received September 15, 1995 / Accepted December 14, 1995
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Sugihara, M. Optimality of the double exponential formula – functional analysis approach –. Numer. Math. 75, 379–395 (1997). https://doi.org/10.1007/s002110050244
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DOI: https://doi.org/10.1007/s002110050244