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The results of the second phase of the Super-Kamiokande solar neutrino measurement are presented and compared to the first phase. −0.15 (sys.)) × 10 6 cm −2 sec −1 and the day-night difference is found to be (−6.3 ± 4.2(stat.) ± 3.7(sys.))%. There is no evidence of systematic tendencies between the first and second phases.

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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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