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Characterization of the speed of convergence of the trapezoidal rule. (English) Zbl 0693.41031

Our aim is to determine the precise space of functions for which the trapezoidal rule converges with a prescribed rate as the number of nodes tends to infinity. Excluding or controlling odd functions in some way it is possible to establish a correspondence between the speed of convergence and regularity properties of the function to be integrated. In this way we characterize Sobolev spaces, certain spaces of infinitely differentiable functions, of functions holomorphic in a strip, of entire functions of order greater than 1 and of entire functions of exponential type by the speed of convergence.

MSC:

41A55 Approximate quadratures
65D32 Numerical quadrature and cubature formulas
46E10 Topological linear spaces of continuous, differentiable or analytic functions
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References:

[1] Bary, N.K.: A treatise on trigonometric series (Vol I and II). Oxford: Pergamon Press 1964 · Zbl 0129.28002
[2] Blakeley, G.R., Borosh, I., Chui, C.K.: A two-dimensional mean problem. J. Approx. Theory22, 11-26 (1973) · Zbl 0375.30013 · doi:10.1016/0021-9045(78)90067-9
[3] Boas, R.P., Jr.: Entire functions. New York: Academic Press 1954 · Zbl 0058.30201
[4] Brass, H.: Umkehrsätze beim Trapezverfahren. Aequationes Math.18, 338-344 (1978) · Zbl 0405.65003 · doi:10.1007/BF03031685
[5] Butzer, P.L., Nessel, R.J.: Fourier analysis and approximation (Vol. I). Basel, Birkhäuser 1971 · Zbl 0217.42603
[6] Davis, P.J., Rabinowitz, P.: Methods of numerical integration (2nd edition). New York: Academic Press 1984 · Zbl 0537.65020
[7] Loxton, J.H., Sanders, J.W.: On an inversion theorem of Möbius. J. Aust. Math. Soc., Ser. A30, 15-32 (1980) · Zbl 0445.10006 · doi:10.1017/S144678870002187X
[8] Loxton, J.H., Sanders, J.W.: The kernel of a rule of approximate integration. J. Aust. Math. Soc., Ser. B21, 257-267 (1980) · Zbl 0418.41026 · doi:10.1017/S0334270000002356
[9] Pólya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis (Vol. I and II, 4th edition), Berlin: Springer 1970 and 1971 · Zbl 0201.38102
[10] Rahman, Q.I., Schmeisser, G.: Characterization of functions in terms of rate of convergence of a quadratur process. (Submitted to Proc. Amer. Math. Soc.) · Zbl 0722.41031
[11] Winter, A.: Diophantine approximations and Hilbert’s space. Am. J. Math.66, 564-578 (1944) · Zbl 0061.24902 · doi:10.2307/2371766
[12] ?ensykbaev, A.A.: Best quadrature formula for some classes of periodic differentiable functions. Math. USSR Izy.11, 1055-1071 (1977) · Zbl 0393.41014 · doi:10.1070/IM1977v011n05ABEH001758
[13] ?ensykbaev, A.A.: Best quadrature formula for the class \(W_{L_2 }^r \) . Anal. Math.3, 83-95 (1977) · Zbl 0374.41011 · doi:10.1007/BF02333255
[14] Zygmund, A.: Trigonometric series (Vol. I and II, 2nd edition). Cambridge: University Press 1968
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