Summary
In this paper we consider the approximate evaluation of\(\int\limits_a^b {K(x)f(x)dx} \), whereK(x) is a fixed Lebesgue integrable function, by product formulas of the form\(\sum\limits_{i = 0}^n {w_i f(x_i )} \) based on cubic spline interpolation of the functionf.
Generally, whenever it is possible, product quadratures incorporate the bad behaviour of the integrand in the kernelK. Here, however, we allowf to have a finite number of jump discontinuities in [a, b]. Convergence results are established and some numerical applications are given for a logarithmic singularity structure in the kernel.
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References
Alaylioglu, A., Lubinsky, D.S., Eyre, D.: Product Integration of Logarithmic Singular Integrands based on Cubic Splines. J. Comput. App. Math.11, 353–366 (1984)
Atkinson, K.E.: The Numerical Solution of Fredholm Integral Equations of the Second Kind. Siam J. Numer. Anal.4, 337–347 (1967)
de Boor, C.: A Practical Guide to Splines. Berlin, Heidelberg, New York: Springer 1972
Cheney, E.W., Schurer, F.: A Note on the Operators Arising in Spline Approximation. J. Approximation Theory1, 94–102 (1968)
Cheney, E.W., Schurer, F.: Convergence of Cubic Splines Interpolants. J. Approximation Theory3, 114–116 (1970)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration (2nd ed.). New York: Academic Press 1984
Dodd, L.R.: Numerical Solution of a Singular Threebody Equation. Proc. Tenth. Internat. Conf. on Few Body Problems in Physics, Karlsruhe, FRG, 1983
Domsta, J.: Approximation by Spline Interpolating Basis. Studia Mathematica58, 223–237 (1976)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products. London: Academic Press 1980
Greville, T.N.E.: Theory and Applications of Spline Functions. New York: Academic Press 1969
Hall, C.A.: On Error bounds for Spline Interpolation. J. Approximation Theory1, 209–218 (1968)
Hall, C.A., Meyer, W.W.: Optimal Error Bounds for Cubic Spline Interpolation. J. Approximation Theory16, 105–122 (1976)
Lucas, T.R.: Error Bounds of Interpolating Cubic Splines under Various End Conditions. SIAM J. Numer. Anal.11, 569–584 (1974)
Marsden, M.: Cubic Spline Interpolation of Continuous Functions. J. Approximation Theory10, 103–111 (1974)
Powell, M.D.J.: Approximation Theory and Methods. London: Cambridge University Press 1981
Rabinowitz, P., Sloan, I.H.: Product Integration in the Presence of a Singularity. SIAM J. Numer. Anal.21, 149–166 (1984)
Schumaker, L.L.: Spline Functions: Basic Theory. New York: Wiley 1981
Schurer, F., Cheney, E.W.: On Interpolating Cubic Splines with Equally-Spaced Nodes. Indag. Math.31, 517–524 (1968)
Sloan, I.H.: On Choosing the Points in Product Integration. J. Math. Phys.21, 1032–1039 (1980)
Sloan, I.H., Smith, W.E.: Properties of Interpolating Product Integration Rules. SIAM J. Numer. Anal.19, 427–442 (1982)
Smith, W.E., Sloan, I.H.: Product Integration Rules Based on the Zeros of Jacobi Polynomials. SIAM J. Numer. Anal.17, 1–13 (1980)
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Dagnino, C., Orsi, A.P. Product integration of piecewise continuous integrands based on cubic splineinterpolation at equally spaced nodes. Numer. Math. 52, 459–466 (1988). https://doi.org/10.1007/BF01462239
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DOI: https://doi.org/10.1007/BF01462239