A systematic introduction to the principal ideas and results of the contemporary theory of approximate integration, this volume approaches its subject from the viewpoint of functional analysis. In addition, it offers a useful reference for practical computations. Its primary focus lies in the problem of approximate integration of functions of a single variable, rather than the more difficult problem of approximate integration of functions of more than one variable.
The three-part treatment begins with concepts and theorems encountered in the theory of quadrature. The second part is devoted to the problem of calculation of definite integrals. This section considers three basic topics: the theory of the construction of mechanical quadrature formulas for sufficiently smooth integrand functions, the problem of increasing the precision of quadratures, and the convergence of the quadrature process. The final part explores methods for the calculation of indefinite integrals, and the text concludes with helpful appendixes.

Contenuti

Preface Translator's Preface Part I. Preliminary Information Chapter 1. Bernoulli Numbers and Bernoulli Polynomials 2. Orthogonal Polynomials 3. Interpolation of Functions 4. Linear Normed Spaces. Linear Operators Part II. Approximate Calculation of Definite Integrals 5. Quadrature Sums and Problems Related to Them. The Remainder in Approximate Quadrature 6. Interpolatory Quadratures 7. Quadratures of the Highest Algebraic Degree of Precision 8. Quadrature Formulas with Least Estimate of the Remainder 9. Quadrature Formulas Containing Preassigned Nodes 10. Quadrature Formulas with Equal Coefficients 11. Increasing the Precision of Quadrature Formulas 12. Convergence of the Quadrature Process Part III. Approximate Calculation of Indefinite Integrals 13. Introduction 14. Integration of Functions Given in Tabular Form 15. Calculation of Indefinte Integrals Using a Small Number of Values of the Integrand 16. Methods Which Use Several Previous Values of the Integral Appendix A. Gaussian Quadrature Formulas for Constant Weight Function Appendix B. Gaussian-Hermite Quadrature Formulas Appendix C. Gaussian-Laguerre Quadrature Formulas Index

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