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Integration of polyharmonic functions. (English) Zbl 0860.31003
The main purpose of the paper is to extend two well-known quadrature formulas, which are precise for all algebraic polynomials of degree \(2m-1\), to the corresponding multivariate analogues, which are exact for the \(m\)-harmonic functions. Observing the analogy between some well-known theorems concerning integration of polyharmonic functions and results about quadrature formulas, the author searches the possibility of extending some Green’s formulas as, for instance \(\int_B u(x)dx= (1/n) \int_S u(x)d\sigma\), for harmonic functions in order to increase the polyharmonic order of precision (a notion introduced in the paper).
The main result of the paper concerns explicit representations and sharp error bounds for such extensions. The principal tool in the proofs is an extension of the notion of univariate monosplines introduced by Schoenberg. The second chapter of the paper is devoted to this problem.

MSC:
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
65D32 Numerical quadrature and cubature formulas
65D07 Numerical computation using splines
35J30 Higher-order elliptic equations
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[1] Nachman Aronszajn, Thomas M. Creese, and Leonard J. Lipkin, Polyharmonic functions, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1983. Notes taken by Eberhard Gerlach; Oxford Science Publications. · Zbl 0514.31001
[2] B. D. Bojanov, H. A. Hakopian, and A. A. Sahakian, Spline functions and multivariate interpolations, Mathematics and its Applications, vol. 248, Kluwer Academic Publishers Group, Dordrecht, 1993. · Zbl 0772.41011
[3] A.Erdélyi et al., Higher Transcendental Functions I, McGraw-Hill, New York, 1953. · Zbl 0051.30303
[4] I. J. Schoenberg, Spline functions, convex curves and mechanical quadrature, Bull. Amer. Math. Soc. 64 (1958), 352 – 357. · Zbl 0085.33701
[5] B. Fuglede, M. Goldstein, W. Haussmann, W. K. Hayman, and L. Rogge , Approximation by solutions of partial differential equations, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 365, Kluwer Academic Publishers Group, Dordrecht, 1992. · Zbl 0744.00033
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