A New Approach to Positivity and Monotonicity for the Trapezoidal Method and Related Quadrature Methods

Q. I. Rahman and G. Schmeisser ()
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Q. I. Rahman: Université de Montréal
G. Schmeisser: University of Erlangen-Nuremberg

A chapter in Progress in Approximation Theory and Applicable Complex Analysis, 2017, pp 463-489 from Springer

Abstract: Abstract For positive integers n let R n [f] be the remainders of a quadrature method applied to a function f. It is of practical importance to know sufficient conditions on f which guarantee that the remainders are non-negative and converge monotonically to zero as n → ∞. For most of the familiar quadrature methods such conditions are known as sign conditions on certain derivatives of f. However, conditions of this type specify only a small subset of the desired functions. In particular, they exclude oscillating functions. In the case of the trapezoidal method, we propose a new approach based on Fourier analysis and the theory of positive definite functions. It allows us to describe much wider classes of functions for which positivity and monotonicity occur. Our considerations include not only the trapezoidal method on a compact interval but also that for integration over the whole real line as well as some related methods.

Keywords: Quadrature; Trapezoidal method; Positive remainders; Monotonically decreasing remainders; Positive definite functions; 41A55; 41A80; 42A82; 65D30; 65D32 (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-319-49242-1_21

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DOI: 10.1007/978-3-319-49242-1_21

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