A generalized Peano Kernel Theorem for distributions of exponential decay

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:433

Article

Blair, Jerome¹  Luttman, Aaron¹  Machorro, Eric¹ 

[1] Natl Secur Technol LLC, Las Vegas, NV 89193 USA

关键词: Peano Kernel Theorem;    Distributions of exponential decay;    Numerical quadrature;   

DOI  :  10.1016/j.jmaa.2015.07.072

来源: Elsevier

【 摘 要 】

The Peano Kernel Theorem (PKT) is a classical representation theorem in numerical integration. The idea is that if T is a quadrature rule that exactly integrates polynomials up to degree n - 1 on a bounded interval [a, b], then there exists a kernel K, depending only on T, such that E(f) = T(f) - integral(b)(a) f (t)dt = integral(b)(a) K(t)f((n))(t) dt, whenever f is an element of C-n([a,b]). In this work, we generalize the PKT from the class of linear functionals on C-n ([a, b]) to the class of Laplace transformable tempered distributions of exponential decay. In particular, it is not necessary that the functionals being approximated have compact support. The generalized result is proven using an approach that provides a formula for computing the kernel K in the Fourier domain, which can be more computationally tractable and efficient in many cases. We conclude with examples of how the generalized Peano Kernel Theorem can be used for error analysis in signal processing. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free