Short communication
A sharp discrete convolution sum estimate

https://doi.org/10.1016/j.cnsns.2022.106923Get rights and content

Abstract

The paper by C. Lubich in Numer. Math. 2(52):129–145, 1988 is widely cited for its analysis of convolution quadrature rules for integrals with weakly singular kernels. This analysis depends on a key technical lemma (an upper bound on a discrete convolution sum) whose proof uses some advanced tools. In the present paper it will be shown that this lemma can be quickly proved in an elementary way; moreover, the new proof includes those cases that were excluded from the 1988 paper, and the bounds obtained are shown to be sharp.

Section snippets

Background

In the research literature on the numerical solution of fractional-derivative differential equations, the technique of convolution quadrature plays a central role. This sophisticated method, pioneered by Lubich in the papers [1], [2] and subsequently used by many researchers, uses Laplace transforms and sectorial operators to derive an elegant and powerful analysis of the numerical errors in the solutions produced by the method. Nevertheless, at the heart of this analysis lies a basic

The discrete convolution sum estimate

Our analysis uses only elementary techniques and is short (though not as short as the original proof of Lemma 1.1). The result proved (Theorem 2.1) is a generalisation of Lemma 1.1 since it includes all values of μ and ν.

Theorem 2.1

As n, the sequence {uv} satisfies |(uv)n|=Onθ1lnnif min{μ,ν}=0,Onθ1in all other cases.

Proof

For each n, the definition of {uv} and Assumption 1.1 imply that |(uv)n||u0vn|+|unv0|+k=1n1|ukvnk|Cnν1+Cnμ1+Ck=1n1kμ1(nk)ν1for some constant C. Now k=1n1kμ1(nk)ν1=k=1

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (6)

  • LubichChristian

    Discretized fractional calculus

    SIAM J Math Anal

    (1986)
  • LubichChristian

    Convolution quadrature and discretized operational calculus. I

    Numer Math

    (1988)
  • LubichChristian

    Convolution quadrature revisited

    BIT

    (2004)
There are more references available in the full text version of this article.

Cited by (0)

The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under grants 12171025 and NSAF-U1930402. The research of Dongling Wang is supported in part by the National Natural Science Foundation of China under grants 11871057 and 91630205.

View full text