Short communicationA sharp discrete convolution sum estimate☆
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 and .
Theorem 2.1 As , the sequence satisfies
Proof For each , the definition of and Assumption 1.1 imply that for some constant . Now
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.
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(2004)
Cited by (0)
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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.