实二次型的正定性：源于卷积算子变步长近似的研究
Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators
AI总结：
本文提出一种新的分析工具，证明在易验证的代数条件下变步长离散卷积核对应的二次型正定，为非局部算子非均匀时间步进格式的稳定性分析提供了首个一般性结果。
摘要：
具有卷积结构的实二次型的正定性在非局部算子时间步进格式的稳定性分析中起重要作用。本文提出了一种新的分析工具，用于处理由卷积算子变步长近似产生的离散卷积核。更确切地说，对于与变步长时间离散化相关的一类离散卷积核，我们证明了在易于验证的代数条件下，其对应的二次型是正定的。证明基于一种基本构造策略，利用了离散正交卷积核和互补卷积核的性质。据我们所知，这是关于变步长离散卷积核正定性简单代数条件的首个一般性结果。利用统一理论，可以直观地获得一些简单非均匀时间步进格式的稳定性。
英文摘要：
The positive definiteness of real quadratic forms with convolution structures plays an important role in stability analysis for time-stepping schemes for nonlocal operators.In this work, we present a novel analysis tool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators. More precisely, for a class of discrete convolution kernels relevant to variable-step time discretizations,we show that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Our proof is based on an elementary constructing strategy using the properties of discrete orthogonal convolution kernels and complementary convolution kernels. To the best of our knowledge, this is the first general result on simple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using the unified theory, the stability for some simple non-uniform time-stepping schemes can be obtained in a straightforward way.
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