扩张矩阵的一些性质

Some properties of expansive matrix

研究了相关于扩张矩阵A的扩张球和拟范数的一些性质.首先通过具体实例及欧氏范数关于A的上下界估计指出扩张矩阵与经典球及欧氏范数匹配不佳,但欧氏范数相关于A仍能保持全局伸缩性.其次研究了相适应于扩张矩阵的扩张球和拟范数关于伸缩性、凸性、可积性、微分估计及傅里叶变换的一些性质.最后通过欧氏范数与相关于扩张矩阵的拟范数的不等式估计证明了相关于拟范数的两类施瓦茨函数空间和相关于欧氏范数的经典施瓦茨函数空间都是等价的.

In this paper, we study some properties of dilated balls and quasi-norms associated with expansive matrix A. Firstly, we point out that expansive matrix can not match very well with classical balls and Euclidean norm via some specific examples and the bounded estimates of Euclidean norm associated with A, but the Euclidean norm still maintains the global flexibility associated with A. Secondly, we study some properties about dilated balls and quasi-norm associated with A in terms of flexibility, convexity, integrability, differential estimate and Fourier transform. Finally, we prove that two kinds of Schwartz function spaces associated with the corresponding quasi-norms are equivalent with the classical Schwartz function space.