Renyi维数与高斯滤波
Renyi Dimension and Gaussian Filtering
摘要：
考虑欧氏d-空间上有限Borel测度μ的Renyi维数理论中的分拆函数S（ε）.这个划分函数S（ε）是应用于将d-空间划分为宽度为ε的d-立方体的测度的q次幂之和。我们进一步Guerin的调查之间的关系，这个分拆函数和勒贝格Lp范数（Lq范数）的卷积μ对一个近似的身份高斯。我们证明了一个关于分拆函数的Lipschitz型估计。这种对分拆函数的限制导致了关于Renyi维数计算的结果。它还表明，分配函数是O-正则变化。我们发现的情况下，可以或不能取代的分拆函数的离散版本。我们发现，最小二乘最佳拟合线性逼近的斜率的分割函数并不总是用来计算上下Renyi维数。
英文摘要：
Consider the partition function S(ε) associated in theory of Renyi dimension to a finite Borel measure μon Euclidean d-space. This partion function S(ε) is the sum of the q-th powers of the measure applied to a partition of d-space into d-cubes of width ε. We further Guerin's investigation of the relation between this partition function and the Lebesgue Lp norm (Lq norm) of the convolution of μagainst an approximate identity of Gaussians. We prove a Lipschitz-type esimate on the partition function. This bound on the partition function leads to results regarding the computation of Renyi dimension. It also shows that the partion function is of O-regular variation. We find situtations where one can or cannot replace the partition function by a discrete version. We discover that the slopes of the least-square best fit linear approximations to the partion function cannot always be used to calculate upper and lower Renyi dimension.
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