论文标题
$φ$ $ - 具有关键粗糙度的经典分形功能的概率方法
A probabilistic approach to the $Φ$-variation of classical fractal functions with critical roughness
论文作者
论文摘要
我们认为Weierstraß和Takagi-van der Waerden的功能具有关键的粗糙度。在这种情况下，这些功能消失了$ p^{\ text {th}} $变化的所有$ p> 1 $，但也无处可区分，因此也不是有限的变化。我们通过表明这些功能具有有限，非零和线性wiener-young $φ$ - 沿$ b $ - adic分区的顺序来解决这个明显的难题，其中$φ（x）= x/\ sqrt { - \ sqrt { - \ log x} $。对于Weierstraß功能，我们的证明基于Martingale Central Limit定理（CLT）。对于Takagi-van der der waerden函数，如果某个参数$ b $奇怪，我们将CLT用于马尔可夫链，而$ b $的标准CLT则使用。
We consider Weierstraß and Takagi-van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing $p^{\text{th}}$ variation for all $p>1$ but are also nowhere differentiable and hence not of bounded variation either. We resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener--Young $Φ$-variation along the sequence of $b$-adic partitions, where $Φ(x)=x/\sqrt{-\log x}$. For the Weierstraß functions, our proof is based on the martingale central limit theorem (CLT). For the Takagi--van der Waerden functions, we use the CLT for Markov chains if a certain parameter $b$ is odd, and the standard CLT for $b$ even.
