AI 中文总结
本文针对广义分形弦发展分布分形泰勒公式与精确级数表示，将留数项表达为狄拉克δ分布的分数阶导数，并联系分形复维数，推进分形表征计划。
AI 中文摘要
受M. L. Lapidus和M. van Frankenhuisjen关于广义分形弦的分形显式公式工作的启发，本文发展了一个带误差项的分布分形泰勒公式，以及广义分形弦的精确分布分形泰勒级数表示。上述作者发展的显式公式提供了一种在适当增长假设（称为languidity假设）下，根据底层分形复维数表达广义分形弦$\eta$的方法。这涉及一个以广义分形弦的复维数为索引的和，对所有$\zeta_{\eta}$的留数求和，并乘以一个缓增或Schwartz测试函数$\phi$的Mellin变换，记为$\widetilde{\phi}$。在languidity假设下，存在误差项。在更强的假设下，即强languidity情形，没有误差项，所得分形显式公式称为精确的。在本文中，我们将分数阶分布显式公式中的留数项表示为狄拉克$\delta$分布的分数$(-\omega)^{\mathrm{th}}$阶分布导数，记为$Y_{\omega}$，应用于形式为$\phi(x) \ln^{k-1}(x)$的修正测试函数，其中$k$从1到$\zeta_{\eta}$的极点$\omega$的重数。该和中的系数用几何zeta函数$\zeta_{\eta}$在$\omega$处Laurent展开主部的系数乘以在$\omega$处评估的伽马函数来表示。本文获得的结果有助于更广泛的计划，即通过涉及底层分形复维数的分形泰勒级数展开来表征分形。
英文摘要
Motivated by the work of M. L. Lapidus and M. van Frankenhuisjen on fractal explicit formulas for generalized fractal strings, we develop in this paper a distributional fractal Taylor's formula with error term and an exact distributional fractal Taylor series representation for generalized fractal strings. The explicit formulas that the aforementioned authors developed provide a way of expressing a generalized fractal string $η$ in terms of the underlying fractal complex dimensions, under suitable growth assumptions on its geometric zeta function, $ζ_η$, known as the languidity assumptions. This involves a sum indexed by the complex dimensions of the generalized fractal string, summing over all residues of $ζ_η$, multiplied by the Mellin transform of a tempered or Schwartz test function $ϕ$, denoted by $\widetildeϕ$. Under languidity assumptions, there is an error term present. Under stronger assumptions, that is, in the case of strong languidity, no error term is present and the resulting fractal explicit formula is said to be exact. In this paper, we express the residue term in the fractional distributional explicit formulas as the fractional $(-ω)^{\mathrm{th}}$-ordered distributional derivative of the Dirac $δ$ distribution, denoted by $Y_ω$, applied to modified test functions of the form $ϕ(x) \ln^{k-1}(x)$, where $k$ runs from $1$ to the multiplicity of the pole $ω$ of $ζ_η$. The coefficients in this sum are written in terms of the coefficients in the principal part of the Laurent expansion at $ω$ of the geometric zeta function $ζ_η$, multiplied by the gamma function evaluated at $ω$. The results obtained in this paper contribute to the broader program of characterizing fractals in terms of fractal Taylor series expansions involving their underlying fractal complex dimensions.
