AI 中文总结
本文精确刻画了固定数字块出现次数下调和和的性质，通过语言分解与随机基数展开建立谱理论，并给出介于欧拉-麦克劳林与泰勒展开之间的块定向插值公式。
AI 中文摘要
我们精确确定了调和和（对具有给定数字块出现次数的整数求和）如何依赖于该出现次数。语言分解与随机基数展开相结合，将这些调和和与一个算子的迭代联系起来，该算子的特征向量是勒贝格测度和奇异测度的分布导数。对偶图景由适当哈代空间中的特征多项式里兹基给出。由此得到的模态展开自然解释为一种块定向欧拉-麦克劳林公式，在欧拉-麦克劳林展开与泰勒展开之间进行插值。
英文摘要
We determine exactly how harmonic sums, taken over integers with a given number of occurrences of a fixed block of digits, depend on that number of occurrences. Language factorizations, combined with stochastic radix expansions, relate these harmonic sums to the iteration of an operator whose eigenvectors are the Lebesgue measure and distributional derivatives of singular measures. The dual picture is given by a Riesz basis of eigenpolynomials in a suitable Hardy space. The resulting modal expansion admits a natural interpretation as a block-directed Euler-Maclaurin formula interpolating between Euler-Maclaurin and Taylor expansions.
Comments100 pages
