完全单调 - 斯蒂尔杰斯 - 伯恩斯坦层次结构中的确定性见证
Determinacy Witnesses in the Completely Monotone--Stieltjes--Bernstein Hierarchy
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中文总结 AI 辅助
研究完全单调 - 斯蒂尔杰斯 - 伯恩斯坦层次结构中的成员问题，通过见证演算，结合具体字典与传输规则，用于多种函数及序列。应用于多个问题，如双伽马桥恒等式等，工作流程统一，每个排除都有精确证书。
中文摘要 AI 辅助
完全单调 - 斯蒂尔杰斯 - 伯恩斯坦层次结构中的成员问题在转换为确定性表示后通常会变得明确。我们将此转换组织为见证演算。对于这里使用的拉普拉斯、斯蒂尔杰斯和豪斯多夫变换，W f 表示唯一可允许的有符号表示数据；一个证书要么证明其正性，要么展示一个障碍。一个针对完全单调、对数完全单调、伯恩斯坦、完全伯恩斯坦和斯蒂尔杰斯函数以及豪斯多夫矩序列的具体字典，与乘积、移位、采样和伯恩斯坦增量的传输规则相结合。应用是这种共同计算的结果。双伽马桥的三重积恒等式控制伽马比率的普通伯恩斯坦和对数完全单调性范围，并给出萨博问题中的严格幂截断。对于整数伽马商，逆拉普拉斯密度是一个阻尼雅可比多项式，因此经典零几何决定了广泛的参数区域。边界符号区分伯恩斯坦和完全伯恩斯坦行为，并将完全单调函数与斯蒂尔杰斯类分开。在离散方面，有符号原子和有限差分解决插值和系数问题。拉马努金积分和分数沃尔泰拉符号提供了正识别示例。在所有应用中，工作流程是相同的：规范化类问题，计算或传输其见证，并解决一个明确的符号问题。每个排除都伴有一个精确的证书。
英文摘要
Membership problems in the completely monotone--Stieltjes--Bernstein hierarchy often become explicit after passage to a determinate representation. We organize that passage as a witness calculus. For the Laplace, Stieltjes, and Hausdorff transforms used here, W f denotes the unique admissible signed representing datum; a certificate either proves its positivity or exhibits an obstruction. A concrete dictionary for completely monotone, logarithmically completely monotone, Bernstein, complete Bernstein, and Stieltjes functions and for Hausdorff moment sequences is combined with transport rules for products, shifts, sampling, and Bernstein increments. The applications are consequences of this common calculation. A triple-product identity for a digamma bridge controls ordinary Bernstein and logarithmic complete monotonicity ranges for gamma ratios and gives the strict power cutoff in Szabo's problem. For integer gamma quotients, the inverse Laplace density is a damped Jacobi polynomial, so classical zero geometry decides broad parameter regions. Boundary signs distinguish Bernstein from complete Bernstein behavior and separate completely monotone functions from the Stieltjes class. On the discrete side, signed atoms and finite differences settle interpolation and coefficient questions. The Ramanujan integral and a fractional Volterra symbol provide positive recognition examples. Across the applications, the workflow is the same: normalize the class question, compute or transport its witness, and decide one explicit sign problem. Every exclusion is accompanied by an exact certificate.
