Abstract In this paper, based on the “fuzzy” calculus covering the continuous range of operations between two couples of arithmetic operations (+, –) and (×, :), a new form of the fractional integral is proposed occupying an intermediate position between the integral and derivative of the first order. This new form of the fractional integral satisfies the C1 criterion according to the Ross classification. The new calculus is tightly related to the continuous values of the continuous spin S = 1 and can generalize the expression for the fractional values of the shifting discrete index. This calculus can be interpreted as the appearance of the hidden states corresponding to unobservable values of S = 1. Many well-known formulas can be generalized and receive a new extended interpretation. In particular, one can factorize any rectangle matrix and receive the “perfect” filtering formula that allows transforming any (deterministic or random) function to another arbitrary function and vice versa. This transformation can find unexpected applications in data transmission, cryptography and calibration of different gadgets and devices. One can also receive the hybrid (”centaur”) formula for the Fourier (F-) transformation unifying both expressions for the direct and inverse F-transformations in one mathematical unit. The generalized Dirichlet formula, which is obtained in the frame of the new calculus to allow selecting the desired resonance frequencies, will be useful in discrete signals processing, too. The basic formulas are tested numerically on mimic data.

中文翻译：

---

“模糊”微积分：量子力学与离散分数算符之间的联系

摘要 在本文中，基于覆盖两对算术运算 (+, –) 和 (×, :) 之间连续运算范围的“模糊”演算，提出了一种新形式的分数积分，它占据了两者之间的中间位置。一阶的积分和导数。根据 Ross 分类，这种新形式的分数积分满足 C1 准则。新的微积分与连续自旋 S = 1 的连续值密切相关，并且可以推广移动离散索引的分数值的表达式。该演算可以解释为对应于 S = 1 的不可观察值的隐藏状态的出现。许多众所周知的公式可以推广并获得新的扩展解释。特别是，人们可以分解任何矩形矩阵并接收“完美”过滤公式，该公式允许将任何（确定性或随机）函数转换为另一个任意函数，反之亦然。这种转换可以在不同小工具和设备的数据传输、加密和校准中找到意想不到的应用。还可以接收傅立叶 (F-) 变换的混合（“centaur”）公式，将直接和逆 F 变换的两种表达式统一在一个数学单位中。在新微积分的框架中获得的广义狄利克雷公式允许选择所需的共振频率，也将在离散信号处理中有用。基本公式在模拟数据上进行了数值测试。