Various fundamental formulas and results for integral transforms on a function space have been studied in many papers. However, there are many limitations with regard to obtaining the fundamental formulas and results with respect to integral transforms on the function space, because generalized Brownian motion has the nonzero mean function. Despite recent attempts address this problem, it has yet to be resolved. In this paper, based on the definitions of the modified generalized integral transform and (generalized) convolution product on function space, we establish the fundamental formulas relating the two. The fundamental formulas obtained via the translation theorem are applied in several examples to demonstrate the usefulness of our fundamental approach.

中文翻译：

---

修正广义积分变换的基本公式

许多论文已经研究了函数空间上积分变换的各种基本公式和结果。然而，由于广义布朗运动具有非零均值函数，因此在获得函数空间上积分变换的基本公式和结果方面存在许多限制。尽管最近尝试解决这个问题，但它仍未得到解决。本文根据修正广义积分变换和（广义）卷积乘积在函数空间上的定义，建立了两者的基本公式。通过平移定理获得的基本公式在几个例子中得到应用，以证明我们的基本方法的有用性。