This study takes a broad approach to the fractal geometry problem and proposes an intrinsic definition of the general box dimensions and the general Hausdorff and packing dimensions by taking into account sums of the type for some prescribed functions , and for all positive real . Our primary aim is to conduct a more comprehensive exploration of the fractal dimensions exhibited by graphs resulting from the products and sums of two continuous functions. This article is entirely devoted to the investigation of this specific problem domain. In the course of our research, we furnish explicit formulas for computing both the general upper box dimension and the general lower box dimension associated with the graphs formed by the products and sums of two continuous functions. Additionally, we shed light on the decomposition of continuous functions into the sum of two continuous functions, employing the framework of the general lower and upper box dimensions. Furthermore, we establish an upper bound for the general upper box dimension applicable to each element within a finite ring of polynomials derived from continuous functions defined over the real number field . Our inquiry also extends to the examination of general Hausdorff and packing dimensions for the graphs generated through both the summation and multiplication of two continuous functions.

中文翻译：

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连续函数的乘积和和图的一般分形维数及其分解

本研究采用广泛的方法来解决分形几何问题，并通过考虑某些规定函数以及所有正实数的类型和，提出了一般盒维数以及一般 Hausdorff 和堆积维数的内在定义。我们的主要目标是对两个连续函数的乘积和和所产生的图形所展示的分形维数进行更全面的探索。本文完全致力于对这个特定问题领域的调查。在我们的研究过程中，我们提供了明确的公式，用于计算与两个连续函数的乘积和和形成的图相关的一般上框尺寸和一般下框尺寸。此外，我们利用一般下框和上框维度的框架，阐明了将连续函数分解为两个连续函数的和。此外，我们为适用于从实数域上定义的连续函数导出的有限多项式环内的每个元素的一般上盒维数建立了上限。我们的探究还扩展到通过两个连续函数的求和和乘法生成的图形的一般豪斯多夫和包装维数的检查。