In this paper, we explore the concept of dimension preserving approximation of continuous multivariate functions defined on the domain ${[0,1]}^q(=[0,1]\times \cdots \times [0,1]$ (q-times) where $q$ is a natural number). We establish a few well-known multivariate constrained approximation results in terms of dimension preserving approximants. In particular, we indicate the construction of multivariate dimension preserving approximants using the concept of $\alpha$-fractal interpolation functions. We also prove the existence of one-sided approximation of multivariate function using fractal functions. Moreover, we provide an upper bound for the fractal dimension of the graph of the $\alpha$-fractal function. Further, we study the approximation aspects of $\alpha$-fractal functions and establish the existence of the Schauder basis consisting of multivariate fractal functions for the space of all real valued continuous functions defined on ${[0,1]}^q$ and prove the existence of multivariate fractal polynomials for the approximation.

在本文中，我们探讨了定义在域上的连续多元函数的维数近似的概念${[0,1]}^q(=[0,1]\times \cdots \times [0,1]$( q次) 其中$q$是自然数）。我们建立了一些著名的多变量约束逼近结果，就保维逼近而言。特别是，我们使用以下概念指示了多元维数保持近似值的构造$\alpha$-分形插值函数。我们还使用分形函数证明了多元函数单边逼近的存在。此外，我们提供了图的分形维数的上限$\alpha$-分形函数。此外，我们研究的近似方面$\alpha$-分形函数，并建立由多元分形函数组成的 Schauder 基的存在，用于定义的所有实值连续函数的空间${[0,1]}^q$并证明近似的多元分形多项式的存在。