Existence of a class of vector-valued and multivariate α-fractal functions on a hype-rectangle of $\mathbb{R}^{q}$

In this article, we first prove the existence of a class of vector-valued and multivariate α-fractal functions on a hype-rectangle of $\mathbb{R}^{q}$. After that, we show that the set of functions with any possible fixed dimension in the continuous function space is dense with respect to uniform norm. Building on this, we also study dimension preserving approximation in this general setting. We then introduce and study some constrained approximation aspects based on dimension-preserving notion. Further, we prove some approximation theoretic results (such as Weierstrass approximation theorem) of the constructed vector-valued and multivariate α-fractal functions on a hype-rectangle of $\mathbb{R}^{q}$. In particular, we show the existence of a dense subset of α-fractal functions in the continuous functions space.