Is there any linear map that lowers the number of variables of functions, namely a map that maps a function of several variables to functions of one variable and at the same time the original function can be reconstructed with its image through an explicit form.  

 For example, $$T: C(\mathbb{R}^n)\rightarrow C(\mathbb{R})$$
 $$f(\mathbb {x})\rightarrow F(t)$$
such that
 $$f(\mathbb {x})=\int\_{\mathbb{R}}K(\mathbf{x},t)F(t)dt,\quad \forall f\in C(\mathbb{R}^n)$$
with some kernel $K$?  

It has not to be this specific form. My key problem is to find transforms that lowers the number of variables of functions and store $f(x)$ separately in $F(t)$, which preserves the information, and $K(\mathbf{x},t)$, which keeps its "location".



tranforms that lowers the number of variables of a function
As your question is not completely precise, this might not really an answer but at least it might be an occasion to modify your question in a more precise way.


If you enlarge the point of view from continuous functions to Borel functions, then the problem is easy: all Polish spaces with no isolated points are Borel-isomorphic, so if for example $x=x(t)$ with inverse $t=t(x)$ is a bijection that
preserves Borel sets you can use $F(t)=f(x(t))$ as transformation $T$ (not only linear, but also multiplication and order preserving; it does not preserve the spaces of continuous functions)


Looking only at continuous functions, you can use also a "dual" of the "space filling curve" approach: the Arnold - Kolmogorov theorem about Hilbert 13th problem. There are versions with only one external function (and, as usual, the internal functions are independent from $f$ so that the representation is really due to a continuous embedding of the $n$-dimensional cube in a larger euclidean space in such a way that continuous functions on the $n$-dimensional cube are identified with continuous functions of one real variable, the sum of the coordinates of the embedding). There are also explicitly constructed such embeddings, for example <http://wissrech.ins.uni-bonn.de/research/pub/braun/remonkoe.pdf> and you can easily search for more.


Edit: I have just seen in [Kolmogorov superposition for smooth functions](https://mathoverflow.net/questions/140859/kolmogorov-superposition-for-smooth-functions) that you already know Vitushkin's theorems that kill the possibility of a general representation with differentiable functions. Even recent activity not cited in that thread confirm the "no-go" status beyond continuity, see for example <http://emis.library.cornell.edu/journals/HOA/FPTA/Volume2010/287647.pdf>


