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.