The vector space $P( \infty )$ can be extended to a Hilbert space over a domain $(a,b)$ by including all square integrable convergent sequences of sums of polynomials. Now if the domain is $(-1,1)$, then the space also contains discontinuous functions such as the step function or non differentiable functions such as the modulus function. How can these functions be written as sum of polynomials? Any sum of polynomials must be infinitely differentiable and continuous. I know that an infinite sum of polynomials might converge to such functions, simliar to the Fourier Transform of a step function.
Edit - The step function can be written as a sum of differentiable and continuous sine functions i.e the Fourier Transform. So I need some intuition about infinite sums of continuous functions converging to discontinuous functions.
If a function $f$ is the point-wise limit of polynomials $p_1,p_2,...$ then $f=p_1+(p_2-p_1)+(p_3-p_2)+...$. If $f$ is not continuous we certainly don't have uniform convergence here. Step functions are point-wise infinite sums of polynomials because they are point-wise limits of sequences of polynomials. I am not sure this is what you wanted. but to get step functions from polynomials you have to take point-wise infinite sums.