A problem on polynomial fitting

krete

I encounter a problem on the fitting ability of a special class of multi-variable polynomials. To be specific, I need find whether a special class of multi-variable polynomials, denoted by p(m), where m is the number of variables, can universally and exactly fit all member in another special class of multi-variable polynomials q(n), where n is also the number of variables. Note that p(m) and q(n) have different forms, respectively. Might you please give me some hints or references on this problem? Thank you in advance for you kind help.

Btw, I tried to read some materials on nonlinear functional analysis and algebra. But I could not find a relevant reference. Please note that the general result (e.g., Weierstrass-Stone Theorem) seems to be too general to guide this problem since the form of p(m) and q(n) are special, and the m and n are related.