Suppose I have a function $h(x_1,x_2)$ which is a polynomial function of degree $2n$ with all even powers; $$h(x_1,x_2) = \sum_{i,j=0}^{n} c_{ij} x_1^i x_2^j$$ where $c_{ij}, x \in \mathbb{R}$. Is there a simple integral transform that I can apply on $h(x_1,x_2)$ such that I can extract the terms with the largest degree, i.e. is there a simple $F(y_1,y_2; x_1,x_2)$ such that $$ g(y_1,y_2) = c_{nn} y_1^n y_2^n + c_{2n,0} y_1^{2n} +c_{0,2n} y_2^{2n} +\dots = \int_{-\infty}^{\infty}dx F(y_1,y_2; x_1,x_2) h(x_1,x_2)?$$ where $\dots$ denote all terms of degree $2n$.
As a simple example, let's set $x_2 =1, x_1=x$ and consider $h(x) = 3x^4 + 7x^2 +2$, does a simple $F(x,y)$ exist such that $g(y) = 3y^4$?
Thanks for any help, my intuition about integral transforms is really bad, I tried some simple test functions for $F(y,x)$ but it didn't work out.