The other day I got curious about different ways to define inner products for functions.
I started investigating the integral
$$I_w(f,g) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} w(x,y)\cdot f(x)\cdot g(y)dx dy$$
For real valued functions: $$(x,y) \to w(x,y) \in \mathbb R\\x\to f(x) \in \mathbb R\\ y\to g(y)\in \mathbb R$$
Can I prove or disprove that it will define an inner product?
Or maybe better question, what restrictions do we need to put on $w(x,y)$ so that it becomes one?
Own Work: I have experimented a bit with odd and even functions and a gaussian-like $$w(x,y) = sign(x)sign(y)\exp(-\pi (x^2+y^2))$$
This seems to numerically give me $$I(x^{odd},y^{even}) \approx 0\\I(x^{odd},y^{odd})>0\\I(x^{even},y^{even})\approx0$$
But I don't know if that would imply anything. The first 16 monomials scaled by a constant gives the following table:
That entire rows and columns are empty tells me this probably does not span the whole space, but only some subset of functions in $\mathbb R$
If $w \geq 0$ then $I(f,g )$ is an inner product on the space $L^{2}(\mu)$ where $\mu (E)=\int_E w(x,y) dxdy$. [$\mu $ is a measure on the Borel sigma algebra of $\mathbb R^{2}$. Note that $f$ and $g$ are considered equal if $f=g \, a.e. [\mu]$].