I am trying to decompose a function $\phi \in L^2(\partial \Omega)$ where $\Omega$ is some bounded domain into a representation on a basis of dirac delta functions. Here is what I have done. Suppose $$\phi(x) = \int_{\partial \Omega} <\phi, \delta_y> \delta_x dy$$
Then we have $$\int_{\partial \Omega} <\phi, \delta_y> \delta_x dy= \int_{\partial \Omega} \bigg[\int_{\partial \Omega} \phi(w) \delta(w - y) dw \bigg] \delta_x dy$$ $$= \int_{\partial \Omega} \phi(y)\delta(y-x) dy$$ $$= \phi(x)$$
Questions:
- I have two different dirac deltas in my decomposition, that is $\delta_x$ and $\delta_y$, does this mean this decomposition is invalid? Normally when decomposing you take the inner product onto some function multiplied by that function. But here I take the projection on $\delta_x$ and multiply it by $\delta_y$. Is this a problem?
- If I want to check if the basis is orthogonal and I take two functions from it have $<\delta_x, \delta_y> = \int_{\partial \Omega} \delta(w - x)\delta(w-y) dw = \delta(x - y)$. So it's $\infty$ if $x = y$ and zero otherwise which implies it is an orthogonal basis. But is it possible to make it orthonormal?
- Am I completely wrong here as the dirac delta function is not in $L^2$? Does this matter or could this decomposition still represent a function in $L^2$? If not, could it viewed as representing them in an approximate sense?