In the theory of distributions, which gives rigorous meaning to statements like $$\int_{-\infty}^{+\infty} f(x) \delta(x) \,dx = f(0),$$ is there also a rigorous notion of, what the physicists would call the radial delta function, with the property $$\int_{0}^{\infty} f(x) \delta_r(x) dx = f(0)?$$ If so, what are the properties of $\delta_r(x)$? What are the suitable test functions $f(x)$?
I would also appreciate some references for further reading.