What would the convolution of a time scaled function with a dirac delta function be?

Question

I know the convolution: $x(t) * \delta(t-t_{0}) = x(t-t_{0})$.

But what would the result be if I have a convolution: $x(\frac{t}{T}) * \delta(t-t_{0})$? (where $T \neq$ 0)

Would it be $x(\frac{t}{T} - t_{0})$ or $x(\frac{t - t_{0}}{T})$?

Answer 1

in case of doubt, do not hesitate to write things down. By definition, of the convolution you have: $$ (x\star\delta(.-t_0))(t)=\int x(\frac{t-z}{T})\delta(z-t_0)dz=x(\frac{t-t_0}{T}) $$