I want to calculate the Fourier transform of the following function $$ \frac{1}{T}e^{i\omega_n t}e^{-i\omega_m t^\prime}\Theta(t-t^\prime), $$ where $\omega_n=\frac{2\pi}{T}n$ ($n\in \mathbb{Z}$) and $\Theta$ is the Heaviside step function $$ \Theta(t-t^\prime)= \begin{cases} 1\, ,\, t-t^\prime\geq 0\\ 0\, ,\, t-t^\prime< 0\, . \end{cases} $$ Since it depends on two variables the transformation reads as follows $$ \frac{1}{T^2}\int_0^Tdt\, dt^\prime \, e^{i(\omega_n-\omega_p) t}e^{i(\omega_l-\omega_m) t^\prime}\Theta(t-t^\prime), $$ the Fourier transform only depends on the frequency differences, so I define $\Delta_{np}\equiv\omega_n-\omega_p$ and $\Delta_{lm}\equiv\omega_l-\omega_m$. The convolution of a periodic function is defined as $$ (f*g)(t)=\frac{1}{\sqrt{T}}\int_0^Tdt^\prime f(t^\prime)g(t-t^\prime). $$ Hence, I interpret the two dimensional Fourier transform as a one dimensional Fourier transform of a convolution where $f(t^\prime)=e^{i(\omega_l-\omega_m) t^\prime}$ and $g(t-t^\prime)=\Theta(t-t^\prime)$. With the substitutions $q=t-t^\prime$ ($dq=dt$) and $t=t^\prime+q$, the integrals can be written as the product of Fourier transforms $$ \frac{1}{T^2}\int_0^Tdq\, \Theta(q)e^{i\Delta_{np} q}\int_0^Tdt^\prime\, e^{i(\Delta_{lm}+\Delta_{np}) t^\prime}. $$ With this substitution, the step function can be omitted because we integrate from $0$ to the fundamental period $T$. Now, if one of the frequency differences are non zero, then the integral vanishes and if both are zero then the integral is rendered $1$. Hence, $$ \frac{1}{T^2}\int_0^Tdq\, \Theta(q)e^{i\Delta_{np} q}\int_0^Tdt^\prime\, e^{i(\Delta_{lm}+\Delta_{np}) t^\prime}=\delta_{n,p}\delta_{m,l}. $$ However, if I go back to my original integral and insert $\Delta_{np}=\Delta_{lm}=0$, I get $$ \frac{1}{T^2}\int_0^Tdt\int_0^T dt^\prime \, \Theta(t-t^\prime)=\frac{1}{T^2}\int_0^Tdt\int_0^t dt^\prime=\frac{1}{2}. $$ Now which result is the correct one? I have been calculating all day now to figure a way out of this ambiguity. How do I calculate this Fourier Transform of the convolution??
Thank you in advance