I have to show that the convolution of a function $f \in L^1(\mathbf{R})$ with the harmonic oscillation $\phi_\omega (t) = \exp(2 \pi i t \omega)$ is equal to the Fourier Transform of $f$, $\hat{f}(\omega)$:
$ f * \phi_\omega = \hat{f} (\omega) = \int_{-\infty}^{\infty} f(t) \exp(-2 \pi i t \omega) dt $
However I cannot quite get it...
Attempt 1:
I tried to show it using the definition of the convolution
$ f * g = \int_{-\infty}^{\infty} f(\tau)g(t-\tau) d \tau, $
this gives me (by exchanging the role of $t$ and $\tau$ in the above equation)
$ \int_{-\infty}^{\infty} f(t)\exp(2 \pi i (\tau - t) \omega) dt = \exp(2 \pi i \tau \omega) \int_{-\infty}^{\infty} f(t)\exp(-2 \pi i t \omega) dt = \exp(2 \pi i \tau \omega) \hat{f}(\omega) $
but this is the fourier transform of $f(t+\tau)$, right?
Attempt 2: Using the convolution theorem
$ \mathcal{F}(f * g) = \mathcal{F}(f) \cdot \mathcal{F}(g) $
we have
$ f * g = \mathcal{F}^{-1}\mathcal{F}(f * \phi_\omega) = \mathcal{F}^{-1}(\mathcal{F}(f) \cdot \mathcal{F}(\phi_\omega)) $
where $\mathcal{F}(f) = \hat{f}(\omega)$ and $\mathcal{F}(\phi_w) = \hat{\phi_w}(\omega) = \delta(\omega-w)$?
Then we have
$ f * g = \mathcal{F}^{-1}(\hat{f}(\omega) \delta(\omega-w)) = \int_{-\infty}^{\infty}\hat{f}(\omega) \delta(\omega-w) \exp(2 \pi i t \omega) d\omega $
And since there only is a nonzero contribution for $\omega = w$ this gives
$ f * g = \exp(2 \pi i t w) \hat{f}(\omega) $
I do have the feeling I messed up some variables and constants :(
What am I missing, or doing wrong?