Harmonic oscillation and Fourier I am back to college after 6 years of professional life and I need help with the following problem


  [...]
  The standard form of the function is $S_{E}(t) = e^{j2\pi f_k t}$, whose derivation results from the convolution theorem:
  $$
h(t)\star e^{j2\pi f_k t} = \int_{-\infty}^{\infty} h(\tau)e^{j2\pi f_k (t-\tau)}d\tau = e^{j2\pi f_k t} \int_{-\infty}^{\infty} h(\tau)e^{-j2\pi f_k \tau}d\tau = e^{j2\pi f_k t} \cdot H(f_k)
$$
  According to the equation above, the transfer function in the time domain can be determined when the channel is excited with an Eigenfunction $s_E(t)=e^{j2\pi f_k t} $. Is this relationship also valid for the harmonic oscillation $s_O(t)= \cos(2\pi f_kt)$? Please provide your derivation (Fourier correspondence table)."


I need some guidance for this problem or some hint for textbooks.
When you express the cosine as $\cos(2\pi f_kt)=\frac{1}{2}(\exp(j2\pi f_c t)+\exp(-j2\pi f_ct)$ and insert it into the convolution integral you get:

$$
h(t)*\cos(2\pi f_ct) = \frac{1}{2}(\exp(j2\pi f_ct)H(f_c)+\exp(-j2\pi f_ct)H(-f_c))
$$

In case your $h(t)$ is real-valued, you have $H(f_c)=H(-f_c)^*=|H(f_c)|\exp(j\phi)$, and you get
$$h(t)*\cos(2\pi f_ct)=\cos(2\pi f_ct+\phi)|H(f_c)|.$$

In case you have a complex impulse response, this does not hold.