Relationship between time average and cross spectrum I have two signals $x(t)$ and $y(t)$ which I can sample at arbitrary $\Delta t$ and $N$.
I am interested to the signals product time average $\langle x(t)y(t)\rangle_t$.
In particular I want  to understand when the time average is zero in term of the discrete Fourier coefficients,  that is when the signals 
can be considered "orthogonal".
For example, if $X(k)$ and $Y(k)$ are the discrete Fourier coefficients, thus the time average is well approximated by:
$$
\langle x(t)y(t)\rangle_t\approx \frac{1}{N^2}  \sum_{k=0}^{N-1}X(k)Y^*(k) 
$$
Thus vanishing cross spectrum represents a sufficient condition in order to have $\langle x(t)y(t)\rangle_t= 0$.
My question is: 


when this also a necessary condition?



Thank you very much
I will infer from the "discrete Fourier coefficients" in your question that we are considering periodic signals, otherwise we should be considering a continuous spectrum (the Continuous-Time Fourier Transform). I will also be using the definition 
$$\left&lt; x(t), y(t) \right&gt;_t = \frac{1}{T}\int_{\alpha}^{\alpha +T} x^*(t) y(t) dt$$ where the superscript $*$ denotes complex conjugation, $T$ is the shortest period that both $x(t)$ and $y(t)$ share, and $\alpha$ is an arbitrary real number. If these definitions are not correct, please alter your question and provide the correct ones.

We know that we can decompose $x(t)$ and $y(t)$ using a Fourier series expansion
$$ x(t) = \sum_{n=-\infty}^{\infty} c_{x,n} \exp\left(\frac{2 \pi n t}{T}\right)$$
and
$$ y(t) = \sum_{n=-\infty}^{\infty} c_{y,n} \exp\left(\frac{2 \pi n t}{T}\right).$$

We know from Parseval's Theorem that 
$$ \sum_{n=-\infty}^{\infty} c^*_{x, n} c_{y, n} = \int_{-\frac{T}{2}}^{\frac{T}{2}} x^*(t) y(t) dt.$$ 
Notice the equality. Also notice that the right hand side is merely the product of $T$ and $\left&lt; x(t), y(t) \right&gt;_t$. This means that whenever the left hand side is zero, so is $\left&lt; x(t), y(t) \right&gt;_t$ and whenever $\left&lt; x(t), y(t) \right&gt;_t$ is zero so is the left hand side. I believe this resolves your question. Let me know if you have objections, and I will try to address them.