What is the integral of this summation

Question

What is the integral of $$\int_0^1 e^{-2 \pi jix} \sum_{k=-\infty}^\infty u_ke^{2 \pi kix} $$ where j,k are integers, i imaginary, and $$u_k$$ in complex numbers, known as Fourier coefficients

Answer 1

The functions are orthonormal, and you can pull the constants out of the integral. The answer is $u_{j}$.

This obviously is only valid under some conditions on the convergence of the series of Fourier coefficients.

$$\int_0^1 e^{-2\pi i j x} \sum_k u_k e^{2\pi i k x} dx =\sum_k u_k \int_0^1 e^{-2\pi i j x} e^{2\pi i k x} dx $$ $$=\sum_k u_k\, \left\langle e^{2 \pi i k x},e^{2\pi i j x}\right\rangle=u_j.$$