Pointwise Convergence

Pointwise convergence implies the series converges to the original function at any point, i.e: the Fourier Series representation of a signal x(t) is said to converge pointwise to the signal x(t) if:

i.e to say  .

Convergence in squared norm

The Fourier Series representation is said to converge in the sense of squared norm to the signal x(t) if

Pointwise convergence implies convergence in squared norm. As convergence in squared norm is a more relaxed condition than pointwise convergence, convergence in the squared norm sense covers a much larger domain of signals than pointwise convergence.

Finally, we now move on to the conditions for these forms of convergence.