Condition for convergence in squared norm sense

If, for a periodic signal x(t) with period T, converges, then its Fourier Series converges to it in the squared norm sense. As is expected, this is a far more relaxed constraint than the Dirichlet conditions.

At this point let us define some terms which will be of use to us later in the course:

is called the instantaneous power or energy density of the signal x(t).

If x(t) is periodic with period T, and , x(t) is called a finite power signal, and the value of the integral is called the power of the signal.

(Thus we can say, if a periodic signal has finite power, we are guaranteed of convergence in squared norm of its Fourier Series)

If x(t) is non-periodic, and , x(t) is said to be a finite energy signal, and the value of the integral is called the energy of the signal.

We now discuss another aspect of the convergence of the Fourier series, the Gibb's Phenomenon