### Condition 1: **The Weak Dirichlet Condition**

For the Fourier Series to exist, the Fourier coefficients must be finite. The Weak Dirichlet Condition guarantees this existence. It essentially says that the integral of the absolute value of the signal must be finite. The limits of integration are different for the Fourier Series case than for the Fourier Transform case. This is a direct result of the differing definitions of the two.

#### Proof

The Fourier Series exists (the coefficients are finite) if

**
Weak Dirichlet Condition for the Fourier Series
**

### Note:

*fails*the above condition.

**The Weak Dirichlet Condition for the Fourier Transform**

#### Condition 2

The Fourier Transform exists if