Distributions

Same scheme can be applied to functions instead of numbers.

1. Consider a function $g \in L_2(0, \pi)$
2. Function $g$ is transformed to the series of its Fourier coefficients $\{a_n\}$
   
   $$g \rightarrow \{a_n\}$$
   
   
   by the Fourier transform:
   
   $$g(x) = \sum _{n=0}^\infty a_n \cos(n x) \quad a_n= {1 \over \pi} \int_0^\pi g(x) \cos(n x) dx$$
   
   

   The sequence $\{a_n\}$ is bounded by the inequality
   

   $$\sum _{n=0}^\infty {a_n^2} <\infty$$ (1)

   
   
   for any function $g \in L_2(0, \pi)$.
3. We may complement $\{a_n\}$ of the type (1) to the set of all sequences, with may have unconstrained sum of the squares of their elements.
4. Applying then the inverse Fourier transform we define the expansion of the functional space $L_2(0, \pi)$ to the space of distributions .