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Same scheme can be applied to functions instead of numbers.
- Consider a function
- Function is transformed to the series of its Fourier coefficients
by the Fourier transform:
The sequence is bounded by the inequality
for any function
- We may complement of the type (1) to the set of all
sequences, with may have unconstrained sum of the squares of their elements.
- Applying then the inverse Fourier transform we define the expansion of the
functional space to the space of distributions .