Hermite Functions and Fourier Series

Using normalized Hermite functions, we construct bases in the space of square integrable functions on unit circle (L2(C)) and l2(Z), which are related to each other by means Fourier transform discrete transform. These relations unitary. The construction orthonormal requires use Gramm–Schmidt method. On both spaces, have provided ladder operators with same properties as for one-dimensional quantum oscillator. linear combinations some multiplication- differentiation-like that, when applied periodic preserve periodicity. Finally, constructed riggings L2(C) so that all mentioned continuous.