Suppose functions $u_{1}(x)$,..$u_{K}(x)$ are a basis of $H[0,1]$( some space of real-valued functions). Define Discrete Fourier transform $ U_{l}(x)=\sum_{j=1}^{K}u_{j}(x)\exp(2\pi i lj/K) $
and suppose functions $U_{1}(x),\ldots,U_{K}(x)$ are orthogonal in $L_{2}$ norm.
Do functions $U_{1}(x),\ldots,U_{K}(x)$ construct a basis of $H[0,1]$
Is it possible to construct a real valued orthogonal basis using $U_{1}(x),\ldots,U_{K}(x)$
Is there a general theory for such cases?