# DFT of basis functions

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.

1. Do functions $U_{1}(x),\ldots,U_{K}(x)$ construct a basis of $H[0,1]$

2. Is it possible to construct a real valued orthogonal basis using $U_{1}(x),\ldots,U_{K}(x)$

3. Is there a general theory for such cases?

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1. By definition, a basis of a vector space is a certain collection of elements of the space. Since the functions $U_1,\dots,U_K$ are not elements of $H[0,1]$, then do not form a basis of $H[0,1]$.
2. Possible, but the only way I can think of is to invert the DFT (returning to $u_j$) and use the Gram-Schmidt on $u_j$.