We study eigenfunction decomposition for periodic 1-dimensional Schroedinger with real singular finite-gap potentials. Many ''real'' inverse spectral data for periodic finite-gap operators(consisting of Riemann Surface with marked ''infinite point'', local parameter and divisors of poles) lead to operators with real but singular coefficients. These operators cannot be considered as self-adjoint in the ordinary (positive) Hilbert spaces of functions of x. In particular, it is true for the special case of Lame' operators with elliptic potential n(n+1)P(x) where eigenfunctions were found in XIX Century by Hermit. However, such Baker-Akhiezer (BA) functions present, right analog of the Discrete and Continuous Fourier Bases on Riemann Surfaces. It turns out that these operators for the nonzero genus are symmetric in the indefinite inner product, described in this work. The analog of Continuous Fourier Transform is an isometry in this inner product. Its image in the space of functions of the real variable x is described. The results presented have been obtained in collaboration with S.P. Novikov.