  This paper discusses a generalization of spectral representations related to
convex one-homogeneous regularization functionals, e.g. total variation or
$\ell^1$-norms. Those functionals serve as a substitute for a Hilbert space
structure (and the related norm) in classical linear spectral transforms, e.g.
Fourier and wavelet analysis. We discuss three meaningful definitions of
spectral representations by scale space and variational methods and prove that
(nonlinear) eigenfunctions of the regularization functionals are indeed atoms
in the spectral representation. Moreover, we verify further useful properties
related to orthogonality of the decomposition and the Parseval identity.
  The spectral transform is motivated by total variation and further developed
to higher order variants. Moreover, we show that the approach can recover
Fourier analysis as a special case using an appropriate $\ell^1$-type
functional and discuss a coupled sparsity example.
