We study signals that are sparse in graph spectral domain and develop explicit algorithms to reconstruct the support set as well as partial components from samples on few vertices of the graph. The number of required samples is independent of the total size of the graph and takes only local properties of the graph into account. Our results rely on an operator based framework for subspace methods and become effective when the spectral eigenfunctions are zero-free or linear independent on small sets of the vertices. The latter has recently been adressed using algebraic methods by the first author.

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