Spectrum degeneracy and impact on extrapolation and sampling for functions on branching lines
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The paper studies functions defined on continuous branching lines connected into a system. A notion of spectrum degeneracy for these functions is introduced. This degeneracy is based on the properties of the Fourier transforms for processes representing functions on the branches that are deemed to be extended onto the real axis. This spectrum degeneracy ensures some opportunities for extrapolation and sampling. The topology of the system is taken into account via a restriction that these processes coincides on certain parts of real axis. It is shown that processes with this spectrum degeneracy are everywhere dense in the set of processes equivalent to functions on the branching lines. Some applications to extrapolation and sampling are considered.
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