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Up to now I've never found a rigorous or a formal definition of what it means for a signal to be sparse other than it means that it has a relatively low number of non-zero entries or that the cardinality of its support is small. The only thing I've found close to a somewhat formal definition of a sparse signal is that a sparse signal is $K$-sparse in a transform basis $\Psi$ if there are exactly K nonzero elements. That definition is found in S. Brunton's book Data Driven Science & Engineering, p. 97.

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This page gives the following definition:

A signal is said to be sparse if it can be represented in a basis or frame (e.g Fourier, Wavelets, Curvelets, etc.) in which the curve obtained by plotting the obtained coefficients, sorted by their decreasing absolute values, exhibits a polynomial decay.

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    $\begingroup$ This definition of sparseness is trivially true for all signals unless the basis is constrained in some way. So I'm not sure how useful it is. $\endgroup$
    – Jazzmaniac
    Feb 26, 2023 at 1:28

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