We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable.
We first give an efficient algorithm for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of $\F_2^n$ (equivalently, for testing whether $f$ is a junta over a small number of parities). We next give an efficient algorithm for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients). In both cases we also prove lower bounds showing that any testing algorithm --- even an adaptive one --- must have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Finally, we give an ``implicit learning'' algorithm that lets us test \emph{any} sub-property of Fourier concision.
Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from~\cite{FGK+:06}.
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