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A tempered distribution is a continuous linear functional on the Schwartz space. Tempered distributions include functions of at most polynomial growth and admit a Fourier transform by duality.

Translation shifts the argument without changing the shape of a function. On the real line, the displayed convention shifts a graph to the right by $h$. Lebesgue measure invariance preserves every finite Lp norm; its Fourier transform is multiplied by $e^{-ih\xi }$.

If a periodic integrable function has all Fourier coefficients zero, it vanishes almost everywhere. Its Fejér sums vanish, whereas the Fejér kernel is an approximate identity and converges to that function in the integrable norm. Applied to a periodized squared Fourier transform, this detects orthogonality of integer translates.

A wavelet has $q$ vanishing moments when these integrals vanish for $0\le r<q$. For compact support, this is equivalent to a zero of order at least $q$ at the origin of its Fourier transform. Such a wavelet annihilates polynomials of degree less than $q$, and a Taylor polynomial bounds coefficients on smooth regions.

The Wigner distribution of two fields is the Fourier transform of a field's two-point product in its separation variable:Its inverse reconstructs $u(\mathbf{z})\overline{v({\mathbf{z}}^{\prime })}$. For vector fields, specify whether the product is an outer product or a scalar contraction; the latter can encode an average over receiver channels.