a tempered distribution
A tempered distribution is a distribution that is tame enough at infinity to have a Fourier transform. Ordinary distributions are allowed to grow wildly far out, and that wildness is exactly what would wreck a transform. By restricting to distributions whose growth is at worst polynomial — bounded by some power of x — we get a family on which the Fourier transform works beautifully. The word 'tempered' means moderated, kept from growing too fast.
The precise definition pairs the larger Schwartz probes against the smaller distributions: a tempered distribution is a continuous linear functional on the Schwartz space S, rather than on the compactly supported test functions. Because Schwartz probes already decay faster than any power, they can absorb a distribution that grows only polynomially, so the pairing stays finite. The space of tempered distributions is written S-prime. It contains the delta, every polynomial, every bounded function, e^(i k x) for real k, and the principal value of 1/x — but it excludes things that blow up exponentially, like e^(x^2), which grows too fast for any Schwartz probe to tame.
Tempered distributions matter because they are the exact home of the Fourier transform of a distribution. Everything you want to transform in PDE theory — point sources, plane waves, polynomials, slowly growing data — is tempered, and the transform maps S-prime onto itself. This is what lets transform methods, dispersion relations, and frequency-space arguments be carried out rigorously rather than formally.
The delta, the constant 1, the Heaviside step, and e^(i k x) are all tempered, so each has a Fourier transform. The Gaussian e^(-x^2) is tempered too. But e^(x^2) is not tempered: it outgrows every Schwartz probe, so the defining pairing diverges and it has no Fourier transform in this theory.
Polynomial growth is allowed; exponential blow-up is not — that boundary is exactly what the Fourier transform needs.
Being tempered is a condition on behaviour at infinity, not on smoothness. The delta is as singular as can be yet is perfectly tempered, while the smooth function e^(x^2) fails — singularity is irrelevant here, only growth.