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Support

In case X is a topological space — for example the real number line — and f is continuous, the support of f is defined in a modified way, as the smallest closed subset of X, outside of which f(x) is zero, i.e., the support is the closure of the set described above.

Functions with compact support on X are those with support that is compact. They are examples of functions that vanish at infinity. In good cases, functions with compact support are dense in the functions that vanish at infinity; but this requires some technical work to justify in a given example.

It is possible also to talk about the support of a distribution, such as the Dirac delta function δ(x) on the real line. In that example, we can consider test functions F, which are smooth functions with support not including the point 0. Since δ(F) (the distribution δ applied as linear functional to F) is 0 for such functions, we can say that the support of δ is {0} only. Since measures on the real line are special cases of distributions, we can also speak of the support of a measure in the same way.

In Fourier analysis in particular, it is interesting to study the singular support of a distribution. This has the intuitive interpretation as the set of points at which a distribution fails to be a function. For example, the Fourier transform of the Heaviside function can up to constant factors be considered to be 1/x (a function) except at x = 0. While this is clearly a special point, it is more accurate to say that the transform qua distribution has singular support {0}: it cannot accurately be expressed as a function in relation to test functions with support including 0.

For distributions in several variables, singular supports allow one to define wave fronts and understand Huygens' principle in terms of mathematical analysis. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails - essentially because the singular supports of two distributions to multiply should be disjoint).
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