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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. Note that every function on a compact space has compact support since every closed subset of a compact space is compact.

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 functionalIn linear algebra, a branch of mathematics, a linear functional is a linear function from a vector space to its field of scalars. Specifically, if V is a vector space over a field k then a linear functional is a linear function from V to k''. The set of a to F) is 0 for such functions, we can say that the support of δ is {0} only. Since measureMeasure can mean: To perform a measurement. In mathematics, a measure is a way to assign non-negative real numbers to subsets of a given set, in order to "measure their sizes or probabilities". See measure (mathematics) for a treatment of the concept.s 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 transformThe Fourier transform named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes"). of the Heaviside step functionThe Heaviside step function named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative inputs and one elsewhere: : The function is used in the mathematics of signal processing to represent a signal that switches on at 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' principleHuygens principle (named for Dutch physicist Christiaan Huygens) is a method of analysis applied to problems of wave propagation. It recognizes that each point of an advancing wave front is in fact the center of a fresh disturbance and the source of a new in terms of mathematical analysisAnalysis is that branch of mathematics which deals with the real numbers and complex numbers and their functions. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in g. 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).