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''This page deals with mathematical distributions. For other meanings of distribution, see
distribution (disambiguation). This article is '''not''' about
probability distributions.''
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In
mathematical analysis, '''distributions''' (also known as '''generalized functions''') are objects which generalize
functions and
probability distributions. They extend the concept of
derivative to all
continuous functions and beyond and are used to formulate generalized solutions of
partial differential equations. They are important in
physics and
engineering where many non-continuous problems naturally lead to differential equations whose solutions are distributions, such as the
Dirac delta distribution.
"Generalized functions" were introduced by
Sergei Sobolev in 1935. They were independently discovered in late 1940s by
Laurent Schwartz, who developed a comprehensive theory of distributions.
Sometimes, people talk of "
probability distribution" when they just mean "probability
measure", especially if it is obtained by taking the product of the
Lebesgue measure by a positive, real-valued measurable function of integral equal to 1.
== Basic idea ==
The basic idea is as follows. If ''f'' : '''R''' → '''R''' is an
integrable function, and φ : '''R''' → '''R''' is a smooth ( = infinitely often
differentiable) function with
compact support (i.e. it is identically zero except on some bounded set), then ∫''f''φd''x'' is a
real number which
linearly and continuously depends on φ. One can therefore think of the function ''f'' as a continuous linear functional on the space which consists of all the "test functions" φ. Similarly, if ''P'' is a probability distribution on the reals and φ is a test function, then ∫φd''P'' is a real number that continuously and linearly depends on φ: probability distributions can thus also be viewed as continuous linear functionals on the space of test functions. This notion of "continuous linear functional on the space of test functions" is therefore used as the definition of a distribution.
Such distributions may be multiplied with real numbers and can by added together, so they form a real
vector space. In general it is not possible to define a multiplication for distributions, but distributions may be multiplied with infinitely often differentiable functions.
To define the derivative of a distribution, we first consider the case of a differentiable and integrable function ''f'' : '''R''' → '''R'''. If <math>\phi</math> is a test function, then we have
:<math>\int_{\mathbf{R}}{}{f'\phi \,dx} = - \int_{\mathbf{R}}{}{f\phi' \,dx}</math>
using
integration by parts (note that φ is zero outside of a bounded set and that therefore no boundary values have to be taken into account). This suggests that if ''S'' is a ''distribution'', we should define its derivative
S' as the linear functional which sends the test function φ to -''S''(φ'). It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes infinitely often differentiable and the usual properties of derivatives hold.
The
Dirac delta (so-called Dirac delta function) is the distribution which sends the test function φ to φ(0). It is the derivative of the
Heaviside step function ''H''(''x'') = 0 if ''x'' < 0 and ''H''(''x'') = 1 if ''x'' ≥ 0. The derivative of the Dirac delta is the distribution which sends the test function φ to -φ'(0). This latter distribution is our first example of a distribution which is neither a function nor a probability distribution.
An alternate definition is the limit of a sequence of functions. For instance the delta function is given by
<math>\delta (x) = \lim_{a \to 0} \delta_a(x)</math>
where δ
a(x) is 1/(2a) if x is between -a and a, and is 0 otherwise.
== Formal definition ==
In the sequel, real-valued distributions on an
open subset ''U'' of '''R'''
''n'' will be formally defined. (With minor modifications, one can also define complex-valued distributions, and one can replace '''R'''
''n'' by any smooth
manifold.) First, the space D(''U'') of test functions on ''U'' needs to be explained. A function φ : ''U'' → '''R''' is said to have ''compact support'' if there exists a
compact subset ''K'' of ''U'' such that φ(''x'') = 0 for all ''x'' in ''U'' \ ''K''. The elements of D(''U'') are the infinitely often differentiable functions φ : ''U'' → '''R''' with compact support. This is a real
vector space. We turn it into a
topological vector space by requiring that a
sequence (or
net) (φ
''k'') converges to 0 if and only if there exists a compact subset ''K'' of ''U'' such that all φ
''k'' are identically zero outside ''K'', and for every ε > 0 and
natural number ''d'' ≥ 0 there exists a
natural number ''k''
0 such that for all ''k'' ≥ ''k''
0 the
absolute value of all ''d''-th derivatives of φ
''k'' is smaller than ε. With this definition, D(''U'') becomes a
complete topological vector space (in fact, a so-called
LF-space).
The
dual space of the topological vector space D(''U''), consisting of all continuous linear functionals ''S'' : D(''U'') → '''R''', is the space of all distributions on ''U''; it is a vector space and is denoted by D'(''U'').
The function ''f'' : ''U'' → '''R''' is called ''locally integrable'' if it is
Lebesgue integrable over every compact subset ''K'' of ''U''. This is a large class of functions which includes all continuous functions. The topology on D(''U'') is defined in such a fashion that any locally integrable function ''f'' yields a continuous linear functional on D(''U'') whose value on the test function φ is given by the Lebesgue integral ∫
''U'' ''f''φ d''x''. Two locally integrable functions ''f'' and ''g'' yield the same element of D(''U'') if and only if they are equal
almost everywhere. Similarly, every
Radon measure μ on ''U'' (which includes the probability distributions) defines an element of D'(''U'') whose value on the test function φ is ∫φ dμ.
As mentioned above, integration by parts suggests that the derivative d''S''/d''x'' of the distribution ''S'' in direction ''x'' should be defined using the formula
:d''S'' / d''x'' (φ) = - ''S'' (dφ / d''x'')
for all test functions φ. In this way, every distribution is infinitely often differentiable, and the derivative in direction ''x'' is a
linear operator on D'(''U'').
The space D'(''U'') is turned into a
locally convex topological vector space
by defining that the sequence (''S''
''k'') converges towards 0 if and only if ''S''
''k''(φ) → 0 for all test functions φ. This is the case if and only if ''S''
''k'' converges uniformly to 0 on all bounded subsets of D(''U''). (A subset of ''E'' of D(''U'') is bounded if there exists a compact subset ''K'' of ''U'' and numbers ''d''
''n'' such that every φ in ''E'' has its support in ''K'' and has its ''n''-th derivatives bounded by ''d''
''n''.) With respect to this topology, differentiation of distributions is a continuous operator; this is an important and desirable property that is not shared by most other notions of differentiation. Furthermore, the test functions (which can themselves be viewed as distributions) are
dense in D'(''U'') with respect to this topology.
If ψ : ''U'' → '''R''' is an infinitely often differentiable function and ''S'' is a distribution on ''U'', we define the product ''S''ψ by (''S''ψ)(φ) = ''S''(ψφ) for all test functions φ. The ordinary product rule of calculus remains valid.
== Compact support and convolution ==
We say that a distribution ''S'' has ''compact support'' if there is a compact subset ''K'' of ''U'' such that for every test function φ whose support is completely outside of ''K'', we have ''S''(φ) = 0. Alternatively, one may define distributions with compact support as continuous linear functionals on the space C
∞(''U''); the topology on C
∞(''U'') is defined such that φ
''k'' converges to 0 if and only if all derivatives of φ
''k'' converge uniformly to 0 on every compact subset of ''U''.
If both ''S'' and ''T'' are distributions on '''R'''
''n'' and one of them has compact support, then one can define a new distribution, the ''convolution'' ''S''*''T'' of ''S'' and ''T'', as follows: if φ is a test function in D('''R'''
''n'') and ''x'', ''y'' elements of '''R'''
''n'', write φ
''x''(''y'') = ''x'' + ''y'', ψ(''x'') = ''T''(φ
''x'') and (''S''*''T'')(φ) = ''S''(ψ).
This generalizes the classical notion of
convolution of functions and is compatible with differentiation in the following sense:
:d/d''x'' (''S'' * ''T'') = (d/d''x'' ''S'') * ''T'' = ''S'' * (d/d''x'' ''T'').
== Tempered distributions and Fourier transform ==
By using a larger space of test functions, one can define the ''tempered distributions'', a subspace of D'('''R'''
''n''). These distributions are useful if one studies the
Fourier transform in generality: all tempered distributions have a Fourier transform, but not all distributions have one.
The space of test functions employed here, the so-called Schwartz-space, is the space of all infinitely differentiable
rapidly decreasing functions, where φ : '''R'''
''n'' → '''R''' is called ''rapidly decreasing'' if any derivative of φ, multiplied with any power of |''x''|, converges towards 0 for |''x''| → ∞. These functions form a complete
topological vector space with a suitably defined family of
seminorms. More precisely, let
:<math> p_{\alpha, \beta} (\phi) = \sup_{x \in \mathbb{R}^n} | x^\alpha D^\beta \phi(x)| </math>
for α, β
multi-indices of size ''n''. Then φ is rapidly-decreasing if all the values
:<math> p_{\alpha, \beta} (\phi) < \infty</math>
The family of seminorms ''p''
α, β defines a
locally convex topology on the Schwartz-space. It is
metrizable and
complete.
The derivative of a tempered distribution is again a tempered distribution.
Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all
square-integrable functions can be viewed as tempered distributions.
To study the Fourier transform, it is best to consider ''complex''-valued test functions and complex-linear distributions. The ordinary
continuous Fourier transform ''F'' yields then an automorphism of Schwartz-space, and we can define the Fourier transform of the tempered distribution ''S'' by (''FS'')(φ) = ''S''(''F''φ) for every test function φ. ''FS'' is thus again a tempered distribution. The Fourier transform is a continuous, linear, bijective operator from the space of tempered distributions to itself. This operation is compatible with differentiation in the sense that
:''F'' (d/d''x'' ''S'') = ''ix'' ''FS''
and also with convolution: if ''S'' is a tempered distribution and ψ is a ''slowly increasing'' infinitely often differentiable function on '''R'''
''n'' (meaning that all derivatives of ψ grow at most as fast as
polynomials), then ''S''ψ is again
a tempered distribution and
:''F''(''S''ψ) = ''FS'' * ''F''ψ.
== Using holomorphic functions as test functions ==
The success of the theory led to investigation of the idea of '''hyperfunction''', in which spaces of
holomorphic functions are used as test functions. A refined theory has been developed, in particular by
Mikio Sato, using
sheaf theory and
several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals.
See also
Colombeau algebra.
==References==
M. J. Lighthill (1958). ''Introduction to Fourier Analysis and Generalised Functions''. Cambridge et. al.: Cambridge University Press.
ISBN 0-521-09128-4
(defines distributions as limits of sequences of functions under integrals)
