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In Mathematical Analysis , distributions (also known as '''generalized functions''') are objects which generalize Function s and Probability Distribution s. They extend the concept of Derivative to all Continuous functions and beyond and are used to formulate generalized solutions of Partial Differential Equation s. 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 introduced in late 1940s by Laurent Schwartz , who developed a comprehensive theory of distributions. Sometimes, people talk of a " 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 ) function with Compact Support (that is, 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 be 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 differentiable functions. To define the derivative of a distribution, we first consider the case of a differentiable and integrable function ''f'' : R → R. If is a test function, then we have : 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 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 : where δa(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 . We turn it into a Topological Vector Space by stipulating 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 topology is called the Strong (operator) Topology . 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'') = φ(''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 with a suitably defined family of Seminorm s. More precisely, let |