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Appendix C Distributions

[page 65, §1]   

[65.1.1] Distributions are generalized functions [31]. [65.1.2] They were invented to overcome the differentiability requirements for functions in analysis and mathematical physics [105, 63]. [65.1.3] Distribution theory has also a physical origin. [65.1.4] A physical observable f can never be measured at a point x\in\mathbb{R}^{d} because every measurement apparatus averages over a small volume around x [115]. [65.1.5] This ‘‘smearing out’’ can be modelled as an integration with smooth ‘‘test functions’’ having compact support.

[65.2.1] Let X denote the space of admissible test functions. [65.2.2] Commonly used test function spaces are C^{{\infty}}(\mathbb{R}^{d}), the space of infinitely often differentiable functions, C_{{\mathrm{c}}}^{{\infty}}(\mathbb{R}^{d}), the space of smooth functions with compact support (see (B.5)), C_{{\mathrm{0}}}^{{\infty}}(\mathbb{R}^{d}), the space of smooth functions vanishing at infinity (see (B.3)), or the so called Schwartz space \mathcal{S}{(\mathbb{R}^{d})} of smooth functions decreasing rapidly at infinity (see (B.21)).

[65.3.1] A distribution \mathit{F}:X\to\mathbb{K} is a linear and continuous mapping that maps \varphi\in X to a real (\mathbb{K}=\mathbb{R}) or complex (\mathbb{K}=\mathbb{C}) number 12 (This is a footnote:) 12For vector valued distributions see [106]. [65.3.2] There exists a canonical correspondence between functions and distributions. [65.3.3] More precisely, for every locally integrable function f\in L^{{1}}_{{\mathrm{loc}}}(\mathbb{R}^{d}) there exists a distribution \mathit{F}_{f}=\langle f,.\rangle (often also denoted with the same symbol f) defined by

\mathit{F}_{f}(\varphi)=\langle f,\varphi\rangle=\int\limits _{{\mathbb{R}^{d}}}f(x)\varphi(x)\;\mathrm{d}x (C.1)

for every test function \varphi\in X. [65.3.4] Distributions that can be written in this way are called regular distributions. [65.3.5] Distributions that are not regular are sometimes called singular. [65.3.6] The mapping f\to\langle f,.\rangle that assigns to a locally integrable f its associated distribution is injective and continuous. [65.3.7] The set of distributions is again a vector space, namely the dual space of the vector space of test functions, and it is denoted as X^{\prime} where X is the test function space.

[page 66, §1]    [66.1.1] Important examples for singular distributions are the Dirac \delta-function and its derivatives. [66.1.2] They are defined by the rules

\displaystyle\int\limits\delta(x)\varphi(x)\mathrm{d}x=\varphi(0) (C.2)
\displaystyle\int\limits\delta^{{(n)}}(x)\varphi(x)=(-1)^{n}\left.\frac{\mathrm{d}^{n}\varphi}{\mathrm{d}x^{n}}\right|_{{x=0}} (C.3)

for every test function \varphi\in X and n\in\mathbb{N}. [66.1.3] Clearly, \delta(x) is not a function, because if it were a function, then \int\limits\delta(x)\varphi(x)\mathrm{d}x=0 would have to hold. [66.1.4] Another example for a singular distribution is the finite part or principal value \mathcal{P}\left\{ 1/x\right\} of 1/x. [66.1.5] It is defined by

\left\langle\mathcal{P}\left\{\frac{1}{x}\right\},\varphi\right\rangle=\lim _{{\varepsilon\to 0+}}\int\limits _{{|x|\geq\varepsilon}}\frac{\varphi(x)}{x}\mathrm{d}x (C.4)

for \varphi\in C_{{\mathrm{c}}}^{{\infty}}(\mathbb{R}). [66.1.6] It is a singular distribution on \mathbb{R}, but regular on \mathbb{R}\setminus\{ 0\} where it coincides with the function 1/x.

[66.2.1] Equation (C.2) illustrates how distributions circumvent the limitations of differentiation for ordinary functions. [66.2.2] The basic idea is the formula for partial integration

\int\limits _{\mathbb{G}}\partial _{i}f(x)\varphi(x)\mathrm{d}x=-\int\limits _{\mathbb{G}}f(x)\partial _{i}\varphi(x)\mathrm{d}x (C.5)

valid for f\in C_{{\mathrm{c}}}^{{1}}(\mathbb{G}), \varphi\in C^{{1}}(\mathbb{G}), i=1,...,d and \mathbb{G}\subset\mathbb{R}^{d} an open set. [66.2.3] The formula is proved by extending f\varphi as 0 to all of \mathbb{R}^{d} and using Leibniz’ product rule. [66.2.4] Rewriting the formula as

\langle\partial _{i}f,\varphi\rangle=-\langle f,\partial _{i}\varphi\rangle (C.6)

suggests to view \partial _{i}f again as a linear continuous mapping (integral) on a space X of test functions \varphi\in X. [66.2.5] Then the formula is a rule for differentiating f given that \varphi is differentiable.

[66.3.1] Distributions on the test function space \mathcal{S}{(\mathbb{R}^{d})} are called tempered distributions. [66.3.2] The space of tempered distributions is the dual space \mathcal{S}{(\mathbb{R}^{d})}^{\prime}. [66.3.3] Tempered distributions generalize locally integrable functions growing at most polynomially for |x|\to\infty. [66.3.4] All distributions with compact support are tempered. Square integrable functions are tempered distributions. [66.3.5] The derivative of a tempered distribution is again a tempered distribution. [66.3.6] \mathcal{S}{(\mathbb{R}^{d})} is dense in L^{{p}}(\mathbb{R}^{d}) for all 1\leq p<\infty but not in L^{{\infty}}(\mathbb{R}^{d}). [66.3.7] The Fourier transform and its inverse are continous maps of the Schwartz space onto itself. [66.3.8] A distribution f belongs to \mathcal{S}{(\mathbb{R}^{d})}^{\prime} if and only if it is the derivative of a continuous function with slow growth, i.e. it is of the form f=\mathrm{D}^{\gamma}[(1+|x|^{2})^{{k/2}}g(x)]

[page 67, §0]    where k\in\mathbb{N}, \gamma\in\mathbb{N}^{d} and g is a bounded continuous function on \mathbb{R}^{d}. [67.0.1] Note that the exponential function is not a tempered distribution.

[67.1.1] A distribution f\in\mathcal{S}{(\mathbb{R}^{d})}^{\prime} is said to have compact support if there exists a compact subset K\subset\mathbb{R}^{d} such that \langle f,\varphi\rangle=0 for all test functions with \mathrm{supp}\,\varphi\cap K=\emptyset. [67.1.2] The Dirac \delta-function is an example. [67.1.3]  Other examples are Radon measures on a compact set K. [67.1.4] They can be described as linear functionals on C^{{0}}(K). [67.1.5] If the set K is sufficiently regular (e.g. if it is the closure of a region with piecewise smooth boundary) then every distribution with compact support in K can be written in the form

f=\sum _{{|\gamma|\leq N}}\mathrm{D}^{\gamma}f_{\gamma} (C.7)

where \gamma=(\gamma _{1},...,\gamma _{d}), \gamma _{j}\geq 0 is a multiindex, |\gamma|=\sum\gamma _{i} and f_{\gamma} are continuous functions of compact support. [67.1.6] Here N\geq 0 and the partial derivatives in \mathrm{D}^{\gamma} are distributional derivatives defined above. [67.1.7] A special case are distributions with support in a single point taken as \{ 0\}. [67.1.8] Any such distributions can be written in the form

f=\sum _{{|\gamma|\leq N}}c_{\gamma}\mathrm{D}^{\gamma}\delta (C.8)

where \delta is the Dirac \delta-function and c_{\gamma} are constants.

[67.2.1] The multiplication of a distribution f with a smooth function g is defined by the formula \langle gf,\varphi\rangle=\langle f,g\varphi\rangle where g\in C^{{\infty}}(\mathbb{G}). [67.2.2] A combination of multiplication by a smooth function and differentiation allows to define differential operators

A=\sum _{{|\gamma|\leq m}}a_{\gamma}(x)\mathrm{D}^{\gamma} (C.9)

with smooth a_{\gamma}(x)\in C^{{\infty}}(\mathbb{G}). [67.2.3] They are well defined for all distributions in C_{{\mathrm{c}}}^{{\infty}}(\mathbb{G})^{\prime}.

[67.3.1] A distribution is called homogeneous of degree \alpha\in\mathbb{C} if

f(\lambda x)=\lambda^{\alpha}f(x) (C.10)

for all \lambda>0. [67.3.2] Here \lambda^{\alpha}=\exp(\alpha\log\lambda) is the standard definition. [67.3.3] The Dirac \delta-distribution is homogeneous of degree -d. [67.3.4] For regular distributions the definition coincides with homogeneity of functions f\in L^{{1}}_{{\mathrm{loc}}}(\mathbb{R}^{d}). [67.3.5] The convolution kernels K_{\pm}^{\alpha} from eq. (2.39) are homogeneous of degree \alpha-1. [67.3.6] Homogeneous distributions remain homogeneous under differentiation. [67.3.7] A homogeneous locally integrable function g on \mathbb{R}^{d}\setminus\{ 0\} of degree \alpha can be extended to homogeneous distributions f on all of \mathbb{R}^{d}. [67.3.8] The degree of homogeneity of f must again be \alpha. [67.3.9] As long as \alpha\neq-d,-d-1,-d-2,... the integral

\langle g_{\beta},\varphi\rangle=\int\limits g\left(\frac{x}{|x|}\right)|x|^{\beta}\mathrm{d}^{d}x (C.11)

[page 68, §0]    which converges absolutely for \mathrm{Re}\,\beta>-d can be used to define f=g_{\alpha} by analytic continuation from the region \mathrm{Re}\,\beta>-d to the point \alpha. [68.0.1] For \alpha=-d,-d-1,..., however, this is not always possible. [68.0.2] An example is the function 1/|x| on \mathbb{R}\setminus\{ 0\}. [68.0.3] It cannot be extended to a homogeneous distribution of degree -1 on all of \mathbb{R}.

[68.1.1] For f\in L^{{1}}_{{\mathrm{loc}}}(\mathbb{G}_{1}) and g\in L^{{1}}_{{\mathrm{loc}}}(\mathbb{G}_{2}) their tensor product is the function (f\otimes g)(x,y)=f(x)g(y) defined on \mathbb{G}_{1}\times\mathbb{G}_{2}. [68.1.2] The function f\otimes g gives a functional

\langle f\otimes g,\varphi(x,y)\rangle=\langle f(x),\langle g(y),\varphi(x,y)\rangle\rangle (C.12)

for \varphi\in C_{{\mathrm{c}}}^{{\infty}}(\mathbb{G}_{1}\times\mathbb{G}_{2}). [68.1.3] For two distributions this formula defines the their tensor product. [68.1.4] An example is a measure \mu(x)\otimes\delta(y) concentrated on the surface y=0 in \mathbb{G}_{1}\otimes\mathbb{G}_{2} where \mu(x) is a measure on \mathbb{G}_{1}. [68.1.5] The convolution of distribution defined in the main text (see eq. (2.52) can then be defined by the formula

\langle f*g,\varphi\rangle=\langle(f\otimes g)(x,y),\varphi(x+y)\rangle (C.13)

whenever one of the distributions f or g has compact support.