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∈Rd 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∞⁢Rd, the space of infinitely often differentiable functions, Cc∞⁢Rd, the space of smooth functions with compact support (see (B.5)), C0∞⁢Rd, the space of smooth functions vanishing at infinity (see (B.3)), or the so called Schwartz space S⁢Rd of smooth functions decreasing rapidly at infinity (see (B.21)).

[65.3.1] A distribution F:X→K is a linear and continuous mapping that maps φ∈X to a real (K=R) or complex (K=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∈Lloc1⁢Rd there exists a distribution Ff=⟨f,.⟩ (often also denoted with the same symbol f) defined by

for every test function φ∈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→⟨f,.⟩ 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′ where X is the test function space.

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

for every test function φ∈X and n∈N. [66.1.3] Clearly, δ⁢x is not a function, because if it were a function, then ∫δ⁢x⁢φ⁢x⁢d⁢x=0 would have to hold. [66.1.4] Another example for a singular distribution is the finite part or principal value P⁢1/x of 1/x. [66.1.5] It is defined by

for φ∈Cc∞⁢R. [66.1.6] It is a singular distribution on R, but regular on R∖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

valid for f∈Cc1⁢G, φ∈C1⁢G, i=1,…,d and G⊂Rd an open set. [66.2.3] The formula is proved by extending f⁢φ as 0 to all of Rd and using Leibniz’ product rule. [66.2.4] Rewriting the formula as

suggests to view ∂i⁡f again as a linear continuous mapping (integral) on a space X of test functions φ∈X. [66.2.5] Then the formula is a rule for differentiating f given that φ is differentiable.

[66.3.1] Distributions on the test function space S⁢Rd are called tempered distributions. [66.3.2] The space of tempered distributions is the dual space S⁢Rd′. [66.3.3] Tempered distributions generalize locally integrable functions growing at most polynomially for x→∞. [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] S⁢Rd is dense in Lp⁢Rd for all 1≤p<∞ but not in L∞⁢Rd. [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 S⁢Rd′ if and only if it is the derivative of a continuous function with slow growth, i.e. it is of the form f=Dγ⁢1+x2k/2⁢g⁢x

[page 67, §0]    where k∈N, γ∈Nd and g is a bounded continuous function on Rd. [67.0.1] Note that the exponential function is not a tempered distribution.

[67.1.1] A distribution f∈S⁢Rd′ is said to have compact support if there exists a compact subset K⊂Rd such that f,φ=0 for all test functions with supp⁢φ∩K=∅. [67.1.2] The Dirac δ-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 C0⁢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

where γ=γ1,…,γd, γj≥0 is a multiindex, γ=∑γi and fγ are continuous functions of compact support. [67.1.6] Here N≥0 and the partial derivatives in Dγ 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

where δ is the Dirac δ-function and cγ are constants.

[67.2.1] The multiplication of a distribution f with a smooth function g is defined by the formula g⁢f,φ=f,g⁢φ where g∈C∞⁢G. [67.2.2] A combination of multiplication by a smooth function and differentiation allows to define differential operators

with smooth aγ⁢x∈C∞⁢G. [67.2.3] They are well defined for all distributions in Cc∞⁢G′.

[67.3.1] A distribution is called homogeneous of degree α∈C if

for all λ>0. [67.3.2] Here λα=exp⁡α⁢log⁡λ is the standard definition. [67.3.3] The Dirac δ-distribution is homogeneous of degree -d. [67.3.4] For regular distributions the definition coincides with homogeneity of functions f∈Lloc1⁢Rd. [67.3.5] The convolution kernels K±α from eq. (2.39) are homogeneous of degree α-1. [67.3.6] Homogeneous distributions remain homogeneous under differentiation. [67.3.7] A homogeneous locally integrable function g on Rd∖0 of degree α can be extended to homogeneous distributions f on all of Rd. [67.3.8] The degree of homogeneity of f must again be α. [67.3.9] As long as α≠-d,-d-1,-d-2,… the integral

[page 68, §0]    which converges absolutely for Re⁢β>-d can be used to define f=gα by analytic continuation from the region Re⁢β>-d to the point α. [68.0.1] For α=-d,-d-1,…, however, this is not always possible. [68.0.2] An example is the function 1/x on R∖0. [68.0.3] It cannot be extended to a homogeneous distribution of degree -1 on all of R.

[68.1.1] For f∈Lloc1⁢G1 and g∈Lloc1⁢G2 their tensor product is the function f⊗g⁢x,y=f⁢x⁢g⁢y defined on G1×G2. [68.1.2] The function f⊗g gives a functional

for φ∈Cc∞⁢G1×G2. [68.1.3] For two distributions this formula defines the their tensor product. [68.1.4] An example is a measure μ⁢x⊗δ⁢y concentrated on the surface y=0 in G1⊗G2 where μ⁢x is a measure on G1. [68.1.5] The convolution of distribution defined in the main text (see eq. (2.52) can then be defined by the formula

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