Appendix C Distributions
[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 SRd 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
.
[65.3.2] There exists a canonical correspondence
between functions and distributions.
[65.3.3] More precisely, for every locally integrable function
f∈Lloc1Rd
there exists a distribution Ff=⟨f,.⟩
(often also denoted with the same symbol f)
defined by
Ffφ=f,φ=∫Rdfxφxdx | | (C.1) |
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
| ∫δxφxdx=φ0 | | (C.2) |
| ∫δnxφx=-1ndnφdxnx=0 | | (C.3) |
for every test function φ∈X and n∈N.
[66.1.3] Clearly, δx is not a function, because if it were a
function, then ∫δxφxdx=0 would have to hold.
[66.1.4] Another example for a singular distribution is
the finite part or principal value P1/x of 1/x.
[66.1.5] It is defined by
P1x,φ=limε→0+∫x≥εφxxdx | | (C.4) |
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
∫G∂ifxφxdx=-∫Gfx∂iφxdx | | (C.5) |
valid for f∈Cc1G, φ∈C1G,
i=1,…,d and G⊂Rd an open set.
[66.2.3] The formula is proved by extending fφ as 0 to all
of Rd and using Leibniz’ product rule.
[66.2.4] Rewriting the formula as
suggests to view ∂if 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 SRd
are called tempered distributions.
[66.3.2] The space of tempered distributions is the dual space
SRd′.
[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] SRd is dense in LpRd 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 SRd′ if and
only if it is the derivative of a continuous function with slow
growth, i.e. it is of the form
f=Dγ1+x2k/2gx
[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∈SRd′ is said to have
compact support
if there exists a compact subset K⊂Rd such that
f,φ=0 for all test functions
with suppφ∩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 C0K.
[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
gf,φ=f,gφ
where g∈C∞G.
[67.2.2] A combination of multiplication by a smooth function
and differentiation allows to define differential
operators
with smooth aγx∈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=expαlogλ 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∈Lloc1Rd.
[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 Reβ>-d can be
used to define f=gα by analytic continuation
from the region Reβ>-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∈Lloc1G1 and g∈Lloc1G2
their tensor product is the function f⊗gx,y=fxgy
defined on G1×G2.
[68.1.2] The function f⊗g gives a functional
f⊗g,φx,y=fx,gy,φx,y | | (C.12) |
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 μx⊗δy concentrated on the
surface y=0 in G1⊗G2 where μx 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
f*g,φ=f⊗gx,y,φx+y | | (C.13) |
whenever one of the distributions f or g has compact support.