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§1.16 Distributions

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§1.16(i) Test Functions

Let ϕ be a function defined on an open interval I=(a,b), which can be infinite. The closure of the set of points where ϕ0 is called the support of ϕ. If the support of ϕ is a compact set (§1.9(vii)), then ϕ is called a function of compact support. A test function is an infinitely differentiable function of compact support.

A sequence {ϕn} of test functions converges to a test function ϕ if the support of every ϕn is contained in a fixed compact set K and as n the sequence {ϕn(k)} converges uniformly on K to ϕ(k) for k=0,1,2,.

The linear space of all test functions with the above definition of convergence is called a test function space. We denote it by 𝒟(I).

A mapping Λ on 𝒟(I) is a linear functional if it takes complex values and

1.16.1 Λ(α1ϕ1+α2ϕ2)=α1Λ(ϕ1)+α2Λ(ϕ2),

where α1 and α2 are real or complex constants. Λ:𝒟(I) is called a distribution if it is a continuous linear functional on 𝒟(I), that is, it is a linear functional and for every ϕnϕ in 𝒟(I),

1.16.2 limnΛ(ϕn)=Λ(ϕ).

From here on we write Λ,ϕ for Λ(ϕ). The space of all distributions will be denoted by 𝒟*(I). A distribution Λ is called regular if there is a function f on I, which is absolutely integrable on every compact subset of I, such that

1.16.3 Λ,ϕ=If(x)ϕ(x)x.

We denote a regular distribution by Λf, or simply f, where f is the function giving rise to the distribution. (If a distribution is not regular, it is called singular.)

Define

1.16.4 Λ1+Λ2,ϕ=Λ1,ϕ+Λ2,ϕ,
1.16.5 cΛ,ϕ=cΛ,ϕ=Λ,cϕ,

where c is a constant. More generally, if α(x) is an infinitely differentiable function, then

1.16.6 αΛ,ϕ=Λ,αϕ.

We say that a sequence of distributions {Λn} converges to a distribution Λ in 𝒟* if

1.16.7 limnΛn,ϕ=Λ,ϕ

for all ϕ𝒟(I).

§1.16(ii) Derivatives of a Distribution

The derivative Λ of a distribution is defined by

1.16.8 Λ,ϕ=-Λ,ϕ,
ϕ𝒟(I).

Similarly

1.16.9 Λ(k),ϕ=(-1)kΛ,ϕ(k),
k=1,2,.

For any locally integrable function f, its distributional derivative is Df=Λf.

§1.16(iii) Dirac Delta Distribution

1.16.10 δ,ϕ =ϕ(0),
ϕ𝒟(I),
1.16.11 δx0,ϕ =ϕ(x0),
ϕ𝒟(I),
1.16.12 δx0(n),ϕ =(-1)nϕ(n)(x0),
ϕ𝒟(I).

The Dirac delta distribution is singular.

§1.16(iv) Heaviside Function

1.16.13 H(x) ={1,x>0,0,x0.
1.16.14 H(x-x0) ={1,x>x0,0,xx0.
1.16.15 DH =δ,
1.16.16 DH(x-x0) =δx0.

Suppose f(x) is infinitely differentiable except at x0, where left and right derivatives of all orders exist, and

1.16.17 σn=f(n)(x0+)-f(n)(x0-).

Then

1.16.18 Dmf=f(m)+σ0δx0(m-1)+σ1δx0(m-2)++σm-1δx0,
m=1,2,.

For α>-1,

1.16.19 x+α=xαH(x)={xα,x>0,0,x0.

For α>0,

1.16.20 Dx+α=αx+α-1.

For α<-1 and α not an integer, define

1.16.21 x+α=1(α+1)nDnx+α+n,

where n is an integer such that α+n>-1. Similarly, we write

1.16.22 ln+x=H(x)lnx={lnx,x>0,0,x0,

and define

1.16.23 (-1)nn!x+-1-n=D(n+1)ln+x,
n=0,1,2,.

§1.16(v) Tempered Distributions

The space 𝒯() of test functions for tempered distributions consists of all infinitely-differentiable functions such that the function and all its derivatives are O(|x|-N) as |x| for all N.

A sequence {ϕn} of functions in 𝒯 is said to converge to a function ϕ𝒯 as n if the sequence {ϕn(k)} converges uniformly to ϕ(k) on every finite interval and if the constants ck,N in the inequalities

1.16.24 |xNϕn(k)|ck,N

do not depend on n.

A tempered distribution is a continuous linear functional Λ on 𝒯. (See the definition of a distribution in §1.16(i).) The set of tempered distributions is denoted by 𝒯*.

A sequence of tempered distributions Λn converges to Λ in 𝒯* if

1.16.25 limnΛn,ϕ=Λ,ϕ,

for all ϕ𝒯.

The derivatives of tempered distributions are defined in the same way as derivatives of distributions.

For a detailed discussion of tempered distributions see Lighthill (1958).

§1.16(vi) Distributions of Several Variables

Let 𝒟(n)=𝒟n be the set of all infinitely differentiable functions in n variables, ϕ(x1,x2,,xn), with compact support in n. If k=(k1,,kn) is a multi-index and x=(x1,,xn)n, then we write xk=x1k1xnkn and ϕ(k)(x)=kϕ/(x1k1xnkn). A sequence {ϕm} of functions in 𝒟n converges to a function ϕ𝒟n if the supports of ϕm lie in a fixed compact subset K of n and ϕm(k) converges uniformly to ϕ(k) in K for every multi-index k=(k1,k2,,kn). A distribution in n is a continuous linear functional on 𝒟n.

The partial derivatives of distributions in n can be defined as in §1.16(ii). A locally integrable function f(x)=f(x1,x2,,xn) gives rise to a distribution Λf defined by

1.16.26 Λf,ϕ=nf(x)ϕ(x)x,
ϕ𝒟n.

The distributional derivative Dkf of f is defined by

1.16.27 Dkf,ϕ=(-1)|k|nf(x)ϕ(k)(x)x,
ϕ𝒟n,

where k is a multi-index and |k|=k1+k2++kn.

For tempered distributions the space of test functions 𝒯n is the set of all infinitely-differentiable functions ϕ of n variables that satisfy

1.16.28 |xmϕ(k)(x)|cm,k,
xn.

Here m=(m1,m2,,mn) and k=(k1,k2,,kn) are multi-indices, and cm,k are constants. Tempered distributions are continuous linear functionals on this space of test functions. The space of tempered distributions is denoted by 𝒯n*.

§1.16(vii) Fourier Transforms of Distributions

Suppose ϕ is a test function in 𝒯n. Then its Fourier transform is

1.16.29 F(x)=F=1(2π)n/2nϕ(t)xtt,

where x=(x1,x2,,xn) and xt=x1t1++xntn. F(x) is also in 𝒯n. For a multi-index α=(α1,α2,,αn), set |α|=α1+α2++αn and

1.16.30 Dα=-|α|Dα=(1x1)α1(1xn)αn,
1.16.31 P(x)=P=cαxα=cαx1α1xnαn,

and

1.16.32 P(D)=cαDα.

Then

1.16.33 1(2π)n/2n(P(D)ϕ)(t)xtt=P(-x)F(x),

and

1.16.34 1(2π)n/2nP(t)ϕ(t)xtt=P(D)F(x).

If u𝒯n* is a tempered distribution, then its Fourier transform (u) is defined by

1.16.35 (u),ϕ=u,F,
ϕ𝒯n,

where F is given by (1.16.29). The Fourier transform (u) of a tempered distribution is again a tempered distribution, and

1.16.36 (P(D)u) =P(-x)(u),
1.16.37 (Pu) =P(D)(u).

In (1.16.36) and (1.16.37) the derivatives in P(D) are understood to be in the sense of distributions.