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§1.16 Distributions


§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)dx.

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.)


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,


1.16.9 Λ(k),ϕ=(-1)kΛ,ϕ(k),

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),
1.16.11 δx0,ϕ =ϕ(x0),
1.16.12 δx0(n),ϕ =(-1)nϕ(n)(x0),

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-).


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

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,

§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)dx,

The distributional derivative Dkf of f is defined by

1.16.27 Dkf,ϕ=(-1)|k|nf(x)ϕ(k)(x)dx,

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,

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 Tempered Distributions

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

1.16.29 (ϕ)(x)=ϕ(x)=1(2π)n/2nϕ(t)eixtdt,

where x=(x1,x2,,xn) and xt=x1t1++xntn. ϕ(x) is also in 𝒯n.


1.16.30 D=(1ix1,1ix2,,1ixn).

For a multi-index α=(α1,α2,,αn), define

1.16.31 P(x)=αcαxα=αcαx1α1xnαn,


1.16.32 P(D)=αcαDα=αcα(1ix1)α1(1ixn)αn.

Here α ranges over a finite set of multi-indices, P(x) is a multivariate polynomial, and P(D) is a partial differential operator. Then

1.16.33 (P(D)ϕ)(x)=P(-x)ϕ(x),


1.16.34 (Pϕ)(x)=P(D)ϕ(x).

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

1.16.35 (u),ϕ=u,(ϕ),

The Fourier transform (u) of a tempered distribution is again a tempered distribution, and

1.16.36 (P(D)u),ϕ=P-(u),ϕ=(u),P-ϕ,
1.16.37 (Pu),ϕ=P(D)(u),ϕ,

in which P-(x)=P(-x); compare (1.16.33) and (1.16.34). In (1.16.36) and (1.16.37) the derivatives in P(D) are understood to be in the sense of distributions, as defined in §1.16(ii) and we also use the convention (1.16.6).

§1.16(viii) Fourier Transforms of Special Distributions

We use the notation of the previous subsection and take n=1 and u=δ in (1.16.35). We obtain

1.16.38 (δ),ϕ=δ,(ϕ)=δ,12π-ϕ(t)eixtdt=12π-ϕ(t)dt=12π1,ϕ,

As distributions, the last equation reads

1.16.39 (δ)=12π,

which is often written conventionally as

1.16.40 -δ(t)eixtdt=1;

see also (1.17.2).

Since 2π(δ)=1, we have

1.16.41 (1),ϕ=2π((δ)),ϕ=2π(δ),(ϕ)=2πδ,((ϕ))=2πδ,ϕ-=2πϕ(0),

in which ϕ-(x)=ϕ(-x). The second to last equality follows from the Fourier integral formula (1.17.8). Since the quantity on the extreme right of (1.16.41) is equal to 2πδ,ϕ, as distributions, the result in this equation can be stated as

1.16.42 (1)=2πδ,

and conventionally it is expressed as

1.16.43 12π-eixtdt=δ(x);

see also (1.17.12).

It is easily verified that

1.16.44 sign(x)=2H(x)-1,

and from (1.16.15) we find

1.16.45 sign=2H=2δ,

where H(x) is the Heaviside function defined in (1.16.13), and the derivatives are to be understood in the sense of distributions. Then

1.16.46 (sign)=(2H)=2(δ)=2π,

and from (1.16.36) with u=sign, P(D)=D, and P-(x)=-ix, we have also

1.16.47 (sign)=xi(sign).

Coupling (1.16.46) and (1.16.47) gives

1.16.48 (sign)=2πix,

that is

1.16.49 (sign),ϕ=i2π-ϕ(x)xdx.

The Fourier transform of H(x) now follows from (1.16.42) and (1.16.48). Indeed, we have

1.16.50 (H)=12(1+sign)=12[(1)+(sign)]=π2(δ+iπx),

that is

1.16.51 (H),ϕ=π2ϕ(0)+i2π-ϕ(x)xdx.

For more detailed discussions of the formulas in this section, see Kanwal (1983) and Debnath and Bhatta (2015).