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Chapter 1

Distributions

The concept of distribution generalises and extends the concept of . A distribution is basically defined by its action on a of test functions. It is indeed a linear functional T

T : F → C φ(x) → T (φ) . (1.1) from a space of functions F of real variable, called the test functions, to C. For example, to any integrable real function f(x) we can associate the linear functional Tf : F → R defined as

R ∞ φ(x) → Tf (φ) = −∞ dx f(x) φ(x) . (1.2)

The idea is that, knowing the value of the integral Tf (φ) on a large enough of test functions φ, we can reconstruct the original function f. Al- lowing for general linear functionals we extend the concept of function. We can define operations on distributions even in cases where they are not well defined on functions. For example, we will be able to define the derivative of a discontinuous function when it is seen as a distribution. In the world of distributions, everything is infinitely differentiable and limits always commute with other operations. The distributions are also called generalized functions.

1 2 CHAPTER 1. DISTRIBUTIONS 1.1 Test functions

In this section we discuss the space F of test functions and its properties. A distribution will be a linear functional on F. For simplicity, we consider test functions of real variable. We want to choose test functions φ ∈ F that are as smooth as possible, in particular that are infinitely differentiable, φ ∈ C∞(R). Moreover, in to guarantee the existence of the integral in (1.2) we require that φ vanishes sufficiently rapidly at infinity. There are two canonical choices for F, the space of bump functions D(R) and the space of rapidly decreasing functions S(R), which we now discuss.

1.1.1 The space D(R) The of a function of real variable is the of the set of points where the function is non-zero

supp(f) = {x ∈ R | f(x) 6= 0} (1.3)

A function with compact support is a function whose support is compact (a closed and bounded of R). We call D(R) the space of real functions of real variable that are infinitely differentiable and with compact support,

∞ D(R) = {φ(x) | φ ∈ C (R) with compact support} (1.4)

D(R) is obviously a . It is also a , but the notion of that is useful for the theory of distribution is not induced by any norm or metric. For our purposes, it is enough to define the notion of convergence of a in the topology of D(R). We say that the sequence {φn ∈ D(R)} converges to the function φ ∈ D(R) if the following conditions are satisfied,

• it exists a bounded interval Iˆ ⊂ R which contains the supports of all the functions φn and φ. In other words, it exists a R such that

φn(x) = 0 and φ(x) = 0 ∀ |x| ≥ R ; (1.5) 1.1. TEST FUNCTIONS 3

• the sequence of the p-th derivatives of φn(x) converges uniformly to the p-th derivative of φ(x) p p d d n−→∞ sup p φn(x) − p φ(x) −→ 0 ∀ p = 0, 1,.... (1.6) x∈R dx dx

In particular, φn(x) converges uniformly to φ(x).

The generalization of the space D(R) to the case of complex-valued func- tions or of functions defined in Rn is straightforward. Example 1.1. The function

( − 1 e 1−x2 |x| < 1 f(x) = (1.7) 0 |x| ≥ 1 belongs to D(R) since it is zero outside the interval [−1, 1], infinitely differ- entiable in (−1, 1) and with vanishing derivatives of all orders in x = ±1. Notice that the function is C∞(R) but not analytic. The Taylor in x = ±1 is identically zero since all derivatives are zero, but the function is not zero in the neighborhood of x = ±1. All the functions in D(R) are necessarily non analytic.

1.1.2 The space S(R) A function f of real variable is called a rapidly decreasing function, or a Schwarz function, if it is infinitely differentiable on R, f ∈ C∞(R), and dq xp f(x) x−→→±∞ 0 ∀ p, q = 0, 1,.... (1.8) dxq In other words, f is infinitely differentiable and f and all its derivatives vanish at infinity more rapidly than any polynomial. The vector space of rapidly decreasing functions is denoted

∞ S(R) = {φ(x) | φ ∈ C (R) and rapidly decreasing} . (1.9)

We define a notion of convergence in S(R) as follows. We say that the sequence of functions {φn ∈ S(R)} converges to the function φ ∈ S(R) if p dq p dq x dxq φn(x) converges uniformly to x dxq φ(x) for all p and q q q p d p d n−→∞ sup x q φn(x) − x q φ(x) −→ 0 ∀ p, q = 0, 1,.... (1.10) x∈R dx dx 4 CHAPTER 1. DISTRIBUTIONS

Example 1.2. The functions

φ(x) = P (x)e−x2 , φ(x) = sin x e−x2 , (1.11) where P (x) is an arbitrary polynomial, are elements of S(R). On the other hand, the functions

1 φ(x) = , φ(x) = e−|x| (1.12) 1 + x2 are not, for the first vanishes as an inverse polynomial at infinity and the second, while vanishing faster than any polynomial at infinity, is not C∞(R).

Example 1.3. Since every function in S(R) is square integrable, S(R) is a vector subspace of L2(R). S(R) is dense in L2(R). In fact, any element of L2(R) is the limit of a sequence of elements of S(R). To see this, it is 2 P∞ enough to expand f ∈ L (R) in the Hermite basis, f = n=0 cnun, where −x2 un(x) ∼ Hn(x)e are obviously rapidly decreasing functions.

1.2 Distributions

We call distribution a functional T : F → C

φ(x) → T (φ) . (1.13) where F = D(R) or F = S(R), with the properties

• T is linear:

T (λ1φ1 + λ2φ2) = λ1T (φ1) + λ2T (φ2) , (1.14)

for all λ1, λ2 ∈ C and φ1, φ2 ∈ F ;

• T is continuous in the following sense: if the sequence φn(x) converges

to φ(x) in F, limn→∞ φn(x) = φ(x), then

lim T (φn) = T (φ) . (1.15) n→∞ 1.2. DISTRIBUTIONS 5

It is also customary to use the notation

(T, φ) ≡ T (φ) (1.16) for the action of a distribution on a test function. In the theory of distributions, the functions φ in D(R) and S(R) are called test functions. The functionals T acting on F = D(R) are simply called distributions and form a space denoted D0(R). The functionals T acting on F = S(R) are called tempered distributions and form a space denoted S0(R). We say that two distributions, T1 et T2, are equal if they act in the same way on all test functions

T1(φ) = T2(φ) ∀ φ ∈ D(R)(S(R)) . (1.17)

The spaces of distributions D0(R) et S0(R) are vector spaces if we define sum and by scalars by

(T1 + T2)(φ) = T1(φ) + T2(φ) , (1.18) (λT )(φ) = λT (φ) , (1.19) for all φ ∈ D0(R)(S0(R)) and all complex λ. The functions with compact support are a subset of the rapidly decreasing functions D(R) ⊂ S(R) . (1.20) This implies the opposite inclusion for the space of distributions

0 0 S (R) ⊂ D (R) . (1.21) since every functional that is defined on all the functions in S(R) is a fortiori defined on the functions in D(R). It is straightforward to extend the previous definitions to the case of multi-variable distributions. In the following we discuss the case of distributions in D0(R), since all properties we will introduce in this chapter are also valid for tempered dis- tributions. More sophisticated operations, like the , can be only defined for tempered distributions. 6 CHAPTER 1. DISTRIBUTIONS

1.2.1 Regular distributions

1 We say that a function f(x) is locally integrable, and we write f ∈ Lloc(R), if its integral on any compact subset K of R is finite Z |f(x)|dx < ∞ . (1.22) K Notice that we are not requiring any nice behavior at infinity. A function |x| 1 1 like f(x) = e is not an element L (R) but it is in Lloc(R). To any locally integrable function f we associate the linear functional Z ∞ ∗ Tf (φ) = (f, φ) = dx f (x) φ(x) . (1.23) −∞ 0 Tf is a distribution Tf ∈ D (R).

Proof. In order to see that Tf is a distribution we need to check that its value on a test function is well defined and that Tf is linear and continuous. Call Kφ the support of φ ∈ D( ). The integral in (1.23) restricts to R dx f ∗(x) φ(x), which R Kφ ∞ is finite since φ is C (R) and f is integrable on any compact. Tf is clearly linear in φ. To see that it is continuous take a sequence of test functions φn converging to φ in the topology of D(R). Recall that this means that the supports of φn and φ are contained in a compact set K, and that φn and all its derivatives converges uniformly to φ. We have Z Z |Tf (φ) − Tf (φn)| ≤ dx |f(x)| |φ(x) − φn(x)| ≤ sup |φ(x) − φn(x)| |f| K x∈K K which converges to zero since f is locally integrable and φn converges uniformly to φ: limn→∞ supx∈K |φ(x) − φn(x)| = 0.

Distributions of the form Tf for a locally integrable function f are called regular distributions. Tf is not in general a tempered distribution, because the integral (1.23) is not necessarily convergent at infinity. Since φ ∈ S(R) vanishes at infinity more rapidly than any polynomials, Tf will be a tempered distribution if f has an algebraic growth, which means that |f(x)| < c|x|k for some k and c for large enough |x|. 1 We can use definition (1.23) to identify the space of functions Lloc(R) with a subset of the space of distributions D(R). In this sense, distributions generalize functions. When no confusion can arise we will make use of the identification f ∼ Tf and write f instead of Tf . 1.2. DISTRIBUTIONS 7

The presence of the complex conjugate of f in the definition (1.23) can be startling but it is dictated by convenience. In particular, with this choice,

Tf (φ) can be formally written as the scalar product (f, φ) of f with φ. This definition also explains the notation (1.16) which is often used for the action of a distribution on a test function.

Example 1.4. The function f(x) = sign(x) is not continuous, it is not differentiable and it is not integrable on R. It is however integrable on any finite interval [a, b] and thus defines the distribution

Z ∞ Z 0 Z ∞ Tf (φ) = dx f(x) φ(x) = − dx φ(x) + dx φ(x) . −∞ −∞ 0

Since f has an algebraic growth, Tf is a tempered distribution. The previous expression is indeed finite for all φ ∈ S(R).

Example 1.5. The Heaviside function ( 1 x ≥ 0 θ(x) = (1.24) 0 x < 0 , is similarly associated to the tempered distribution

Z ∞ Z ∞ Tθ(φ) = dx θ(x) φ(x) = dx φ(x) . −∞ 0

Example 1.6. By identifying locally integrable functions with the associ- ated distribution, we have the following inclusion: C∞(R) ⊂ D0(R). The elementary functions sin(x), cos(x), log(x),P (x), where P (x) is a polynomial are tempered distributions. The function ex is a distribution, but growing faster than a polynomial, is not tempered.

Example 1.7. A notable chain of inclusions is

2 0 0 D(R) ⊂ S(R) ⊂ L (R) ⊂ S (R) ⊂ D (R) . (1.25)

In particular we will see that each space in this chain is dense in those containing it. 8 CHAPTER 1. DISTRIBUTIONS

1.2.2 Singular distributions

Most distributions are not of the form (1.23) for any locally integrable func- tion. They are called singular and are the truly new objects. We give various examples. In order to see that a functional is a distribution we always need to check that its value on a test function is well defined and it is linear and continuous. We will often leave as an exercise to the reader to verify the continuity. This can be done with the same method used for regular distributions.

Example 1.8. The . The most famous example of singular distribution is given by the Dirac delta function which is not a function, as the name would suggest, but the distri- bution defined by

δx0 (φ) = φ(x0) . (1.26)

In other words, δx0 associates to the test function φ(x) its value at the point x0. It is obvious that the functional δx0 is well defined and linear. It is also continuous since for a sequence φn → φ in D(R)

|δx0 (φn) − δx0 (φ)| = |φn(x0) − φ(x0)| (1.27) converges to zero for n → ∞ because φn converges uniformly, and therefore also pointwise, to φ. δx0 is obviously well defined also for φ ∈ S(R) and it is therefore a tempered distribution. In physics it is customary to use the following notation Z ∞

δx0 (φ) ≡ δ(x − x0)φ(x) dx , (1.28) −∞ as if there were a function δ(x − x0) such that Z ∞ δ(x − x0)φ(x) dx = φ(x0) . (1.29) −∞

Unfortunately a function δ(x − x0) which satisfies the previous equation for all φ does not exist. Even if we will use the notation (1.29) we must keep in mind that δx0 is not a function but a truly singular distribution. When x0 = 0 we simply write δ or δ(x). 1.2. DISTRIBUTIONS 9

Example 1.9. The Principal Value. The function 1/x is not locally integrable in the neighborhood of x = 0 and it cannot be considered as a distribution. We can however define a tempered distribution T which is close enough to 1/x using the Principal Value

Z − φ(x) Z ∞ φ(x)  T (φ) = lim dx + dx . (1.30) →0 −∞ x  x The limit always exists since the sum of integrals can be written as

Z − φ(x) Z ∞ φ(x) Z ∞ φ(x) − φ(−x) dx + dx = dx , (1.31) −∞ x  x  x

φ(x)−φ(−x) and the integrand x is continuous in the neighborhood of x = 0 for all C∞(R) functions. We can thus take the  → 0 limit on the right hand side of (1.31) and write

Z ∞ φ(x) − φ(−x) T (φ) = dx . (1.32) 0 x

1 The distribution T is also written with some abuse of notations as P x and 1 can be viewed as a regularization of the function x in the neighborhood of x = 0.

Example 1.10. Other regularizations of the function 1/x are given by the distributions Z +∞ φ(x) T±(φ) = lim dx . (1.33) →0 −∞ x ± i The original singularity of the integrand is avoided by shifting the position of the singularity in the complex plane by a small imaginary part. The action of T± on a test function is often denoted with some abuse of notations as

Z φ(x) T (φ) = dx . (1.34) ± x ± i0 and the distributions T± themselves as 1/(x ± i0). The distributions T± are 1 related to P x by the identities 1 1 = P ∓ πiδ(x) . (1.35) x ± i0 x 10 CHAPTER 1. DISTRIBUTIONS

We can indeed write Z +∞ x Z +∞   T±(φ) = lim 2 2 φ(x)dx ∓ i 2 2 φ(x)dx (1.36) →0 −∞ x +  −∞ x +  The first integral on the right hand side can be written as

Z ∞ x2 φ(x) − φ(−x) 2 2 dx , (1.37) 0 x +  x and thus converges for  → 0 to the principal value (1.32). The second integral can be written, by the change of variables x = y, as

Z +∞  Z +∞ 1 ∓ i 2 2 φ(x)dx = ∓i 2 φ(y)dy , (1.38) −∞ x +  −∞ y + 1 which converges for  → 0 to ∓iφ(0) R dy = ∓iπφ(0). We then have R y2+1 Z φ(x) Z φ Z dx = P − πi δ(x)φ(x)dx (1.39) x + i0 x which, being valid for all test functions φ, gives us the identity (1.35).

Example 1.11. The finite part. The principal value for a function which behaves as 1/x2 is not well defined. We can modify the expression (1.32) as follows

Z ∞ φ(x) + φ(−x) − 2xφ0(0) T (φ) = 2 dx . (1.40) 0 x which is now well defined since the numerator is O(x2) in the neighborhood of x = 0 by Taylor theorem. T in (1.40) defines a distribution, called the 1 2 finite part and denoted as fp x2 , which regularizes the function 1/x in the neighborhood of x = 0.

A distribution is only defined by its action on test functions and, even if we often denotes distributions as ”functions” like in the case of δ(x) or 1/(x+i0), we must remember that these are formal expressions and distributions are not defined pointwise. The concept of value of the distribution T at the point x is meaningless. Only for regular distributions Tf we can identify the distribution with the function f and define the value of T at a point x as 1.3. LIMITS OF DISTRIBUTIONS 11 the value f(x), if it exists. More generally, we can study the behavior of a distribution T on a subset I of R by considering the action of T on test functions with support in I. If, on all test functions with support in I, the action of T is the same of the action of a regular distribution Tf we can say that T is equal to f in the interval I and we can talk about the values of T at points of I. For example, δ(x) is identified with the function zero and P 1/x and 1/(x ± i0) with the function 1/x on all intervals that do not contain x = 0. We can define the support of T as the closure of the complement of the biggest subset of R where it can be identified with the zero function. The support of δx0 is the point x0.

Example 1.12. Compactly supported distributions. In general, the action of a distribution T ∈ D0(R) is only defined on C∞ functions with compact support. However if T itself has compact support, it effectively truncate the domain of definition of any functions is acting on. A distribution T with compact support can act on any C∞ function. In practice, if K = Supp(T ) and φ ∈ C∞(R), we can always find φ˜ ∈ D(R) which coincides with φ in K and define T (φ) ≡ T (φ˜).

1.3 Limits of distributions

We can define a notion of convergence for distributions. We say that the 0 sequence {Tn} ∈ D (R) converges to T in the sense of distributions and we write limn→∞ Tn = T , if, for all test functions φ ∈ D(R), we have

lim Tn(φ) = T (φ) . (1.41) n→∞

When the distributions Tn are regular and associated with locally integrable functions fn, the distributional limit of Tn is not necessarily the same of the p 0 pointwise (or uniform or L ) limit of the fn. The limit in D (R) is often called limit in weak sense, since it is defined by considering the action of Tn on test functions.

inx Example 1.13. The sequence of locally integrable functions fn(x) = e for n = 1, 2, 3,... converges pointwise to 1 for x = 2πk avec k ∈ Z and is not 12 CHAPTER 1. DISTRIBUTIONS convergent for all other x. On the other hand, the sequence of distributions

Tfn converge to 0 as we see with an integration by parts Z +∞ Z +∞ −inx 1 −inx 0 n→∞ (Tfn , φ) = e φ(x)dx = e φ (x)dx −→ 0 . (1.42) −∞ in −∞ Example 1.14. The sequence of locally integrable functions

( 1 2 k k < x < k fk(x) = k ≥ 1 (1.43) 0 elsewhere converges pointwise to 0. The associated distributions converge instead to the Dirac delta function,

2 Z k

lim (Tfk , φ) = lim k dx φ(x) = lim φ(x0) = φ(0) = (δ0, φ) (1.44) k→∞ k→∞ 1 k→∞ k

1 2 where x0 is a point of the interval [ k , k ] and we used the mean value theorem for the C∞ function φ.

Example 1.15. The Dirac delta function δ(x) can be obtained as a limit of any of the following of functions   n 2 2 1 sin nx 1 sin nx2 1 1 f (x) = √ e−n x , , , (1.45) n π π x nπ x nπ x2 + 1/n2

All these functions are obtained by rescaling: fn(x) = nf(nx), where f(x) R ∞ is a locally integrable function of unit area, −∞ f(x)dx = 1. Given a test function φ, we have

∞ ∞ ∞ Z x=y/n Z n→∞ Z fn(x)φ(x)dx = f(y)φ(y/n)dy −→ φ(0) f(y)dy = φ(0) . −∞ −∞ −∞ if f is sufficiently regular. It follows that lim fn(x) = δ(x) in distributional sense. Notice that the pointwise limit of all the functions fn(x) is 0 for x 6= 0 and +∞ for x = 0. It is sometime said that the Dirac delta function δ(x) is vanishing for x 6= 0 and goes to infinity for x = 0 and it is not uncommon on physics textbook to find the writing ( 0 x 6= 0 δ(x) = (1.46) ∞ x = 0 1.4. OPERATIONS ON DISTRIBUTIONS 13

We stress again that the concept of pointwise value of a distribution f(x) is not well defined. In particular the value of δ in x = 0 makes no sense. What it is true is that δ(x) is the limit of functions fn which become more and more supported around x = 0 and whose value in the origin increases when n becomes large.

The previous example shows that the weak limit of locally integrable func- tions can be a singular distribution. In fact, one can show that all singular distributions can be obtained as a limit of locally integrable functions. One can show indeed that D(R) is dense in D0(R).

1.4 Operations on distributions

Many operations that are defined for functions can be extended to distribu- tions. The idea is to let the act on the test function. One usually defines the operation for regular distributions, move the operation on the test function and extend the definition to the singular distributions. Since test functions are C∞(R), it follows, for example, that all distributions are differentiable.

1.4.1 Change of variables

Distributions are not defined point-wise but we can nevertheless define the concept of change of variables. This is easy to do for a regular distribution

Tf , associated with the locally integrable function f(x). We want to consider now the function g(x) = f(u(x)), or in short g = f ◦ u, where u(x) ∈ C∞(R) and is one-to-one map of R to itself. We have Z +∞ Z +∞ y=u(x) f(y) −1 Tg(φ) = dxf(u(x))φ(x) = dy 0 −1 φ(u (y)) , (1.47) −∞ −∞ |u (u (y))| or  φ ◦ u−1  T (φ) = T . (1.48) f◦u f |u0 ◦ u−1| What we did was to move the operation (the change of variable) from T to the test function. Now that the operation is only acting on the test function 14 CHAPTER 1. DISTRIBUTIONS we can generalize the previous formula and define, for any distribution T , the distribution T ◦ u, defined as

T ◦ u(φ) := T (φ ◦ u−1/|u0 ◦ u−1|) . (1.49)

The definition makes sense since φ ◦ u−1/|u0 ◦ u−1| is a test function, being C∞(R) and with compact support. One can easily check that the functional T ◦ u is continuous. The above definition may look complicated and it is better to demonstrate it by examples.

Example 1.16. Translation

Given x0 ∈ R and the distribution δx0 we want to find the distribution translated by x → Ta(x) = x + a. Using (1.49) we have

Z ∞ −1 δx0 ◦ Ta(φ) = δx0 (φ ◦ Ta ) = δ(x − x0)φ(x − a) dx = φ(x0 − a) (1.50) ∞

and we find, not unexpectedly, that the translation δx0 ◦ Ta is δx0−a.

Example 1.17. Change of scale Given λ ∈ R and the distribution δ we want to find the distribution obtained by a dilatation x → Dλ(x) = λx. Using (1.49) we have

Z ∞ −1 φ(x/λ) δ ◦ Dλ(φ) = δx(φ ◦ Dλ ) = δ(x) dx = φ(0)/|λ| (1.51) ∞ |λ| and we find that δ ◦ Dλ = δ/|λ|. With the usual abuse of notation we can δ(x) write the more transparent expression δ(λx) = |λ| .

Example 1.18. Change of variables in the delta function For a more general change of variables x → u(x), with u monotonic and with a single zero in x = x0, we would obtain from (1.49)

−1 δ(u (0)) δ(x0) δ(u(x)) = 0 −1 = 0 (1.52) |u (u (0))| |u (x0)|

Notice that that δ(u(x)) evaluates the test function in the zeros of u(x), as expected, but it also multiplies it by a constant. Since the delta function is vanishing outside x = 0 we can generalize the construction by using functions 1.4. OPERATIONS ON DISTRIBUTIONS 15 u(x) which are not necessarily monotonic but have a finite number of zeros 0 xi with u (xi) 6= 0. In this case we have

X δ(xi) δ(u(x)) = (1.53) |u0(x )| i i

2 δ(x−1) δ(x+1) For example, δ(x − 1) = 2 + 2 .

1.4.2 Multiplication by a C∞ function

We define the product of a distribution T by a function g ∈ C∞ as

(gT )(φ) := T (gφ) . (1.54)

The definition makes sense since gφ is a test function, being C∞(R) and with compact support. Notice that it is important to have g ∈ C∞ in order to have gφ ∈ D(R). One can easily check that the functional gT is continuous. The definition is motivated by the case of a regular distribution Tf where indeed Z ∞ (Tgf , φ) = dx g(x)f(x)φ(x) = (Tf , gφ) . (1.55) −∞ Example 1.19. We want to show that 1 xP = 1 (1.56) x in D0(R). Notice that 1 is an element of D0(R), being locally integrable. 1 Denoting T = P x we have indeed Z xφ(x) Z ∞ xT (φ) = T (xφ) = lim dx = dxφ(x) = T1(φ) (1.57) →0 |x|≥ x −∞ since the integral in principal value of the C∞ function φ is the same as the ordinary integral. Similarly we prove 1 x = 1 . (1.58) x + i0 Example 1.20. The general solution of the algebraic equation xT = 1 for T in D0(R) is 1 T = P + cδ(x) (1.59) x 16 CHAPTER 1. DISTRIBUTIONS where c is an arbitrary constant. In fact xδ = 0 identically in D0(R) since Z ∞ xδ(φ) = δ(xφ) = δ(x)xφ(x) = 0 (1.60) −∞

1 For c = ∓iπ we have T = x±i0 , as Example 1.10 shows. Notice that the inverse in the space of distributions D0(R) is not unique.

1.4.3 Derivative of a distribution

Given a locally integrable function f whose derivative f 0 exists and it is 0 locally integrable, we can consider the distribution Tf 0 ∈ D (R) and write Z +∞ Z +∞ 0 +∞ 0 Tf 0 (φ) = dxf (x)φ(x) = [f(x)φ(x)]−∞ − dxf(x)φ (x) −∞ −∞ Z +∞ 0 0 = − dxf(x)φ (x) = −(Tf , φ ) , (1.61) −∞ where we moved the operation (the derivative) on the test function by inte- grating by parts. It is important to notice that we can do that since φ is C∞ and the boundary terms vanish since φ ∈ D(R) has compact support. This computation suggests to define the derivative T 0 for an arbitrary distribution T ∈ D0(R) as

T 0(φ) := −(T, φ0) . (1.62)

The previous definition makes sense since φ0 is still a test function. It is easy to check that T 0 is continuous. Since test functions are C∞, we can iterate the operation of derivation an infinite number of times: the n-th derivative of T will be given by

T (n)(φ) = (−1)n T (φ(n)) , (1.63) where φ(n) denotes the n-th derivative of the test function φ. Distributions are infinitely differentiable! The derivative of distributions has properties similar to those of functions.

If T1 and T2 are two distributions we have

0 0 0 (λ1T1 + λ2T2) = λ1T1 + λ2T2 ∀ λ1, λ2 ∈ C , (1.64) 1.4. OPERATIONS ON DISTRIBUTIONS 17

0 0 (T (x − x0)) = T (x − x0) ∀x0 ∈ R , (1.65) (T (ax))0 = aT 0(ax) . (1.66)

All locally integrable functions f, even if not differentiable in ordinary 0 sense in one or more points, have a derivative Tf when considered as distri- butions.

Example 1.21. The derivative of the distribution associated with the Heav- iside function θ(x) of Example 1.5 is Z ∞ Z ∞ 0 0 0 0 Tθ(φ) = −Tθ(φ ) = − dx θ(x)φ (x) = − dx φ (x) = φ(0) . (1.67) −∞ 0

0 We thus find Tθ = δ. The standard derivative of the Heaviside function does not exist, but the distributional derivative does and it is given by the delta function. As expected, the distributional derivative of θ(x) vanishes outside x = 0. The discontinuity (jump) in x = 0 gives rise to a delta function. With 0 0 some abuse of language we will also write θ instead of Tθ. More generally, the distributional derivative allows to make sense of the derivative of discontinuous functions. Given a function f which is continuous and differentiable with continuous derivative everywhere except at the point 0 1 x0 where it has a jump discontinuity, we define the element f ∈ Lloc(R) as the (discontinuous) locally integrable function that coincides with the ordinary 1 derivatives of f(x) in R − {x0} . We have Z ∞ 0 0 Tf (φ) = − dx f(x)φ (x) = −∞ Z x0 Z ∞ − f(x)φ(x)|x0 + dx f 0(x)φ(x) − f(x)φ(x)|∞ + dx f 0(x)φ(x) −∞ x0 −∞ x0 Z ∞ 0 + − = dx f (x)φ(x) + [f(x0 ) − f(x0 )]φ(x0) (1.68) −∞ where we have integrated by parts in the intervals [−∞, x0] and [x0, +∞]. + − By defining the discontinuity disc(f, x0) = f(x0 ) − f(x0 ), we can write

0 Tf = Tf0 + disc(f, x0)δx0 . (1.69)

1 0 We can define the value of f in x0 arbitrarily: the value in x0 does not matter for integration and functions in L1 which differs at a point are identified in the L1. 18 CHAPTER 1. DISTRIBUTIONS

Example 1.22. The function ln |x| is locally integrable in the neighborhood of x = 0 and thus defines a regular distribution. As a function it is not differentiable in ordinary sense in x = 0. In D0(R) we have the identity d ln |x| 1 = P (1.70) dx x as the following computation shows

Z +∞ Z 0 0 0 Tln |x|(φ) = − dx ln |x|φ (x) = − lim dx ln |x|φ (x) = −∞ →0 |x|≥  Z 1  Z ∞ 1 lim ln  (φ() − φ(−)) + dx φ(x) = P dx φ(x) . →0 |x|≥ x −∞ x

The boundary terms vanish since

φ() − φ(−) lim( ln ) ∼ 0(2φ0(0)) = 0 . (1.71) →0 

Similarly we can show that in D0(R) d2 ln |x| 1 = −fp . (1.72) dx2 x2

1.4.4 of distributions

Distributions are not defined pointwise and therefore we cannot multiply two distributions. For example, the product δ(x)2 does not make any sense: formally we would write Z ∞ δ(x)δ(x)φ(x)dx = δ(0)φ(0) = ∞ . (1.73) −∞ The natural product on distributions is the convolution. The convolution of two functions f and g is defined as the integral Z ∞ (f ∗ g)(x) = dx0f(x0)g(x − x0) . (1.74) −∞

One can show that if f, g ∈ L1(R) then f ∗ g ∈ L1(R). More generally, if f and g are locally integrable and one of them has compact support, then f ∗ g is again locally integrable. It is easy to see that the convolution of functions 1.4. OPERATIONS ON DISTRIBUTIONS 19

satisfies the following properties. Given three functions f1, f2 et f3, two of which at least with compact support, and two complex numbers λ, µ we have

f1 ∗ f2 = f2 ∗ f1 , (1.75)

(f1 ∗ f2) ∗ f3 = f1 ∗ (f2 ∗ f3) , (1.76)

f1 ∗ (λf2 + µf3) = λ(f1 ∗ f2) + µ(f1 ∗ f3) , (1.77) 0 0 0 (f1 ∗ f2) = f1 ∗ f2 = f1 ∗ f2 , (1.78) the last equation being a consequence of the obvious identity

Z ∞ Z ∞ dx0f(x0)g(x − x0) = dx0f(x − x0)g(x0) , (1.79) −∞ −∞ obtained by changing variables x0− > x − x0 in the integral. As usual, we try to find a convenient definition for the convolution of two distributions by considering first the case of regular distributions. Consider then two locally integrable functions f and g with at least one of them with compact support. The regular distribution Tf∗g acts as

Z ∞ Z ∞  Tf∗g(φ) = dx dyf(y)g(x − y) φ(x) −∞ −∞ Z ∞ Z ∞  = dy f(y) dx g(x)φ(x + y) . (1.80) −∞ −∞

We can interpret the last expression as follows. We first act with the distri- bution Tg of variable x on the function φ(x + y) considering y as a parameter and then we act on the result, which is a function of y, with the distribution

Tf . Using the notation (1.16) and abusing it we can write

(Tf (y), (Tg(x), φ(x + y)) , (1.81) where we indicated explicitly the variables the distributions are acting on. ∞ ∞ Notice that (Tg(x), φ(x + y)) is a C function of y, since φ is C . If g has compact support, also (Tg(x), φ(x + y)) is and it is a test function in D(R) to which we can apply Tf . If g is not compactly supported, (Tg(x), φ(x + y)) is not a test function and the previous expression does not make sense.

However, we assumed that if g is not compactly supported, f is. Tf can 20 CHAPTER 1. DISTRIBUTIONS

∞ then act on any C function because the support of Tf effectively reduce the integral to a compact set (see Example 1.12). We are now ready to generalize the construction to distributions. Recall from Section 1.2.2 and Example 1.12 that the concept of support of a distri- bution is well defined and that distributions with compact support can act on any C∞ function. Given two distributions T and S, one of them with compact support, we define the convolution T ∗ S ∈ D(R) as T ∗ S(φ) = (T (y), (S(x), φ(x + y)) . (1.82) One can show that (S(x), φ(x + y)) is a C∞ function of y and that it is compactly supported if S is. If S has compact support, (S(x), φ(x+y)) is then a test function and the previous definition makes sense for any distribution T . If S is not compactly supported T will be and its action on (S(x), φ(x + y)) is well defined even if the function is only C∞. One can show that the convolution for distributions satisfies the proper- ties (1.75)-(1.78). Example 1.23. The delta function is an for the convolution product δ ∗ T = T ∗ δ = T. (1.83) With an obvious abuse of notation we have indeed Z ∞ Z ∞ (δ ∗ T, φ) = dx dy δ(x)T (y)φ(x + y) = dy T (y)φ(y) = (T, φ) . −∞ −∞

1.4.5 Fourier Transform of Distributions

We can actually extend the Fourier Transform to tempered distributions. As usual, we first look at the case of regular distributions. The regular distribution Tgˆ corresponding to the Fourier Transform of a function g ∈ S(R) acts on an element ϕ ∈ S(R) as Z ∞ (Tgˆ, ϕ) = dp gˆ(p) ϕ(p) . (1.84) −∞ By inserting the explicit form of the Fourier Transform of g, and doing formal manipulations, one can also write Z ∞ Z ∞ 1 −ipx (Tgˆ, ϕ) = √ dx g(x) dp ϕ(p) e , (1.85) 2π −∞ −∞ 1.4. OPERATIONS ON DISTRIBUTIONS 21

which can be interpreted as the action of the regular distribution Tg on the Fourier Transform of ϕ. In formulae, Z ∞ Z ∞ (Tgˆ, ϕ) = dp gˆ(p) ϕ(p) = dx g(x)ϕ ˆ(x) = (Tg, ϕˆ). (1.86) −∞ −∞ A generalization of (1.86) leads to the definition of the Fourier Transfom of a generic distribution T ∈ S0(R) Tˆ(ϕ) = T (ϕ ˆ) . (1.87)

This definition extends the Fourier Transform from S(R) to S0(R). It is not hard to prove that the extension is a linear, bijective and bicontinuous operator from S0(R) to S0(R). Notice that we cannot define the Fourier Transform for all distributions: the definition (1.87) would fail for T ∈ D0(R) since the Fourier Transform of φ ∈ D(R) is not necessarily in D(R). Example 1.24. The Fourier Transform of the Dirac delta function is a con- stant: δˆ = √1 . Indeed 2π 1 Z ∞ 1 (δ,ˆ φ) = (δ, φˆ) = φˆ(0) = √ dxφ(x) = (√ , φ) . (1.88) 2π −∞ 2π Notice that we could have obtained the same result by the formal manipula- tion 1 Z +∞ 1 δˆ(p) = √ dx δ(x) e−ipx = √ . (1.89) 2π −∞ 2π Consequently, the inverse theorem tells us that the Fourier Transform of the √ function 1 is F(1)(λ) = 2πδ(p). This can be interpreted as an integral representation of the delta function, namely 1 Z +∞ δ(x) = dp eipx . (1.90) 2π −∞ This integral obviously does not exist in conventional sense, however it makes sense in distributional sense, namely when applied to a test function. Example 1.25. Any locally integrable function with algebraic growth is in S0(R) and has a well defined Fourier Transform. Take for example a Pn k polynomial P (x) = k=0 akx . We find n √ n X k k X k (k) F(P )(p) = aki ∂p F(1)(p) = 2π aki δ (p) , (1.91) k=0 k=0 22 CHAPTER 1. DISTRIBUTIONS which is a distribution with support in the point p = 0. Vice-versa, the Fourier Transform of a linear combination of δ(x) and its derivatives is a polynomial. One can show more generally that the Fourier Transform of any distribution with compact support is a C∞ function with algebraic growth.

1 Example 1.26. Consider the Fourier Transform of the distribution x+i0 . We can easily compute it with the residue theorem 1 Z ∞ e−ixp √ √ lim √ = − 2πi lim e−pθ(p) = − 2πiθ(p) , (1.92) →0 2π −∞ x + i →0 where we close the contour in the upper (lower) half plane for p < 0 (p > 0). The inverse Fourier Transform gives us a useful integral representation i Z ∞ = eixpdp . (1.93) x + i0 0 It is interesting to observe that this formula can be obtained as a formal limit for  → 0 of i Z ∞ = eixp−pdp , (1.94) x + i 0 where now the integral converges in traditional sense for all  > 0. Some of the properties of Fourier Transform of functions extend to the Fourier Transform of distributions. For instance

F(T (n))(p) = (ip)n FT (p) (1.95) dn F(xnT )(p) = in FT (p) (1.96) dpn

F(T1 ∗ T2) = FT1 FT2 (1.97)

The last property deserves some comments. First of all, we need to assume that at least one of the two distributions T1 and T2 has compact support, since, as we saw in Section 1.4.4, this a necessary condition for the convolution

T1 ∗T2 to exist. Under this hypothesis, one can show that T1 ∗T2 is a tempered distribution, so that the left hand side of (1.97) is well defined. Moreover, as we already mentioned, the Fourier Transform of a tempered distribution with compact support is actually a C∞ function with algebraic growth. It is only this property that allows to make sense of the right hand side of (1.97) as the product of a tempered distribution with a C∞ function. Remember that the product of two distributions is generically not defined. 1.5. EXERCISES 23 1.5 Exercises

Exercise 1.1. Show that the k-th derivative of the delta function acts as (δ(k), φ) = (−1)kφ(k)(0).

2 R ∞ R ∞ Exercise 1.2. Define a D(R ) distribution by (f, φ) = 0 0 φ(x, y)dxdy. Com- pute ∂2f/∂x∂y.

d2 ln |x| 1 Exercise 1.3. Show that dx2 = −fp x2 in D(R).

Exercise 1.4. Define the distribution F = log(x + i0) in D(R) by (F, φ) = R ∞ lim→0 −∞ log(x + i)φ(x)dx. 1) Show that log(x + i0) = log |x| + iπθ(−x). Hint: analyze log argument in the complex plane. d 2) Show that dx log(x + i0) = 1/(x + i0) 3) Check consistency of 1) and 2) with (1.35) and Example 1.22.

Exercise 1.5. Solve for T the algebraic equation P (x)T = Q(x) in D(R), where P and Q are real polynomials with distinct zeros.

2 Exercise 1.6. Solve for T the differential equation x dT/dx = 1 in D(R). Hint: solve it first for functions T and add delta-function like ambiguities.

P∞ Exercise 1.7. Show that T = n=−∞ δ(x − n) is a tempered distribution. 1) Expand formally the periodic distribution T in Fourier series in [0, 2π] 2) Show that the result in 1) is equivalent to the Poisson resummation formula valid P∞ ˆ P∞ ˆ R ∞ −2πix for functions in S(R): k=−∞ φ(k) = n=−∞ φ(n) where φ(p) = −∞ φ(x)e is the Fourier transform of φ.