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24. The Analytic Continuation of Distributions
114
11. DISTRIBUTIONS
24. The Analytic Continuation of Distributions We have made a canonical identification of locally integrable functions f ( x ) with distributions, and this identification is at the base of our theory. However, there are functions arising quite naturally in analysis which are not locally integrable, and it is therefore desirable to extend the relation between functions and distributions to a wider class of functions. Let us suppose that the functions in question have isolated singularities. It is therefore sufficient to suppose that we are concerned with a function f ( x ) , locally integrable except in a neighborhood of the origin; we shall also suppose that the singularity is not too bad, more exactly, that for some integer N the function IxlNf(x)is integrable over 1x1 5 1. Iff(x) is to be identified with the distribution T , then, for testfunctions rp whose support does not contain the origin, we ought to have
The same formula ought to be valid whenever the testfunction q ( x ) is such that the integral above exists, in particular, then, if rp(x) satisfies an inequality of the form Icp(x)l S ClxlN.Making use, then, of the Taylor expansion for rp, we have rp(4= P(x)x(x) + K4
+
9
where the testfunction ~ ( xequals ) 1 on a suitable sphere 1x1 6 R, and where I$(x)l 5 C(xINif the degree of the polynomial P(x) is N - 1. Since T has already been defined for functions of the type $(x), it remains only to define it for functions of the type P(x)x(x),and here we will have
the sum being taken over all indices tl with ItlI 6 N - 1, and therefore T(P1) = C,DU6)(rp),where C, = (l/a!)T(xdX(x)).It follows that we have very little choice in determining a distribution T which is to correspond to f ( x ) ; T is completely determined on sets not containing the origin, and we are at liberty to determine only the coefficients C, in the polynomial P(D ) = c C , D " which is applied to 6. Moreover, these coefficients cannot be determined arbitrarily, for we should want to assign a distribution to the function f ( x ) in a way consistent with, say, differentiation: the function D"f has a singularity at the origin, and should correspond to a distribution S for which
(c
24.
THE ANALYTIC CONTINUATION OF DISTRIBUTIONS
115
D"T = S. Hence we seek a recipe which will provide a consistent determination of a distribution T to correspond to a functionf(x) having a certain type of singularity. While no solution to this problem exists in general, those functions f ( x ) having singularities of an analytic nature do admit such natural extensions to distributions, the extension being found by a process of analytic continuation to which this section is devoted. Let us remark that we have already twice encountered this kind of problem: the derivative of the function g(x) which vanished for x < 0 and was given by x-'" for x > 0 was given in the sense of distributions by a function f ( x ) which coincided with the usual derivative away from the origin, but had a nonintegrable singularity at the origin. We also encountered this problem when we sought to divide a distribution in one variable by the function a(x) = x. We proceed to the definition of a distribution depending analytically on a parameter, that is, a distribution-valued analytic function. Let G be a region in the complex 1-plane and for every 1in G let TAbe a distribution in B'(0). This function is said to be analytic, if for every testfunction cp in B(0) the numerically valued function TA(cp)is analytic for 1in G. The usual results concerning analytic functions may now be extended to the functions TA, for example, we consider the derivative with respect to 1
which is evidently a linear functional in cp; it is a distribution, since it is the limit of a sequence of difference quotients TA+,(cp)- TA(cp) h and we have already established that the space of distributions is weak-star sequentially complete. It is also evident that the derivative is itself an analytic distribution-valued function in G, and so are the higher derivatives. In the same way, we can consider the Taylor expansion: for any testfunction cp, we have TA(q)= ak(cp)(rl -
1
the series converging in the largest circle about 1, in G. The coefficients are given by 1 dkTA ak(q) = - -(cp) taken at i= I,. k ! dLk These coefficientsare distributions, and so TAis the limit of the distributions which are the partial sums of the series. Hence 1 dkTA TA =C - -( I - I,)k k! dlk
116
11. DISTRIBUTIONS
with the series converging in the space of distributions in the largest circle about 1, which is contained in G. Similar arguments, all based on the theorem that the pointwise limit of a sequence of distributions is a distribution, permit us to speak of a Laurent expansion of a distribution and of certain integrals of distributions along a path in G. We also obtain the concept of analytic continuation: if H i s a region containing G as a proper subset, and if the numerically valued functions T,(q) are all analytically continuable to H , then the distribution T, can be continued to a distribution valued function analytic in the larger domain; all that we need for the argument is to notice that the circles of convergence of the Taylor series are now larger: they are the largest circles about their centers which are contained in H. Let P ( D ) be a polynomial in the differential operator with Cm-coefficients. The distribution P ( D ) T , is evidently also analytic in 1and if we make an analytic continuation of T, from G to H, the function P(D)T, is similarly continuable, and the continuation is still the result of applying the differential operator to the continuation of T,. A similar assertion can be made if we consider the multiplication of T, by the smooth function a(x) in &(R): a(x)T, is analytically continuable and its continuation is the result of multiplying the continuation of TAby the function a(x). Finally, if we suppose that R = Rnand I a suitable linear transformation of that space onto itself, we find that the analytic continuation of T , 1 coincides with the analytic continuation of T, composed with the mapping 1. In particular, if T, is homogeneous of order k, so is its analytic continuation, and indeed, if k is itself an analytic function of A the extension is homogeneous of order k(1). We give two illustrations of this important topic. Consider first R = R' and the distribution which corresponds to the function which vanishes for x < 0 and is given by x A - l for x > 0; if Re[A] > 0, the function is locally integrable and the distribution is clearly analytic in the right half-plane of A. For any testfunction q ( x ) we pass to the Taylor expansion about the origin, writing 0
q(x)= P(x)
+ xNg(x).
We then have
The function on the left-hand side we know to be analytic in the right halfplane; the first term on the right-hand side is an entire function of 1. The
24.
THE ANALYTIC CONTINUATION OF DISTRIBUTlONS
117
middle term is analytic in the half-plane Re[1] > -N, and the last term is a rational function of 1 which we can compute explicitly: it is
x-- 1 + k
N-l
1 Dkcp(0)
k=Ok!
*
Since cp and N were arbitrary, it is evident that we can continue the distribution from the right half-plane to the whole plane, with the exception of simple poles at the origin and the negative integers. Since these poles are exactly those of the Gamma function, we find it advantageous to consider, instead, the distribution TA= x"-'/r(A) with the convention that the distribution is 0 on the left half of the real axis; TAadmits an analytic continuation to an entire distribution-valued function. For 1 > 1 we obviously have Ti = TA-, where the prime denotes differentiation with respect to x; by analytic continuation, then, that relation holds everywhere. For 1 = 1, Ti = Tl = Y ( x ) , the Heaviside function, equal to + 1 for x > 0 and equal to 0 for x c 0; we know its derivative, Y' = 6, and hence infer that To = 6 , and therefore T-k = Dk6 for all k 2 0. We note that this is consistent with the general rule that TA is homogeneous of degree 1- 1. For our second example, we consider the function r'-" on the space R" where r = 1x1; if Re[1] > 0, this is a locally integrable function which is homogeneous of degree 1 - n. As in the previous example, the distribution may be continued analytically over into the left half-plane, although poles will appear at some of the negative integers. For any testfunction cp, we may write
where P ( x ) is the Taylor expansion of cp(x) about the origin, taken to all terms of order N - 1. The first term is entire as a function of 1,the second is analytic in the half-plane Re[A] > -N, and the third term is a rational function of 1 which we may compute explicitly:
where the sum is taken over all indices for which la1
The coefficient reduces to
5 N - 1 and
118
11. DISTRIBUTIONS
and it is easy to see that the integral vanishes if the multi-index a contains an odd integer. Hence a must be of the form 2p and our distribution can only have poles at the origin and the even negative integers. It is therefore convenient to pass to the distribution defined by the equation T A= ra-"/l-(l/2)to obtain a distribution analytic in the entire plane. It is easy to verify that we have ATA= 2(L - n)T,-, and since T2= (2 - n)w, E where E is the fundamental solution for the Laplacian satisfying AE = 6, we infer that To = (0,/2)6 and that T - z k is a constant multiple of Ak6 for integers k > 0. Here we are supposing n 3, so that E is the fundamental solution for the Laplacian. For some applications, it is desirable to pass to the Riesz kernel: the distribution
which now has simple poles at values l of the form n + 2k for integers k 2 0. This analytic distribution satisfies the equation - A R , = R , - 2 and R , = - E. Thus R - 2 k= ( -A)kd if k is an integer 20.When 1 is real and in the interval (0, n), the kernel R , is a positive locally integrable function. In conclusion, we should remark that the simple formulas which we have written to obtain the analytic continuation of a distribution are rarely the best: the continuation is independent of the particular formula with which we compute, and the astute selection of such a formula will always be profitable in any particular case.
25. The Convolution of a Distribution with a Testfunction In this section we consider only distributions defined on the whole space R". We have already introduced a special notation for the testfunctions cp o I and the distributions T 0 I when I is the reflection of R" through the origin, v " namely, T 1 = T and cp 1 = 4; it is desirable also to introduce F h = r - h and d.= (- 1)l"lD". It is then easy to verify the identities 0
0
(D'P)"
=fib@,
(rP(Ph)"=yh@,
(D"T)" = 3?, (FhT)" = yhf, and it will be a general rule that the reflection of a product is the product of the reflections.
11. DISTRIBUTIONS
24. The Analytic Continuation of Distributions We have made a canonical identification of locally integrable functions f ( x ) with distributions, and this identification is at the base of our theory. However, there are functions arising quite naturally in analysis which are not locally integrable, and it is therefore desirable to extend the relation between functions and distributions to a wider class of functions. Let us suppose that the functions in question have isolated singularities. It is therefore sufficient to suppose that we are concerned with a function f ( x ) , locally integrable except in a neighborhood of the origin; we shall also suppose that the singularity is not too bad, more exactly, that for some integer N the function IxlNf(x)is integrable over 1x1 5 1. Iff(x) is to be identified with the distribution T , then, for testfunctions rp whose support does not contain the origin, we ought to have
The same formula ought to be valid whenever the testfunction q ( x ) is such that the integral above exists, in particular, then, if rp(x) satisfies an inequality of the form Icp(x)l S ClxlN.Making use, then, of the Taylor expansion for rp, we have rp(4= P(x)x(x) + K4
+
9
where the testfunction ~ ( xequals ) 1 on a suitable sphere 1x1 6 R, and where I$(x)l 5 C(xINif the degree of the polynomial P(x) is N - 1. Since T has already been defined for functions of the type $(x), it remains only to define it for functions of the type P(x)x(x),and here we will have
the sum being taken over all indices tl with ItlI 6 N - 1, and therefore T(P1) = C,DU6)(rp),where C, = (l/a!)T(xdX(x)).It follows that we have very little choice in determining a distribution T which is to correspond to f ( x ) ; T is completely determined on sets not containing the origin, and we are at liberty to determine only the coefficients C, in the polynomial P(D ) = c C , D " which is applied to 6. Moreover, these coefficients cannot be determined arbitrarily, for we should want to assign a distribution to the function f ( x ) in a way consistent with, say, differentiation: the function D"f has a singularity at the origin, and should correspond to a distribution S for which
(c
24.
THE ANALYTIC CONTINUATION OF DISTRIBUTIONS
115
D"T = S. Hence we seek a recipe which will provide a consistent determination of a distribution T to correspond to a functionf(x) having a certain type of singularity. While no solution to this problem exists in general, those functions f ( x ) having singularities of an analytic nature do admit such natural extensions to distributions, the extension being found by a process of analytic continuation to which this section is devoted. Let us remark that we have already twice encountered this kind of problem: the derivative of the function g(x) which vanished for x < 0 and was given by x-'" for x > 0 was given in the sense of distributions by a function f ( x ) which coincided with the usual derivative away from the origin, but had a nonintegrable singularity at the origin. We also encountered this problem when we sought to divide a distribution in one variable by the function a(x) = x. We proceed to the definition of a distribution depending analytically on a parameter, that is, a distribution-valued analytic function. Let G be a region in the complex 1-plane and for every 1in G let TAbe a distribution in B'(0). This function is said to be analytic, if for every testfunction cp in B(0) the numerically valued function TA(cp)is analytic for 1in G. The usual results concerning analytic functions may now be extended to the functions TA, for example, we consider the derivative with respect to 1
which is evidently a linear functional in cp; it is a distribution, since it is the limit of a sequence of difference quotients TA+,(cp)- TA(cp) h and we have already established that the space of distributions is weak-star sequentially complete. It is also evident that the derivative is itself an analytic distribution-valued function in G, and so are the higher derivatives. In the same way, we can consider the Taylor expansion: for any testfunction cp, we have TA(q)= ak(cp)(rl -
1
the series converging in the largest circle about 1, in G. The coefficients are given by 1 dkTA ak(q) = - -(cp) taken at i= I,. k ! dLk These coefficientsare distributions, and so TAis the limit of the distributions which are the partial sums of the series. Hence 1 dkTA TA =C - -( I - I,)k k! dlk
116
11. DISTRIBUTIONS
with the series converging in the space of distributions in the largest circle about 1, which is contained in G. Similar arguments, all based on the theorem that the pointwise limit of a sequence of distributions is a distribution, permit us to speak of a Laurent expansion of a distribution and of certain integrals of distributions along a path in G. We also obtain the concept of analytic continuation: if H i s a region containing G as a proper subset, and if the numerically valued functions T,(q) are all analytically continuable to H , then the distribution T, can be continued to a distribution valued function analytic in the larger domain; all that we need for the argument is to notice that the circles of convergence of the Taylor series are now larger: they are the largest circles about their centers which are contained in H. Let P ( D ) be a polynomial in the differential operator with Cm-coefficients. The distribution P ( D ) T , is evidently also analytic in 1and if we make an analytic continuation of T, from G to H, the function P(D)T, is similarly continuable, and the continuation is still the result of applying the differential operator to the continuation of T,. A similar assertion can be made if we consider the multiplication of T, by the smooth function a(x) in &(R): a(x)T, is analytically continuable and its continuation is the result of multiplying the continuation of TAby the function a(x). Finally, if we suppose that R = Rnand I a suitable linear transformation of that space onto itself, we find that the analytic continuation of T , 1 coincides with the analytic continuation of T, composed with the mapping 1. In particular, if T, is homogeneous of order k, so is its analytic continuation, and indeed, if k is itself an analytic function of A the extension is homogeneous of order k(1). We give two illustrations of this important topic. Consider first R = R' and the distribution which corresponds to the function which vanishes for x < 0 and is given by x A - l for x > 0; if Re[A] > 0, the function is locally integrable and the distribution is clearly analytic in the right half-plane of A. For any testfunction q ( x ) we pass to the Taylor expansion about the origin, writing 0
q(x)= P(x)
+ xNg(x).
We then have
The function on the left-hand side we know to be analytic in the right halfplane; the first term on the right-hand side is an entire function of 1. The
24.
THE ANALYTIC CONTINUATION OF DISTRIBUTlONS
117
middle term is analytic in the half-plane Re[1] > -N, and the last term is a rational function of 1 which we can compute explicitly: it is
x-- 1 + k
N-l
1 Dkcp(0)
k=Ok!
*
Since cp and N were arbitrary, it is evident that we can continue the distribution from the right half-plane to the whole plane, with the exception of simple poles at the origin and the negative integers. Since these poles are exactly those of the Gamma function, we find it advantageous to consider, instead, the distribution TA= x"-'/r(A) with the convention that the distribution is 0 on the left half of the real axis; TAadmits an analytic continuation to an entire distribution-valued function. For 1 > 1 we obviously have Ti = TA-, where the prime denotes differentiation with respect to x; by analytic continuation, then, that relation holds everywhere. For 1 = 1, Ti = Tl = Y ( x ) , the Heaviside function, equal to + 1 for x > 0 and equal to 0 for x c 0; we know its derivative, Y' = 6, and hence infer that To = 6 , and therefore T-k = Dk6 for all k 2 0. We note that this is consistent with the general rule that TA is homogeneous of degree 1- 1. For our second example, we consider the function r'-" on the space R" where r = 1x1; if Re[1] > 0, this is a locally integrable function which is homogeneous of degree 1 - n. As in the previous example, the distribution may be continued analytically over into the left half-plane, although poles will appear at some of the negative integers. For any testfunction cp, we may write
where P ( x ) is the Taylor expansion of cp(x) about the origin, taken to all terms of order N - 1. The first term is entire as a function of 1,the second is analytic in the half-plane Re[A] > -N, and the third term is a rational function of 1 which we may compute explicitly:
where the sum is taken over all indices for which la1
The coefficient reduces to
5 N - 1 and
118
11. DISTRIBUTIONS
and it is easy to see that the integral vanishes if the multi-index a contains an odd integer. Hence a must be of the form 2p and our distribution can only have poles at the origin and the even negative integers. It is therefore convenient to pass to the distribution defined by the equation T A= ra-"/l-(l/2)to obtain a distribution analytic in the entire plane. It is easy to verify that we have ATA= 2(L - n)T,-, and since T2= (2 - n)w, E where E is the fundamental solution for the Laplacian satisfying AE = 6, we infer that To = (0,/2)6 and that T - z k is a constant multiple of Ak6 for integers k > 0. Here we are supposing n 3, so that E is the fundamental solution for the Laplacian. For some applications, it is desirable to pass to the Riesz kernel: the distribution
which now has simple poles at values l of the form n + 2k for integers k 2 0. This analytic distribution satisfies the equation - A R , = R , - 2 and R , = - E. Thus R - 2 k= ( -A)kd if k is an integer 20.When 1 is real and in the interval (0, n), the kernel R , is a positive locally integrable function. In conclusion, we should remark that the simple formulas which we have written to obtain the analytic continuation of a distribution are rarely the best: the continuation is independent of the particular formula with which we compute, and the astute selection of such a formula will always be profitable in any particular case.
25. The Convolution of a Distribution with a Testfunction In this section we consider only distributions defined on the whole space R". We have already introduced a special notation for the testfunctions cp o I and the distributions T 0 I when I is the reflection of R" through the origin, v " namely, T 1 = T and cp 1 = 4; it is desirable also to introduce F h = r - h and d.= (- 1)l"lD". It is then easy to verify the identities 0
0
(D'P)"
=fib@,
(rP(Ph)"=yh@,
(D"T)" = 3?, (FhT)" = yhf, and it will be a general rule that the reflection of a product is the product of the reflections.