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25. The Convolution of a Distribution with a Test Function
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.
25.
THE CONVOLUTION OF A DISTRIBUTION WITH A TESTFUNCTION
119
When cp is a testfunction and T a distribution, we define the convolution of T with rp as the function (or distribution) which follows:
49 = (T * cp)(x) = T ( F xG)
.
We have already seen that a(x) belongs to the class b(R")of C"-functions on the space. It is then immediate that ( y h
* cp)(x)
a)(x) = ( y h
= ( T * y h cp)(x)
and passing to difference quotients we obtain, in general (D"u)(x)= (D"T * cp)(x) = (T
* D"cp)(X).
Another useful and easy identity is ( T * cp)" = ? * @. When T has a compact support T * cp also has a compact support, since for large 1x1 the testfunction YxG has a support disjoint from that of T ; the convolution is therefore another testfunction. Another obvious but important fact is the following:
4 0 ) = (T * cp)(O) = T(,Fofj) = T(i0) ;
hence T(cp)= (T * G)(O). When the distribution T is given by the locally integrable function f ( x ) , we have
( T * cp)(x> = /S(X
- Y>cp(Y> dY,
which is the usual definition for the convolution of two functions. If we convolute two testfunctions cp and rl/ we obtain a third: x ( x ) = (cp =f
* *>(XI
d x - Y)*(Y) dY.
Of course, the integral above is taken only over a compact set, since the support of II/ is compact, and we find it convenient to approximate the integral by a sequence of Riemann sums formed in the following way. The domain of integration F is written as a finite union of disjoint measurable sets F i , each
120
11. DISTRIBUTIONS
of diameter smaller than mating sum is
E;
a point y i is chosen in each Fi, and the approxi-
= S,(X)
.
The approximating sums are themselves all testfunctions, being finite linear combinations of translates of cp, and there exists a fixed compact set K which supports all the testfunctions S, ,namely, the set of all points x whose distance from F is at most twice the diameter of the support of cp. The sums S,(x) con) every x as E diminishes to 0 and those sums are uniformly verge to ~ ( x at bounded :
Moreover, the derivatives D"S,(x) are the corresponding Riemann sums for the convolution (D'cp * i,b)(x) and are therefore uniformly bounded and converge pointwise. Thus the Arzela-Ascoli theorem guarantees that the test) the space 9 ( R " ) ;it is this circumstance that functions s , ( ~ converge ) to ~ ( xin enables us to prove the following theorem.
Theorem:
( T * c p ) * J I =T*(cp*Il/).
PROOF: ( T * x)(x)
= T(Y,i)
=
jv * cp)(x - Y ) W ) d y *
= ((T cp)
*W)
We have earlier defined the regularization of a locally integrable function
f ( x ) : we took a testfunction cp(x) which was even and positive, and for which jcp(x)dx = 1, and defined cp,(x) as E-"((P 1;') = (l/e")cp(x/~).The regularization was the C"-function &(x) = (f*rp&) and we showed that the regularizations converge tof(x) in any reasonable sense. In particular, whenf((x) was since a testfunction, the regularizations converged tof(x) in the topology of 9, 0
25.
THE CONVOLUTIONOF A DISTRIBUTION WITH A TESTFUNCTION
121
a sequence of them had a fixed compact support and converged uniformly, as well as all of the derivatives. We can now extend the idea of regularization to general distributions, not just locally integrable functions as follows: T, = T,(x) = ( T * cp,)(x) ;
this is a Cm-function, and for any testfunction IcI(x), T,($)
* bm) * cp, * Il/)(O) = T * (cp, * Il/)(O) = T(cp, * IcI) = (T,
= (T
and this converges with diminishing E to T($). Thus, the regularizations converge to Tin the space of distributions. Let the distribution T be fixed: it is then clear that the mapping which carries the testfunction cp(x) into the C"-function a(x) = (T * cp)(x) is a linear mapping of 9 into d which commutes with translation and which is sequentially continuous, a convergent sequence in 9 being carried into a convergent sequence in 6. The next theorem assures us that every such mapping is a convolution. Let 2 be a linear mapping of 9 ( R " )into b(R")which comTheorem: mutes with translation: r h ( y q ) = y(rhq), and which is sequentially continuous: (P, converging to 0 in 9 implies 2cp,converging to 0 in &; then there exists a unique distribution T such that 9 c p = T * cp for all cp. PROOF: The evaluation functional ycp(0) is a linear functional orP9 and from the continuity of the mapping it is even a sequentially continuous linear functional on 9, hence a distribution. We write, then, Ycp(0)= i'(cp), and now (dRcp)(x)= 2'(F-x cp)(O) = F(2Cxcp)= T ( Y x ;P) = ( T * cp)(x). The distribution T is uniquely determined since its regularizations are, T, being the image under 2 of the function cp,, and these regularizations converge to T as the E converges to 0. When the distribution T has compact support, the corresponding convolution mapping carries the space of testfunctions into itself. The previous theorem admits an easy extension to one asserting that the linear, translational invariant mappings of 9 into itself which are sequentially continuous are convolutions with distributions in 6'. In fact, the only point in the proof which is not immediate is the compactness of the support of T, however, if that set were not compact, there would be a sequence of points x, in the support of ?
122
11. DISTRIBUTIONS
having no finite limit point, and each x, would be surrounded by a neighborhood of small diameter supporting a testfunction $, for which ?($,) = I . We 19, , a system of translate these testfunctions to the origin, forming cp, = .Yxn testfunctions supported by the unit sphere. For a suitable choice of constants c, converging rapidly to 0, the sequence c, rp, converges to 0 in La, while the system of their convolutions with T is not supported by any fixed compact, hence does not converge to 0 in 9. Let Tbe the 6-distribution and cp a testfunction;(6 * cp)(x) = 6(Yx@)= cp(x) and so 6 corresponds to the identity mapping of 9 into itself. More, generally, then, for any polynomial P with constant coefficients,the convolution with the distribution P ( D ) 6 is the mapping which carries cp into P(D)rp. In a similar way we see that the translation operator y h itself corresponds to convolution with the distribution y h 6 ,and this distribution is the measure consisting of a unit mass at the point x = h. If the distribution T has compact support and a(x) is a function in the class 6 we can obviously form the convolution (T * u)(x) = T ( F x a), whether or not a(x) is in La; the convolution is again in 6 and it is easy to verify that Du(T* a) = (D"T* a ) = T * ( P a ) as well as fh(T*U)=FhT*U = T*yhU.
If a sequence a, converges to 0 in the metric space 8, then the sequence T * a, also converges to 0. We also have another consequence of the previous theorem.
'
Corollary: Every linear mapping9 from 6 into I which is continuous and commutes with translation is of the form 2Za = T * a for some uniquely determined distribution T with compact support. PROOF: The restriction o f 9 t o 9 satisfies the hypothesis of thetheorem, hence corresponds to a distribution. The continuity of the linear form 9a(O) on the space 6 makes the distribution one with compact support. We consider finally the support of the convolution T * cp, where rp is a testfunction. Theorem:
supp T * rp E supp T + supp cp.
PROOF: It should be noted that the fact that supp rp is compact makes it easy to show that the sum supp T + supp rp is closed. It will be enough to
26.
THE CONVOLUTION OF DISTRIBUTIONS
123
show that any point x for which ( T * cp)(x) is not 0 is of the form x = y + z with y in supp T and z in supp cp. Since T ( Y X4) is not 0, there is a point y in the support of T which is also in the support o f y , 4, and this support is the set supp 6 + x; hence y = x z for some z in the support of cp.
-
26. The Convolution of Distributions In this section we define the convolution of two distributions, one of which has compact support; later we will extend the definition a little further. It should be made clear, however, that it is not possible to define the convolution of a pair of arbitrary distributions. Let T be a distribution on R" and S another distribution with compact support. The distribution S defines a mapping of the space of testfunctions into itself; this mapping is sequentially continuous and commutes with translation. The distribution T also defines a mapping of the testfunctions into €' which is sequentially continuous and commutes with translation, and it therefore follows that the composition of these mappings is sequentially continuous from 9 to d and commutes with translation, hence, corresponds to a distribution which we write T * S and take as the definition of the convolution of those distributions. We could have considered the mappings in another order: convolution with T would carry 9 into d and convolution with S would carry d into 8 ; the composed mapping would be continuous from 9 to 8,would commute with translation, and would therefore correspond to a uniquely determined distribution which we write S * T. It is important to show that S * T = T * S, that is, that the composed mappings, in either order, are the same. For this purpose we need the following lemma. Lemma:
If S is in b', cp in 9, and a in b then ( S * (cp
* a ) ) ( x ) = ( ( S * cp) * a ) ( x )
*
PROOF: Choose r > 0 so large that the supports of both S and cp are contained in a sphere of radius r about the origin, then choose a large M and a testfunction ~ ( xwhich ) is equal to 1 on a sphere of radius M + 4r. We then write a ( x ) as a sum:
4 x 1 = x(x)a(x) + (1 - x(x))a(x) = a,(x)
+ az(x).
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.
25.
THE CONVOLUTION OF A DISTRIBUTION WITH A TESTFUNCTION
119
When cp is a testfunction and T a distribution, we define the convolution of T with rp as the function (or distribution) which follows:
49 = (T * cp)(x) = T ( F xG)
.
We have already seen that a(x) belongs to the class b(R")of C"-functions on the space. It is then immediate that ( y h
* cp)(x)
a)(x) = ( y h
= ( T * y h cp)(x)
and passing to difference quotients we obtain, in general (D"u)(x)= (D"T * cp)(x) = (T
* D"cp)(X).
Another useful and easy identity is ( T * cp)" = ? * @. When T has a compact support T * cp also has a compact support, since for large 1x1 the testfunction YxG has a support disjoint from that of T ; the convolution is therefore another testfunction. Another obvious but important fact is the following:
4 0 ) = (T * cp)(O) = T(,Fofj) = T(i0) ;
hence T(cp)= (T * G)(O). When the distribution T is given by the locally integrable function f ( x ) , we have
( T * cp)(x> = /S(X
- Y>cp(Y> dY,
which is the usual definition for the convolution of two functions. If we convolute two testfunctions cp and rl/ we obtain a third: x ( x ) = (cp =f
* *>(XI
d x - Y)*(Y) dY.
Of course, the integral above is taken only over a compact set, since the support of II/ is compact, and we find it convenient to approximate the integral by a sequence of Riemann sums formed in the following way. The domain of integration F is written as a finite union of disjoint measurable sets F i , each
120
11. DISTRIBUTIONS
of diameter smaller than mating sum is
E;
a point y i is chosen in each Fi, and the approxi-
= S,(X)
.
The approximating sums are themselves all testfunctions, being finite linear combinations of translates of cp, and there exists a fixed compact set K which supports all the testfunctions S, ,namely, the set of all points x whose distance from F is at most twice the diameter of the support of cp. The sums S,(x) con) every x as E diminishes to 0 and those sums are uniformly verge to ~ ( x at bounded :
Moreover, the derivatives D"S,(x) are the corresponding Riemann sums for the convolution (D'cp * i,b)(x) and are therefore uniformly bounded and converge pointwise. Thus the Arzela-Ascoli theorem guarantees that the test) the space 9 ( R " ) ;it is this circumstance that functions s , ( ~ converge ) to ~ ( xin enables us to prove the following theorem.
Theorem:
( T * c p ) * J I =T*(cp*Il/).
PROOF: ( T * x)(x)
= T(Y,i)
=
jv * cp)(x - Y ) W ) d y *
= ((T cp)
*W)
We have earlier defined the regularization of a locally integrable function
f ( x ) : we took a testfunction cp(x) which was even and positive, and for which jcp(x)dx = 1, and defined cp,(x) as E-"((P 1;') = (l/e")cp(x/~).The regularization was the C"-function &(x) = (f*rp&) and we showed that the regularizations converge tof(x) in any reasonable sense. In particular, whenf((x) was since a testfunction, the regularizations converged tof(x) in the topology of 9, 0
25.
THE CONVOLUTIONOF A DISTRIBUTION WITH A TESTFUNCTION
121
a sequence of them had a fixed compact support and converged uniformly, as well as all of the derivatives. We can now extend the idea of regularization to general distributions, not just locally integrable functions as follows: T, = T,(x) = ( T * cp,)(x) ;
this is a Cm-function, and for any testfunction IcI(x), T,($)
* bm) * cp, * Il/)(O) = T * (cp, * Il/)(O) = T(cp, * IcI) = (T,
= (T
and this converges with diminishing E to T($). Thus, the regularizations converge to Tin the space of distributions. Let the distribution T be fixed: it is then clear that the mapping which carries the testfunction cp(x) into the C"-function a(x) = (T * cp)(x) is a linear mapping of 9 into d which commutes with translation and which is sequentially continuous, a convergent sequence in 9 being carried into a convergent sequence in 6. The next theorem assures us that every such mapping is a convolution. Let 2 be a linear mapping of 9 ( R " )into b(R")which comTheorem: mutes with translation: r h ( y q ) = y(rhq), and which is sequentially continuous: (P, converging to 0 in 9 implies 2cp,converging to 0 in &; then there exists a unique distribution T such that 9 c p = T * cp for all cp. PROOF: The evaluation functional ycp(0) is a linear functional orP9 and from the continuity of the mapping it is even a sequentially continuous linear functional on 9, hence a distribution. We write, then, Ycp(0)= i'(cp), and now (dRcp)(x)= 2'(F-x cp)(O) = F(2Cxcp)= T ( Y x ;P) = ( T * cp)(x). The distribution T is uniquely determined since its regularizations are, T, being the image under 2 of the function cp,, and these regularizations converge to T as the E converges to 0. When the distribution T has compact support, the corresponding convolution mapping carries the space of testfunctions into itself. The previous theorem admits an easy extension to one asserting that the linear, translational invariant mappings of 9 into itself which are sequentially continuous are convolutions with distributions in 6'. In fact, the only point in the proof which is not immediate is the compactness of the support of T, however, if that set were not compact, there would be a sequence of points x, in the support of ?
122
11. DISTRIBUTIONS
having no finite limit point, and each x, would be surrounded by a neighborhood of small diameter supporting a testfunction $, for which ?($,) = I . We 19, , a system of translate these testfunctions to the origin, forming cp, = .Yxn testfunctions supported by the unit sphere. For a suitable choice of constants c, converging rapidly to 0, the sequence c, rp, converges to 0 in La, while the system of their convolutions with T is not supported by any fixed compact, hence does not converge to 0 in 9. Let Tbe the 6-distribution and cp a testfunction;(6 * cp)(x) = 6(Yx@)= cp(x) and so 6 corresponds to the identity mapping of 9 into itself. More, generally, then, for any polynomial P with constant coefficients,the convolution with the distribution P ( D ) 6 is the mapping which carries cp into P(D)rp. In a similar way we see that the translation operator y h itself corresponds to convolution with the distribution y h 6 ,and this distribution is the measure consisting of a unit mass at the point x = h. If the distribution T has compact support and a(x) is a function in the class 6 we can obviously form the convolution (T * u)(x) = T ( F x a), whether or not a(x) is in La; the convolution is again in 6 and it is easy to verify that Du(T* a) = (D"T* a ) = T * ( P a ) as well as fh(T*U)=FhT*U = T*yhU.
If a sequence a, converges to 0 in the metric space 8, then the sequence T * a, also converges to 0. We also have another consequence of the previous theorem.
'
Corollary: Every linear mapping9 from 6 into I which is continuous and commutes with translation is of the form 2Za = T * a for some uniquely determined distribution T with compact support. PROOF: The restriction o f 9 t o 9 satisfies the hypothesis of thetheorem, hence corresponds to a distribution. The continuity of the linear form 9a(O) on the space 6 makes the distribution one with compact support. We consider finally the support of the convolution T * cp, where rp is a testfunction. Theorem:
supp T * rp E supp T + supp cp.
PROOF: It should be noted that the fact that supp rp is compact makes it easy to show that the sum supp T + supp rp is closed. It will be enough to
26.
THE CONVOLUTION OF DISTRIBUTIONS
123
show that any point x for which ( T * cp)(x) is not 0 is of the form x = y + z with y in supp T and z in supp cp. Since T ( Y X4) is not 0, there is a point y in the support of T which is also in the support o f y , 4, and this support is the set supp 6 + x; hence y = x z for some z in the support of cp.
-
26. The Convolution of Distributions In this section we define the convolution of two distributions, one of which has compact support; later we will extend the definition a little further. It should be made clear, however, that it is not possible to define the convolution of a pair of arbitrary distributions. Let T be a distribution on R" and S another distribution with compact support. The distribution S defines a mapping of the space of testfunctions into itself; this mapping is sequentially continuous and commutes with translation. The distribution T also defines a mapping of the testfunctions into €' which is sequentially continuous and commutes with translation, and it therefore follows that the composition of these mappings is sequentially continuous from 9 to d and commutes with translation, hence, corresponds to a distribution which we write T * S and take as the definition of the convolution of those distributions. We could have considered the mappings in another order: convolution with T would carry 9 into d and convolution with S would carry d into 8 ; the composed mapping would be continuous from 9 to 8,would commute with translation, and would therefore correspond to a uniquely determined distribution which we write S * T. It is important to show that S * T = T * S, that is, that the composed mappings, in either order, are the same. For this purpose we need the following lemma. Lemma:
If S is in b', cp in 9, and a in b then ( S * (cp
* a ) ) ( x ) = ( ( S * cp) * a ) ( x )
*
PROOF: Choose r > 0 so large that the supports of both S and cp are contained in a sphere of radius r about the origin, then choose a large M and a testfunction ~ ( xwhich ) is equal to 1 on a sphere of radius M + 4r. We then write a ( x ) as a sum:
4 x 1 = x(x)a(x) + (1 - x(x))a(x) = a,(x)
+ az(x).