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34. Periodic Distributions in Several Variables
34.
165
PERIODIC DISTRIBUTIONS IN SEVERAL VARIABLES
Hence there exists a constant C, so that this sum is bounded by C, hZN where h = 2'*-"' is smaller than 1, since CL > 3. Thus (ckl is finite. Bernstein's theorem is the best possible in the following sense: there exists a functionf(x) that is Lipschitzian of order 4 with a Fourier series that does not converge absolutely.
+
1
34. Periodic Distributions in Several Variables The distributions T which we consider in this section are defined on R" and have n linearly independent periods. That is to say, we suppose the existence of n linearly independent vectors h , , h, , . . . , h, in R", such that Yhk T = T for k = 1, 2, . . . ,n. There will then exist a linear transformation I of R" into itself, such that / ( & ) = hk for all k, where ek is the unit vector in the direction of the kth coordinate axis. It will follow that if S = To f then F z S = S for all z in Z " , the lattice of points with integer coordinates. We shall presently show that S is temperate, from which it will follow that T is also temporate. Let $(x) be a testfunction which is equal to 1 on the cube C defined by the inequalities lxil 5 1, i = 1, 2, . . . , n ; we suppose further that 0 5 $(x) S 1 and that $(x) vanishes outside a small neighborhood of C. The sum
+
is a periodic C"-function which never vanishes, and the ratios cPZ(4
=5
2
$(x)/4(4
= Y2 c P o ( 4
form a partition of unity. Accordingly, the distribution S may be written as a F 2 ( p 0 S ) each , term of which has compact support and is sum: S = CzsZn therefore temperate. The argument of the previous section shows that the sum converges in the space of temperate distributions, hence that S is temperate. Moreover, the equation .T2S = S, which holds for all z in Z" implies A
*
that (e-izS - 1)s= 0 for all such z, and therefore, S is supported by the lattice of points of the form 27rc where [ has integer coordinates. It should also be A
clear that S must be a measure.
*
The distribution T = S 0 I-' therefore has the Fourier transform T which is a measure supported by the lattice 1,(27rZ").The simplest illustration gives rise to the Poisson summation formula: the distribution P which consists of a unit mass at every point of the lattice Z " satisfies the equation ( 1 - eiznCr)P= 0
166
11. DISTRIBUTIONS A
A
for every [ in that lattice, and from this it follows that YZnC P = P and therefore h
A
that P is periodic. This means that P consists of the measure which puts the same mass m at every point of 2nZ". Accordingly, for every function cp(x) in 9,
Since cp(x) may be the Gaussian, m is clearly positive, while for cp = $ 0 Ze we have
and taking formula.
E
= J% we find m = ( 2 7 ~ ) "This ~ ~ . establishes the following
Poisson Summation Formula (2) :
The study of the periodic distributions in R" is, there.are, exactly para :I to that in R 1; as in the previous section, one shows that iff(x) is a locally integrable function such that every point in the lattice 2nZ" is a period of f(x), then its Fourier transform is a measure supported by 2".At the point z of that lattice, the measure has the mass
jo.-*jo e - i ( X i Z ' f ( x , ,x 2 , . . . , x,) d x , d x ,
2n 2 n
c, = (2n)-"12J'o
2n
dx,.
When the total mass is finite, the function is given by the absolutely convergent series
and when the function is locally L2,the Parseval equation is valid:
1 IcL12 = 1 1 . * . j o z n l f ( x l . x 2 , . . . ,x,)12 d x , d x , ...dx,. Zn Zn
z EZ"
0
0
The Poisson summation formula makes possible an alternate proof for the Minkowski Lemma stated in Section 14. Following C. L. Siegel, we suppose that I/ is a parallelepiped lxil < bi , i = 1,2, . . . ,n, in R" of volume m( V ) = b, which contains no point different from the origin with integer coordinates. We show that m( V ) =< 2".
35.
SPHERICAL HARMONICS
167
Let C be the cube [ x i [< 1, i = 1,2, . . . ,n in R" and l a linear transformation which maps V onto C ; then ldet flm(V ) = 2". If t ( s ) is defined on the real axis as the triangle function, vanishing for Is1 > 1 and given by 1 - Is1 on [ - 1, 11, then T ( x )= t(xk) is a continuous function on R" supported by C. Its Fourier transform is readily computed: it is the positive, integrable function
n;=l
The composed function (T 0 I)@) = T(l(x))is supported by V and so vanishes for all x other than the origin with integer coordinates. Even though this the Poisson summation formula may be applied function is not in the class 9, to it in view of an obvious regularization argument. Accordingly, C ( T 0 I)(k)= (2.)"'2C(To
1)^(2nk),
where all terms in these sums are nonnegative and the summation is taken over all k in the lattice 2" of points with integer coordinates. The left-hand side reduces to the single term corresponding to k = 0, hence,
A
A
and because ( T 0 I,)(O) = T(0)= (27r)-"/2,the first term ofthe series is I/ldet I). We therefore find that 1 = ldet 1 I - I + R where the remainder, R, is nonnegative, and it follows that m( V ) 5 2". We should also note that if equality holds in this inequality, the remainder above must vanish, and since it is a sum of nonnegative terms, every term T(1,(2nk))vanishes fork not the origin. Since the zeros of $are exactly the lattice 2nZ", the transformation I , must carry that lattice into itself. We see then that I maps Z " into itself and so the vertices of the parallelepiped V = l - ' ( C ) are points of 2". Thus V must contain C, the smallest parallelepiped with vertices in Z " containing the origin in its interior. Now since ldet fI = 1, it follows that V coincides with C if m( V ) = 2" and V intersects Z " only in the origin.
35. Spherical Harmonics For m 2 0, we consider the space Il, consisting of homogeneous polyP U, nomials P ( x ) of degree m defined on R" for n 2 3. Thus, P 0 le = E ~ and is obviously a vector space of dimension d(m) = (m + n - l)!/m!(n - l)!, a basis for the space being given by the monomials xa for lctl = m.
165
PERIODIC DISTRIBUTIONS IN SEVERAL VARIABLES
Hence there exists a constant C, so that this sum is bounded by C, hZN where h = 2'*-"' is smaller than 1, since CL > 3. Thus (ckl is finite. Bernstein's theorem is the best possible in the following sense: there exists a functionf(x) that is Lipschitzian of order 4 with a Fourier series that does not converge absolutely.
+
1
34. Periodic Distributions in Several Variables The distributions T which we consider in this section are defined on R" and have n linearly independent periods. That is to say, we suppose the existence of n linearly independent vectors h , , h, , . . . , h, in R", such that Yhk T = T for k = 1, 2, . . . ,n. There will then exist a linear transformation I of R" into itself, such that / ( & ) = hk for all k, where ek is the unit vector in the direction of the kth coordinate axis. It will follow that if S = To f then F z S = S for all z in Z " , the lattice of points with integer coordinates. We shall presently show that S is temperate, from which it will follow that T is also temporate. Let $(x) be a testfunction which is equal to 1 on the cube C defined by the inequalities lxil 5 1, i = 1, 2, . . . , n ; we suppose further that 0 5 $(x) S 1 and that $(x) vanishes outside a small neighborhood of C. The sum
+
is a periodic C"-function which never vanishes, and the ratios cPZ(4
=5
2
$(x)/4(4
= Y2 c P o ( 4
form a partition of unity. Accordingly, the distribution S may be written as a F 2 ( p 0 S ) each , term of which has compact support and is sum: S = CzsZn therefore temperate. The argument of the previous section shows that the sum converges in the space of temperate distributions, hence that S is temperate. Moreover, the equation .T2S = S, which holds for all z in Z" implies A
*
that (e-izS - 1)s= 0 for all such z, and therefore, S is supported by the lattice of points of the form 27rc where [ has integer coordinates. It should also be A
clear that S must be a measure.
*
The distribution T = S 0 I-' therefore has the Fourier transform T which is a measure supported by the lattice 1,(27rZ").The simplest illustration gives rise to the Poisson summation formula: the distribution P which consists of a unit mass at every point of the lattice Z " satisfies the equation ( 1 - eiznCr)P= 0
166
11. DISTRIBUTIONS A
A
for every [ in that lattice, and from this it follows that YZnC P = P and therefore h
A
that P is periodic. This means that P consists of the measure which puts the same mass m at every point of 2nZ". Accordingly, for every function cp(x) in 9,
Since cp(x) may be the Gaussian, m is clearly positive, while for cp = $ 0 Ze we have
and taking formula.
E
= J% we find m = ( 2 7 ~ ) "This ~ ~ . establishes the following
Poisson Summation Formula (2) :
The study of the periodic distributions in R" is, there.are, exactly para :I to that in R 1; as in the previous section, one shows that iff(x) is a locally integrable function such that every point in the lattice 2nZ" is a period of f(x), then its Fourier transform is a measure supported by 2".At the point z of that lattice, the measure has the mass
jo.-*jo e - i ( X i Z ' f ( x , ,x 2 , . . . , x,) d x , d x ,
2n 2 n
c, = (2n)-"12J'o
2n
dx,.
When the total mass is finite, the function is given by the absolutely convergent series
and when the function is locally L2,the Parseval equation is valid:
1 IcL12 = 1 1 . * . j o z n l f ( x l . x 2 , . . . ,x,)12 d x , d x , ...dx,. Zn Zn
z EZ"
0
0
The Poisson summation formula makes possible an alternate proof for the Minkowski Lemma stated in Section 14. Following C. L. Siegel, we suppose that I/ is a parallelepiped lxil < bi , i = 1,2, . . . ,n, in R" of volume m( V ) = b, which contains no point different from the origin with integer coordinates. We show that m( V ) =< 2".
35.
SPHERICAL HARMONICS
167
Let C be the cube [ x i [< 1, i = 1,2, . . . ,n in R" and l a linear transformation which maps V onto C ; then ldet flm(V ) = 2". If t ( s ) is defined on the real axis as the triangle function, vanishing for Is1 > 1 and given by 1 - Is1 on [ - 1, 11, then T ( x )= t(xk) is a continuous function on R" supported by C. Its Fourier transform is readily computed: it is the positive, integrable function
n;=l
The composed function (T 0 I)@) = T(l(x))is supported by V and so vanishes for all x other than the origin with integer coordinates. Even though this the Poisson summation formula may be applied function is not in the class 9, to it in view of an obvious regularization argument. Accordingly, C ( T 0 I)(k)= (2.)"'2C(To
1)^(2nk),
where all terms in these sums are nonnegative and the summation is taken over all k in the lattice 2" of points with integer coordinates. The left-hand side reduces to the single term corresponding to k = 0, hence,
A
A
and because ( T 0 I,)(O) = T(0)= (27r)-"/2,the first term ofthe series is I/ldet I). We therefore find that 1 = ldet 1 I - I + R where the remainder, R, is nonnegative, and it follows that m( V ) 5 2". We should also note that if equality holds in this inequality, the remainder above must vanish, and since it is a sum of nonnegative terms, every term T(1,(2nk))vanishes fork not the origin. Since the zeros of $are exactly the lattice 2nZ", the transformation I , must carry that lattice into itself. We see then that I maps Z " into itself and so the vertices of the parallelepiped V = l - ' ( C ) are points of 2". Thus V must contain C, the smallest parallelepiped with vertices in Z " containing the origin in its interior. Now since ldet fI = 1, it follows that V coincides with C if m( V ) = 2" and V intersects Z " only in the origin.
35. Spherical Harmonics For m 2 0, we consider the space Il, consisting of homogeneous polyP U, nomials P ( x ) of degree m defined on R" for n 2 3. Thus, P 0 le = E ~ and is obviously a vector space of dimension d(m) = (m + n - l)!/m!(n - l)!, a basis for the space being given by the monomials xa for lctl = m.