- Email: [email protected]
12. The Orthogonal Group
58
I. INTRODUCTION
that is, the derivatives of the regularizations are the regularizations of the derivatives. Of course we must require that the distance from x to the boundary of G is greater than E ; then, since the differentiation under the integral sign is legitimate
Since
the differential operator D" may be taken as acting in the variable y, hence
and this may now be integrated by parts. The integrated terms vanish, since the function (P( {x - y } / e ) vanishes outside a sphere of radius E about x, and so finally
as required. The hypothesis of our assertion could be slightly weakened: we need only require k - 1 continuous derivatives for f(x), the derivatives of order k - 1 being Lipschitzian.
12, The Orthogonal Group In this section we consider linear transformations 1 of R" into itself; these are the transformations which satisfy
for all vectors x and y and all scalars a and b. When a coordinate system is fixed, such transformations are usually written as matrices. The general linear group, GL(n), is the class of linear transformations on R" which have inverses, and obviously forms a group. There are two remarkable subgroups, the group of homothetic transformations and the orthogonal group. The homothetic
12.
THE ORTHOGONAL GROUP
59
transformations are of the form I 1 where I > 0 and Z is the identity transformation; we write the transformation I,. Evidently I,l, = l,, and I-' = l,,,, I , =identity and the group is isomorphic to the multiplicative group of positive reals. The orthogonal group consists of those linear transformations which are isometries: (l(x)l = 1x1. The surface of the unit ball is then mapped by 1 onto itself in a one-to-one way. The group is denoted by the symbol O(n). Iff(x) is a function on R",the composition offwith an element of GL(n) is defined as the function
(f W ) =f ( W. 0
Clearly ( f o 1') 1" = f 0 (1'1"). If there exists a number k such that f o 1, = for all E > 0, thenf(x) is said to be homogeneous of degree k. A transformation 1 has an adjoint I* defined by the equation 0
~''f
(W, Y ) = (x, l*(Y)) for all x and y in R";here it should be emphasized that the inner product is a real inner product since the vectors are elements of a real vector space. When I is written as a matrix, I* is given by the transposed matrix. It is not difficult to show that if I has an inverse, then I* also has an inverse and (l*)-' = (I-')*. We write 1, for the adjoint of the inverse, and this notation will be particularly = useful when the Fourier transform is studied. It is easy to see that (I(x),I&) (x, y ) for all vectors x and y . When I belongs to the orthogonal group, 1 = 1,. There are many occasions when it is desirable to be able to form certain averages of functions defined on the orthogonal group, and this circumstance leads naturally to the development of a theory of integration over that group. Rather than consider the general theory of integration on locally compact groups, we confine ourselves here to the theory of integration developed by W. Maak for compact metric groups and present that theory for the particular group O(n). First note that O(n) occurs naturally as a compact metric space, the metric being defined in the following way. For any element 1 in O(n) define its displacement d(1) by
d ( l ) = sup Ix - I(x)l 1x1 = I
this supremum is attained for at least one point xo of the unit sphere since that sphere is compact and the function Ix - I(x)( is continuous. For any x in R" and any pair of group elements 1, and l2 Ix - 1112(X)l
s Ix - Wl + 112(x) - l,lz(X)l 5 4 1 2 ) + 411)
60
1. INTRODUCTION
and therefore d(l,12)S d(I,) + 41,). Obviously d(l) 5 2 for all I in O(n) and d(I) = 0 only when I is the identity. It is also clear that d(1-l) = d(I). If xo is a point of the unit sphere where the function J x- 1,12(x)lattains its maximum, then, if y o = Iz(xo), 4412)
= 1x0
- I*12(Xo)l - I2(I,I2
= lI,(XO)
X0)l
= IYO
- ~,~I(YO)l
64
1,)
2
and since the transformations I, and I , are arbitrary, d(I,lz)= 41, I l ) . For the metric on O(n) we take P ( l , 1,) = d(l,1;
*
I
Evidently p ( I , , 1,) = 0 if and only if I , = I,, while p ( l , , I,) = p(l, ,I,) since = (I,/;')-'. The triangle inequality is easily verified:
121;l
PUl3
4)= 4W) = d(l,lY'12 l;,)
d(fJY1)+ d ( l J 3
=PUl,
12)
+ PUZ 4) 9
*
It is important to note that the metric is invariant under the group operation: if I , , I , , and I , are any three elements of the group, then d l I l 2 > 11l3)
=d
l 2
9
l3)
moreover, PUI,
I,) =d l ? , 1Y1)
and from these equations it is clear that multiplication by a group element I , is a homeomorphism, indeed an isometry of O(n) onto itself. It is also easy to show that O(n) is compact under this metric: a sequence 1k of group elements is a sequence of homeomorphisms of the unit ball 1x1 5 1 in R";since the 1, are isometries, they are necessarily Lipschitzian with Lipschitz constant + 1, and the Ascoli-Arzela theorem guarantees that there exists a subsequence Ik, converging uniformly on the unit ball. The limit is evidently a linear isometry lo of R" onto itself, and the numbers p(&, , lo) converge to 0. It is therefore established that O(n) is a compact metric space and also a topological group since the group operations are (jointly) continuous for this metric. Before proceeding to Maak's theory of integration, we first follow that author in his solution to the so-called Marriage Problem. Let there be given
12. THE ORTHOGONAL GROUP
61
two finite sets B and G of boys and girls respectively, and a relation of acquaintance between them. We shall say that a solution to the Marriage Problem exists if it is possible to marry off the boys, each to a girl of his acquaintance. Since the marriages are to be monogamous, it is obvious that a necessary condition for such a mass marriage to be possible is that there exist no subset of k boys whose total acquaintance among the girls involves fewer than k girls. The result of Maak is that this condition is also sufficient for a solution to the Marriage Problem. The proof is by induction on N , the cardinal of B. The proof for N = 1 or N = 2 is immediate, and is here omitted. The assertion being supposed true for N , the passage to N 1 is established by the following argument. Suppose, first, there exists a proper subset H of B consisting of k boys, whose total acquaintance consists of exactly k girls forming a set W. By the inductive hypothesis, the boys in H can be married to the girls in W. The remaining N 1 - k boys have sufficiently many acquaintances in the complement of W, for if there exists a subset M o f j remaining boys having a total acquaintance among the unmarried girls which consists of fewer than j girls, the union H u M is a set of k + j boys, with a total acquaintance of at most k + j - 1 girls, contradicting the hypothesis of the Maak theorem. Hence, the existence of a proper subset of k boys, acquainted with exactly k girls leads to a solution of the marriage problem for N 1. More generally, then, given N + I boys, to solve the Marriage Problem we marry off one boy to a girl of his acquaintance. The remaining N boys can be married off to the remaining girls by the inductive hypothesis, since otherwise there exists a subset of k of them with a total acquaintance of at most k - 1 unmarried girls, and in this latter case we divorce the married couple, and adjoin the divorcte to those k - 1 unmarried girls, obtaining a set of k boys whose total acquaintance consists of exactly k girls. This is the case considered in the first part of our argument, which is now complete. It is now possible to construct a Haar (or Maak) measure on the group O(n).Given E > 0, let Hi be a decomposition of O(n)into a finite union of sets of diameter at most E ; such a decomposition is minimal when the total number N = N ( E )of sets in the decomposition is a minimum. We consider only minimal decompositions of O(n). For any such minimal decomposition, choose a point li in each H iand put the mass 1/N at that point; in this way there is determined a measure p of total mass 1 consisting of N equal point masses more or less uniformly distributed through O(n). Let E approach 0 through a countable set of values; there results a sequence pk of measures on O(n) to which Helly’s theorem is applicable, and hence a subsequence converging weakly to a measure p,, . I t is a remarkable fact that p,, is independent of the particular choice of the measures pk . To show this, we first consider any two minimal decompositions H l and H; associated with a particular E > 0; there are equally many sets in each
+
+
+
62
1. INTRODUCTION
decomposition, namely, N = N(E).Let p’ and p” be the two associated measures, consisting of equal point masses at points I,’ in H iand IJ in H J respectively. If we think of the Hfas boys and the H J as girls, the hypotheses of the Marriage Problem are satisfied if the H,‘ and H J are said to be acquainted if and only if they have a nonempty intersection. For if there exists a subset of k sets Hi whose union intersects at most k - 1 sets H J , the decomposition H,‘ could not have been minimal, since the substitution of the k - 1 sets H J for the k sets Hf results in a decomposition of O(n) into N - 1 sets of diameter at most E . It follows, therefore, that the subscripts may be assigned to the N sets H J in such a way that Hfalways has a nonempty intersection with the H J having the same subscript. Suppose, next, that F ( I ) is any continuous function on O(n); since the group is compact, the function is uniformly continuous, and we let ~ ( t ) denote the modulus of continuity. Now
I
N
5 W(2E) since p ( l ; , If’)< 2.5, because the sets Hf and HI’have diameter at most E and a nonempty intersection. It follows immediately that F ( I ) dpk(l) is a Cauchy
s
s
sequence of numbers, converging, of course, to F ( I )dpo(l).Thusp, isuniquely determined and it was not necessary to pass to a subsequence in the application of Helly’s theorem. Moreover, if lo is a fixed group element, F ( x ) continuous on O(n) and pk a sequence of approximating measures of the type constructed above which converges to p o , then
=
1 lim - C F(li)
N-+W
N
Thus the measure p o is invariant under the group operation: if A is a Bore1 set and lo a group element, then p 0 [ 4 = p0Cl0 A]. The approximating sums ( l / N ) F ( l i ) and the corresponding measures pk introduced above for the construction of the Haar measure are themselves
13.
SECOND-ORDER DIFFERENTIAL OPERATORS
63
particularly useful in applications; we shall call them Maak sums and Maak measures, in analogy to Riemann sums. One application is immediate. Let F(x) be a function defined and continuous on some ball 1x1 S R in R". The corresponding function F(l(x))= (F l)(x) for fixed x is a continuous function on O(n) and hence is integrable. Now 0
is a spherical average of F ( x ) ;it satisfies the equation G(l(x))= G(x) for all 1 in O(n) and hence G(x) is a function of radius 1x1 only. However, it is possible to find a spherical average of F(x) in another way: the function F(x) is written in terms of the spherical coordinates F(r, 0) and the average is H ( x ) = h(r) = l F ( r , O)dw(0).Evidently both F ( x ) and ( F ol)(x) give rise to the same spherical average H ( x )in this way. So also do the functions ( I / N ) C (F /i)(x)and hence H ( x ) = j G ( r , 0) dw(0)and therefore H ( x ) = G(x). 0
Accordingly, the Lebesgue integral
llXl sRF(x)dx
=
jlxl s R F ( l ( ~ dx, ) ) which is
G(x)dx = lORjF(r,0) dw(O)r"-l drw, and it independent of I , is equal to !lxlS R follows finally that the Lebesgue measure dx in R" coincides with the product measure w, dwr"-' dr as was asserted in Section 7.
13. Second-Order Differential Operators In this section we prove a general existence and uniqueness theorem for solutions of a second-order linear differential equation. The equations are considered relative to a closed, finite interval [a, b] and are of the form d
--(P(x) dX
2)+
R ( x ) y ( x ) = h(x) .
We shall require that h(x) and R(x) be integrable over the interval, and that the reciprocal of P ( x ) be also integrable over that interval. A solution y ( x ) must be such that the terms of the equation make sense, at least almost everywhere; hence we require y ( x ) to be absolutely continuous. It will then be differentiable almost everywhere and the product P ( x ) dy/dx will then make sense almost everywhere. We shall say that this product is absolutely continuous when we mean that there exists an absolutely continuous function on [a, b] which coincides almost everywhere with P ( x ) dy/dx; we will then be able to speak of the value of P ( x ) dy/dx at an arbitrary point c of the interval.
I. INTRODUCTION
that is, the derivatives of the regularizations are the regularizations of the derivatives. Of course we must require that the distance from x to the boundary of G is greater than E ; then, since the differentiation under the integral sign is legitimate
Since
the differential operator D" may be taken as acting in the variable y, hence
and this may now be integrated by parts. The integrated terms vanish, since the function (P( {x - y } / e ) vanishes outside a sphere of radius E about x, and so finally
as required. The hypothesis of our assertion could be slightly weakened: we need only require k - 1 continuous derivatives for f(x), the derivatives of order k - 1 being Lipschitzian.
12, The Orthogonal Group In this section we consider linear transformations 1 of R" into itself; these are the transformations which satisfy
for all vectors x and y and all scalars a and b. When a coordinate system is fixed, such transformations are usually written as matrices. The general linear group, GL(n), is the class of linear transformations on R" which have inverses, and obviously forms a group. There are two remarkable subgroups, the group of homothetic transformations and the orthogonal group. The homothetic
12.
THE ORTHOGONAL GROUP
59
transformations are of the form I 1 where I > 0 and Z is the identity transformation; we write the transformation I,. Evidently I,l, = l,, and I-' = l,,,, I , =identity and the group is isomorphic to the multiplicative group of positive reals. The orthogonal group consists of those linear transformations which are isometries: (l(x)l = 1x1. The surface of the unit ball is then mapped by 1 onto itself in a one-to-one way. The group is denoted by the symbol O(n). Iff(x) is a function on R",the composition offwith an element of GL(n) is defined as the function
(f W ) =f ( W. 0
Clearly ( f o 1') 1" = f 0 (1'1"). If there exists a number k such that f o 1, = for all E > 0, thenf(x) is said to be homogeneous of degree k. A transformation 1 has an adjoint I* defined by the equation 0
~''f
(W, Y ) = (x, l*(Y)) for all x and y in R";here it should be emphasized that the inner product is a real inner product since the vectors are elements of a real vector space. When I is written as a matrix, I* is given by the transposed matrix. It is not difficult to show that if I has an inverse, then I* also has an inverse and (l*)-' = (I-')*. We write 1, for the adjoint of the inverse, and this notation will be particularly = useful when the Fourier transform is studied. It is easy to see that (I(x),I&) (x, y ) for all vectors x and y . When I belongs to the orthogonal group, 1 = 1,. There are many occasions when it is desirable to be able to form certain averages of functions defined on the orthogonal group, and this circumstance leads naturally to the development of a theory of integration over that group. Rather than consider the general theory of integration on locally compact groups, we confine ourselves here to the theory of integration developed by W. Maak for compact metric groups and present that theory for the particular group O(n). First note that O(n) occurs naturally as a compact metric space, the metric being defined in the following way. For any element 1 in O(n) define its displacement d(1) by
d ( l ) = sup Ix - I(x)l 1x1 = I
this supremum is attained for at least one point xo of the unit sphere since that sphere is compact and the function Ix - I(x)( is continuous. For any x in R" and any pair of group elements 1, and l2 Ix - 1112(X)l
s Ix - Wl + 112(x) - l,lz(X)l 5 4 1 2 ) + 411)
60
1. INTRODUCTION
and therefore d(l,12)S d(I,) + 41,). Obviously d(l) 5 2 for all I in O(n) and d(I) = 0 only when I is the identity. It is also clear that d(1-l) = d(I). If xo is a point of the unit sphere where the function J x- 1,12(x)lattains its maximum, then, if y o = Iz(xo), 4412)
= 1x0
- I*12(Xo)l - I2(I,I2
= lI,(XO)
X0)l
= IYO
- ~,~I(YO)l
64
1,)
2
and since the transformations I, and I , are arbitrary, d(I,lz)= 41, I l ) . For the metric on O(n) we take P ( l , 1,) = d(l,1;
*
I
Evidently p ( I , , 1,) = 0 if and only if I , = I,, while p ( l , , I,) = p(l, ,I,) since = (I,/;')-'. The triangle inequality is easily verified:
121;l
PUl3
4)= 4W) = d(l,lY'12 l;,)
d(fJY1)+ d ( l J 3
=PUl,
12)
+ PUZ 4) 9
*
It is important to note that the metric is invariant under the group operation: if I , , I , , and I , are any three elements of the group, then d l I l 2 > 11l3)
=d
l 2
9
l3)
moreover, PUI,
I,) =d l ? , 1Y1)
and from these equations it is clear that multiplication by a group element I , is a homeomorphism, indeed an isometry of O(n) onto itself. It is also easy to show that O(n) is compact under this metric: a sequence 1k of group elements is a sequence of homeomorphisms of the unit ball 1x1 5 1 in R";since the 1, are isometries, they are necessarily Lipschitzian with Lipschitz constant + 1, and the Ascoli-Arzela theorem guarantees that there exists a subsequence Ik, converging uniformly on the unit ball. The limit is evidently a linear isometry lo of R" onto itself, and the numbers p(&, , lo) converge to 0. It is therefore established that O(n) is a compact metric space and also a topological group since the group operations are (jointly) continuous for this metric. Before proceeding to Maak's theory of integration, we first follow that author in his solution to the so-called Marriage Problem. Let there be given
12. THE ORTHOGONAL GROUP
61
two finite sets B and G of boys and girls respectively, and a relation of acquaintance between them. We shall say that a solution to the Marriage Problem exists if it is possible to marry off the boys, each to a girl of his acquaintance. Since the marriages are to be monogamous, it is obvious that a necessary condition for such a mass marriage to be possible is that there exist no subset of k boys whose total acquaintance among the girls involves fewer than k girls. The result of Maak is that this condition is also sufficient for a solution to the Marriage Problem. The proof is by induction on N , the cardinal of B. The proof for N = 1 or N = 2 is immediate, and is here omitted. The assertion being supposed true for N , the passage to N 1 is established by the following argument. Suppose, first, there exists a proper subset H of B consisting of k boys, whose total acquaintance consists of exactly k girls forming a set W. By the inductive hypothesis, the boys in H can be married to the girls in W. The remaining N 1 - k boys have sufficiently many acquaintances in the complement of W, for if there exists a subset M o f j remaining boys having a total acquaintance among the unmarried girls which consists of fewer than j girls, the union H u M is a set of k + j boys, with a total acquaintance of at most k + j - 1 girls, contradicting the hypothesis of the Maak theorem. Hence, the existence of a proper subset of k boys, acquainted with exactly k girls leads to a solution of the marriage problem for N 1. More generally, then, given N + I boys, to solve the Marriage Problem we marry off one boy to a girl of his acquaintance. The remaining N boys can be married off to the remaining girls by the inductive hypothesis, since otherwise there exists a subset of k of them with a total acquaintance of at most k - 1 unmarried girls, and in this latter case we divorce the married couple, and adjoin the divorcte to those k - 1 unmarried girls, obtaining a set of k boys whose total acquaintance consists of exactly k girls. This is the case considered in the first part of our argument, which is now complete. It is now possible to construct a Haar (or Maak) measure on the group O(n).Given E > 0, let Hi be a decomposition of O(n)into a finite union of sets of diameter at most E ; such a decomposition is minimal when the total number N = N ( E )of sets in the decomposition is a minimum. We consider only minimal decompositions of O(n). For any such minimal decomposition, choose a point li in each H iand put the mass 1/N at that point; in this way there is determined a measure p of total mass 1 consisting of N equal point masses more or less uniformly distributed through O(n). Let E approach 0 through a countable set of values; there results a sequence pk of measures on O(n) to which Helly’s theorem is applicable, and hence a subsequence converging weakly to a measure p,, . I t is a remarkable fact that p,, is independent of the particular choice of the measures pk . To show this, we first consider any two minimal decompositions H l and H; associated with a particular E > 0; there are equally many sets in each
+
+
+
62
1. INTRODUCTION
decomposition, namely, N = N(E).Let p’ and p” be the two associated measures, consisting of equal point masses at points I,’ in H iand IJ in H J respectively. If we think of the Hfas boys and the H J as girls, the hypotheses of the Marriage Problem are satisfied if the H,‘ and H J are said to be acquainted if and only if they have a nonempty intersection. For if there exists a subset of k sets Hi whose union intersects at most k - 1 sets H J , the decomposition H,‘ could not have been minimal, since the substitution of the k - 1 sets H J for the k sets Hf results in a decomposition of O(n) into N - 1 sets of diameter at most E . It follows, therefore, that the subscripts may be assigned to the N sets H J in such a way that Hfalways has a nonempty intersection with the H J having the same subscript. Suppose, next, that F ( I ) is any continuous function on O(n); since the group is compact, the function is uniformly continuous, and we let ~ ( t ) denote the modulus of continuity. Now
I
N
5 W(2E) since p ( l ; , If’)< 2.5, because the sets Hf and HI’have diameter at most E and a nonempty intersection. It follows immediately that F ( I ) dpk(l) is a Cauchy
s
s
sequence of numbers, converging, of course, to F ( I )dpo(l).Thusp, isuniquely determined and it was not necessary to pass to a subsequence in the application of Helly’s theorem. Moreover, if lo is a fixed group element, F ( x ) continuous on O(n) and pk a sequence of approximating measures of the type constructed above which converges to p o , then
=
1 lim - C F(li)
N-+W
N
Thus the measure p o is invariant under the group operation: if A is a Bore1 set and lo a group element, then p 0 [ 4 = p0Cl0 A]. The approximating sums ( l / N ) F ( l i ) and the corresponding measures pk introduced above for the construction of the Haar measure are themselves
13.
SECOND-ORDER DIFFERENTIAL OPERATORS
63
particularly useful in applications; we shall call them Maak sums and Maak measures, in analogy to Riemann sums. One application is immediate. Let F(x) be a function defined and continuous on some ball 1x1 S R in R". The corresponding function F(l(x))= (F l)(x) for fixed x is a continuous function on O(n) and hence is integrable. Now 0
is a spherical average of F ( x ) ;it satisfies the equation G(l(x))= G(x) for all 1 in O(n) and hence G(x) is a function of radius 1x1 only. However, it is possible to find a spherical average of F(x) in another way: the function F(x) is written in terms of the spherical coordinates F(r, 0) and the average is H ( x ) = h(r) = l F ( r , O)dw(0).Evidently both F ( x ) and ( F ol)(x) give rise to the same spherical average H ( x )in this way. So also do the functions ( I / N ) C (F /i)(x)and hence H ( x ) = j G ( r , 0) dw(0)and therefore H ( x ) = G(x). 0
Accordingly, the Lebesgue integral
llXl sRF(x)dx
=
jlxl s R F ( l ( ~ dx, ) ) which is
G(x)dx = lORjF(r,0) dw(O)r"-l drw, and it independent of I , is equal to !lxlS R follows finally that the Lebesgue measure dx in R" coincides with the product measure w, dwr"-' dr as was asserted in Section 7.
13. Second-Order Differential Operators In this section we prove a general existence and uniqueness theorem for solutions of a second-order linear differential equation. The equations are considered relative to a closed, finite interval [a, b] and are of the form d
--(P(x) dX
2)+
R ( x ) y ( x ) = h(x) .
We shall require that h(x) and R(x) be integrable over the interval, and that the reciprocal of P ( x ) be also integrable over that interval. A solution y ( x ) must be such that the terms of the equation make sense, at least almost everywhere; hence we require y ( x ) to be absolutely continuous. It will then be differentiable almost everywhere and the product P ( x ) dy/dx will then make sense almost everywhere. We shall say that this product is absolutely continuous when we mean that there exists an absolutely continuous function on [a, b] which coincides almost everywhere with P ( x ) dy/dx; we will then be able to speak of the value of P ( x ) dy/dx at an arbitrary point c of the interval.