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b(Rep)r(orting)e(Service,)j(pro) n(viding)f(that)f(the)g(base)g(fee)f(of)h($50.00)d(p)r(er)j(cop)n(y)g (is)h(paid)g(directly)h(to)d(CCC,)i(222)d(Rosew)n(o)r(o)r(d)h(Driv)n (e,)j(Dan)n(v)n(ers,)0 5040 y(MA)24 b(01923.)2028 5239 y Fp(47)p eop %%Page: 48 2 48 1 bop 0 83 a Fl(where)45 b Fk(f)j Fm(2)39 b Fk(L)533 53 y Ff(1)570 83 y Fp(\()p Fk(G)p Fp(\()p Fk(i;)14 b(d)p Fp(\)\))p Fl(,)47 b Fk(g)42 b Fm(2)d Fk(L)1179 53 y Ff(1)1216 83 y Fp(\()p Fk(G)p Fp(\()p Fk(j;)14 b(d)p Fp(\)\))p Fl(,)49 b Fk(\030)43 b Fm(2)c Fk(G)p Fp(\()p Fk(j;)14 b(d)p Fp(\))p Fl(,)48 b Fk(\021)42 b Fm(2)d Fk(G)p Fp(\()p Fk(i;)14 b(d)p Fp(\))p Fl(,)48 b Fk(\033)s Fp(\()p Fm(\001)p Fp(\))c Fl(is)g(the)g(unique)g(rotation)e(in)m(v)-5 b(arian)m(t)0 272 y(measure)44 b(on)f(the)i(appropriate)d(in)m(tegration)g(space)j (with)e(total)f(mass)i(1.)76 b(Suc)m(h)45 b(Radon)e(transforms)g (\014nd)0 461 y(n)m(umerous)33 b(applications)e(in)h(con)m(v)m(ex)i (geometry)-8 b(,)33 b(see)g([3])g({)f([5],)h([8],)f([9].)0 655 y(W)-8 b(e)33 b(mak)m(e)g(use)g(of)f(the)h(follo)m(wing)d (notation:)166 849 y Fk(L)p Fm(f)p Fk(a)309 861 y Ff(1)345 849 y Fk(;)14 b(:)g(:)g(:)g(;)g(a)574 861 y Fj(k)614 849 y Fm(g)23 b Fp(=)32 b Fl(the)h Fk(k)s Fl(-\015at)f(spanned)i(b)m(y) f(the)g(v)m(ectors)h Fk(a)2254 861 y Ff(1)2291 849 y Fk(;)14 b(:)g(:)g(:)g(;)g(a)2520 861 y Fj(k)2561 849 y Fl(,)166 1043 y Fk(V)19 b(ol)300 1013 y Ff(\()p Fj(k)q Ff(\))392 1043 y Fp(\()p Fk(a)468 1055 y Ff(1)506 1043 y Fk(;)14 b(:)g(:)g(:)f(;)h(a)734 1055 y Fj(k)775 1043 y Fp(\))23 b(=)29 b Fl(the)g(non-orien)m(ted)g(v)m(olume)g(of)f(the)i (parallelepip)s(ed)c(spanned)31 b(b)m(y)f Fk(a)3431 1055 y Ff(1)3468 1043 y Fk(;)14 b(:)g(:)g(:)g(;)g(a)3697 1055 y Fj(k)3766 1043 y Fl(\(w)m(e)30 b(shall)166 1237 y(often)j(omit)d(the) j(dimension)e(of)i(the)g(v)m(olume\),)166 1431 y Fk(V)19 b(ol)300 1401 y Ff(\()p Fj(k)q Ff(\))392 1431 y Fp(\()p Fk(\030)t Fp(\))24 b(=)32 b Fl(the)h(non-orien)m(ted)g(v)m(olume)f(of)g 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y Fl(,)32 b(suc)m(h)h(that)e Fk(\030)c Fp(=)c Fk(L)p Fm(f)p Fk(a)3190 2412 y Ff(1)3226 2400 y Fk(;)14 b(:)g(:)g(:)f(;)h(a)3454 2412 y Fj(k)3495 2400 y Fm(g)31 b Fl(for)f Fk(\030)e Fm(2)23 b Fk(G)p Fp(\()p Fk(k)s(;)14 b(d)p Fp(\))p Fl(.)0 2654 y Fn(1.2.)55 b(Main)36 b(results.)57 b Fl(In)37 b(Section)h(4,)g(w)m(e)h(study)f(the)g(action) f(of)g Fk(R)2477 2666 y Fj(ij)2572 2654 y Fl(and)h Fk(R)2830 2666 y Fj(j)s(i)2926 2654 y Fl(on)f(the)h(functions)f(of)g(the)h(t)m (yp)s(e)0 2843 y Fk(f)9 b Fp(\()p Fk(\021)s Fp(\))24 b(=)g([)p Fk(\021)s(;)14 b(\020)411 2855 y Ff(0)448 2843 y Fp(])471 2813 y Fj(\013)552 2843 y Fl(\()p Fk(f)9 b Fp(\()p Fk(\021)s Fp(\))24 b(=)f([)p Fk(\021)927 2813 y Fg(?)984 2843 y Fk(;)14 b(\020)1057 2855 y Ff(0)1094 2843 y Fp(])1117 2813 y Fj(\013)1197 2843 y Fl(in)33 b(the)g(case)h(of)f(the)g(dual)g(Radon)g(transform\))f(for)g Fk(\021)k Fl(from)c Fk(G)p Fp(\()p Fk(i;)14 b(d)p Fp(\))34 b Fl(\(or)f Fk(G)p Fp(\()p Fk(j;)14 b(d)p Fp(\))p Fl(,)0 3032 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Fi(Prop)s(osition)32 b(1.1.)43 b(Let)33 b Fk(i)23 b(<)g(j)28 b Fm(\024)22 b Fk(k)k(<)d(d)p Fi(,)33 b Fk(d)23 b Fm(\025)f Fp(3)p Fi(.)43 b(Then)33 b(fo)m(r)f Fk(\013)24 b(>)f Fp(0)32 b Fi(the)h(follo)m(wing)f(relation)g(holds:) 1466 374 y Fp(\()p Fk(R)1561 386 y Fj(j)s(i)1620 374 y Fp([)p Fm(\001)1666 340 y Fg(?)1722 374 y Fk(;)14 b(\020)1795 386 y Ff(0)1833 374 y Fp(])1856 340 y Fj(\013)1903 374 y Fp(\)\()p Fk(\021)s Fp(\))24 b(=)f Fk(c)2191 340 y Fg(\003)2229 374 y Fp(\()p Fk(\013)p Fp(\)[)p Fk(\021)2413 340 y Fg(?)2470 374 y Fk(;)14 b(\020)2543 386 y Ff(0)2580 374 y Fp(])2603 340 y Fj(\013)2651 374 y Fk(;)1295 b Fp(\(1.3\))0 665 y Fi(where)34 b Fk(\021)26 b Fm(2)d Fk(G)p Fp(\()p Fk(i;)14 b(d)p Fp(\))p Fi(,)1372 878 y Fk(c)1408 844 y Fg(\003)1446 878 y Fp(\()p Fk(\013)p Fp(\))24 b(=)1675 775 y Fj(d)p Fg(\000)p Fj(k)q Fg(\000)p Ff(1)1726 799 y Fh(Y)1726 978 y Fj(l)p Ff(=0)1907 792 y Fp(\000)1973 725 y Fh(\000)2020 759 y Fj(d)p Fg(\000)p Fj(i)p Fg(\000)p Fj(l)p 2020 773 184 4 v 2096 821 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b(W)-8 b(e)27 b(denote)f(b)m(y)h Fk(\030)2330 95 y Fj(j)s Ff(+1)2449 83 y Fk(;)14 b(:)g(:)g(:)g(;)g (\030)2670 95 y Fj(d)2735 83 y Fl(a)25 b(certain)h(orthonormal)d(basis) j(of)g Fk(\030)4057 53 y Fg(?)4113 83 y Fl(.)0 272 y(Then)g(the)f(v)m (ectors)h Fk(\030)766 284 y Fj(i)p Ff(+1)878 272 y Fk(;)14 b(:)g(:)g(:)f(;)h(\030)1098 284 y Fj(d)1162 272 y Fl(form)23 b(an)h(orthonormal)e(basis)j(in)f Fk(\021)2443 242 y Fg(?)2499 272 y Fl(.)41 b(If)24 b Fp([)p Fk(\030)t(;)14 b(\020)2792 284 y Ff(0)2830 272 y Fp(])23 b(=)f(0)p Fl(,)k(i.e.)40 b Fk(dim)3382 205 y Fh(\000)3420 272 y Fk(\030)3460 242 y Fg(?)3535 272 y Fm(\\)18 b Fk(\020)3650 242 y Fg(?)3644 293 y Ff(0)3707 205 y Fh(\001)3768 272 y Fk(>)23 b Fp(0)p Fl(,)i(then)0 461 y Fp([)p Fk(\021)s(;)14 b(\020)140 473 y Ff(0)178 461 y Fp(])23 b(=)f(0)k Fl(for)h(all)d Fk(\021)i Fm(\032)d Fk(\030)t Fl(,)28 b(since)f Fk(\030)1175 431 y Fg(?)1254 461 y Fm(\032)c Fk(\021)1386 431 y Fg(?)1468 461 y Fl(and)k Fk(dim)1811 394 y Fh(\000)1849 461 y Fk(\021)1893 431 y Fg(?)1967 461 y Fm(\\)19 b Fk(\020)2083 431 y Fg(?)2077 482 y Ff(0)2140 394 y Fh(\001)2201 461 y Fk(>)j Fp(0)p Fl(.)41 b(Hence,)30 b(form)m(ula)24 b(\(1.1\))i(holds)g(automatically) -8 b(.)0 650 y(It)33 b(means)f(that)h(in)e(the)i(follo)m(wing)d(it)i (su\016ces)j(to)d(pro)m(v)m(e)i(\(1.1\))e(for)g(the)h(case)g Fp([)p Fk(\030)t(;)14 b(\020)3013 662 y Ff(0)3051 650 y Fp(])23 b Fm(6)p Fp(=)g(0)p Fl(,)32 b(i.e.)1564 921 y Fk(\030)1604 887 y Fg(?)1679 921 y Fm(\\)19 b Fk(L)p Fm(f)p Fk(e)1891 933 y Fj(k)q Ff(+1)2014 921 y Fk(;)14 b(:)g(:)g(:)g(;)g(e)2238 933 y Fj(d)2276 921 y Fm(g)23 b Fp(=)f Fm(f)p Fp(0)p Fm(g)p Fk(:)1392 b Fp(\(4.2\))0 1192 y Fl(The)34 b(follo)m(wing)29 b(relation)i(holds:)1336 1463 y Fp([)p Fk(\021)s(;)14 b(\020)1476 1475 y Ff(0)1514 1463 y Fp(])23 b Fm(\021)g Fk(V)c(ol)r Fp(\()p Fk(\030)1850 1475 y Fj(i)p Ff(+1)1962 1463 y Fk(;)14 b(:)g(:)g(:)f(;)h(\030)2182 1475 y Fj(d)2221 1463 y Fk(;)g(e)2297 1475 y Fj(k)q Ff(+1)2422 1463 y Fk(;)g(:)g(:)g(:)f(;)h(e)2645 1475 y Fj(d)2684 1463 y Fp(\))23 b(=)641 1734 y(=)f Fk(V)d(ol)r Fp(\()p Fk(\030)930 1746 y Fj(j)s Ff(+1)1050 1734 y Fk(;)14 b(:)g(:)g(:)f(;)h (\030)1270 1746 y Fj(d)1309 1734 y Fk(;)g(e)1385 1746 y Fj(k)q Ff(+1)1510 1734 y Fk(;)g(:)g(:)g(:)f(;)h(e)1733 1746 y Fj(d)1771 1734 y Fp(\))19 b Fm(\001)g Fk(V)g(ol)2011 1667 y Fh(\000)2049 1734 y Fk(P)12 b(r)2151 1751 y Fj(L)2197 1735 y Fc(?)2246 1751 y Fg(f)p Fj(\030)2310 1759 y Fd(j)r Fb(+1)2412 1751 y Fj(;:::)o(;\030)2541 1760 y Fd(d)2576 1751 y Fj(;e)2627 1760 y Fd(k)q Fb(+1)2734 1751 y Fj(;:::)o(;e)2864 1760 y Fd(d)2899 1751 y Fg(g)2937 1734 y Fm(f)p Fk(\030)3015 1746 y Fj(i)p Ff(+1)3127 1734 y Fk(;)i(:)g(:)g(:)g(;)g(\030)3348 1746 y Fj(j)3383 1734 y Fm(g)3425 1667 y Fh(\001)3476 1734 y Fk(;)0 1974 y Fl(or,)32 b(brie\015y)-8 b(,)1679 2163 y Fp([)p Fk(\021)s(;)14 b(\020)1819 2175 y Ff(0)1856 2163 y Fp(])23 b(=)g([)p Fk(\030)t(;)14 b(\020)2126 2175 y Ff(0)2164 2163 y Fp(])p Fk(Q)p Fp(\()p Fk(\030)t(;)g(\021)s Fp(\))p Fk(;)1508 b Fp(\(4.3\))0 2402 y Fl(where)36 b Fk(Q)p Fp(\()p Fk(\030)t(;)14 b(\021)s Fp(\))35 b Fl(denotes)h(the)f Fp(\()p Fk(j)24 b Fm(\000)c Fk(i)p Fp(\))p Fl(-dimensional)31 b(v)m(olume)j(of)g(the)h(parallelepip)s(ed)d(spanned)k(b)m(y)g(pro)5 b(jections)0 2591 y(of)32 b Fk(\030)147 2603 y Fj(i)p Ff(+1)259 2591 y Fk(;)14 b(:)g(:)g(:)g(;)g(\030)480 2603 y Fj(j)547 2591 y Fl(\(cf.)44 b(\(4.1\)\))32 b(on)m(to)g(the)h(plane) 1262 2862 y Fk(L)1319 2828 y Fg(?)1375 2862 y Fm(f)p Fk(\030)1453 2874 y Fj(j)s Ff(+1)1572 2862 y Fk(;)14 b(:)g(:)g(:)f(;)h(\030)1792 2874 y Fj(d)1831 2862 y Fk(;)g(e)1907 2874 y Fj(k)q Ff(+1)2032 2862 y Fk(;)g(:)g(:)g(:)f(;)h(e)2255 2874 y Fj(d)2294 2862 y Fm(g)22 b Fp(=)h Fk(L)2503 2828 y Fg(?)2558 2862 y Fp(\()p Fk(\030)2630 2828 y Fg(?)2687 2862 y Fk(;)14 b(\020)2766 2828 y Fg(?)2760 2883 y Ff(0)2822 2862 y Fp(\))p Fk(:)1092 b Fp(\(4.4\))0 3133 y Fl(Th)m(us,)34 b(b)m(y)g(\(4.3\))e(w)m(e)h(ha)m(v)m(e)1520 3209 y Fh(Z)1343 3451 y Fj(\021)r Fg(2)p Fj(G)p Ff(\()p Fj(i;d)p Ff(\):)p Fj(\021)r Fg(\032)p Fj(\030)1763 3322 y Fp([)p Fk(\021)s(;)14 b(\020)1903 3334 y Ff(0)1940 3322 y Fp(])1963 3288 y Fj(\013)2011 3322 y Fk(\033)s Fp(\()p Fk(d\021)s Fp(\))24 b(=)f Fk(c)2360 3334 y Fj(\013)2407 3322 y Fp(\()p Fk(\030)t Fp(\)[)p Fk(\030)t(;)14 b(\020)2647 3334 y Ff(0)2685 3322 y Fp(])2708 3288 y Fj(\013)2756 3322 y Fk(;)1451 3646 y(c)1487 3658 y Fj(\013)1534 3646 y Fp(\()p Fk(\030)t Fp(\))24 b(=)1909 3533 y Fh(Z)1732 3775 y Fj(\021)r Fg(2)p Fj(G)p Ff(\()p Fj(i;d)p Ff(\):)p Fj(\021)r Fg(\032)p Fj(\030)2165 3646 y Fk(Q)p Fp(\()p Fk(\030)t(;)14 b(\021)s Fp(\))2416 3611 y Fj(\013)2464 3646 y Fk(\033)s Fp(\()p Fk(d\021)s Fp(\))p Fk(:)1281 b Fp(\(4.5\))0 3931 y Fl(W)-8 b(e)33 b(write)f Fk(Q)p Fp(\()p Fk(\030)t(;)14 b(\021)s Fp(\))33 b Fl(in)f(a)g(di\013eren)m(t)h(form.)42 b(First)32 b(w)m(e)i(giv)m(e)e(another)h(represen)m(tation)g(to)f(the)h(plane)f (\(4.4\):)1696 4202 y Fk(L)1753 4168 y Fg(?)1808 4202 y Fp(\()p Fk(\030)1880 4168 y Fg(?)1937 4202 y Fk(;)14 b(\020)2016 4168 y Fg(?)2010 4222 y Ff(0)2072 4202 y Fp(\))24 b(=)e Fk(\030)h Fm(\\)c Fk(\020)2384 4214 y Ff(0)2421 4202 y Fk(:)1525 b Fp(\(4.6\))0 4473 y Fl(Indeed,)34 b(if)e Fk(\034)h Fm(2)24 b Fk(L)639 4443 y Fg(?)694 4473 y Fp(\()p Fk(\030)766 4443 y Fg(?)823 4473 y Fk(;)14 b(\020)902 4443 y Fg(?)896 4494 y Ff(0)958 4473 y Fp(\))p Fl(,)33 b(then)g Fk(\034)h Fm(?)22 b(f)p Fk(\030)1507 4485 y Fj(j)s Ff(+1)1626 4473 y Fk(;)14 b(:)g(:)g(:)g(;)g(\030)1847 4485 y Fj(d)1886 4473 y Fm(g)32 b Fl(and)h Fk(\034)g Fm(?)23 b(f)p Fk(e)2388 4485 y Fj(k)q Ff(+1)2512 4473 y Fk(;)14 b(:)g(:)g(:)f(;)h(e)2735 4485 y Fj(d)2774 4473 y Fm(g)p Fl(.)43 b(Hence,)34 b Fk(\034)f Fm(2)24 b Fk(L)3408 4443 y Fg(?)3464 4473 y Fm(f)p Fk(\030)3542 4485 y Fj(j)s Ff(+1)3661 4473 y Fk(;)14 b(:)g(:)g(:)f(;)h(\030)3881 4485 y Fj(d)3920 4473 y Fm(g)23 b Fp(=)g Fk(\030)t Fl(,)0 4662 y Fk(\034)33 b Fm(2)23 b Fk(L)204 4632 y Fg(?)260 4662 y Fm(f)p Fk(e)341 4674 y Fj(k)q Ff(+1)465 4662 y Fk(;)14 b(:)g(:)g(:)f(;)h(e)688 4674 y Fj(d)727 4662 y Fm(g)22 b Fp(=)h Fk(\020)915 4674 y Ff(0)952 4662 y Fl(,)32 b(or)e Fk(\034)j Fm(2)23 b Fk(\030)d Fm(\\)15 b Fk(\020)1437 4674 y Ff(0)1475 4662 y Fl(.)43 b(Th)m(us,)32 b Fk(L)1874 4632 y Fg(?)1930 4662 y Fp(\()p Fk(\030)2002 4632 y Fg(?)2058 4662 y Fk(;)14 b(\020)2137 4632 y Fg(?)2131 4683 y Ff(0)2194 4662 y Fp(\))23 b Fm(\022)g Fk(\030)c Fm(\\)d Fk(\020)2499 4674 y Ff(0)2536 4662 y Fl(.)43 b(Since)31 b Fk(dim)3018 4595 y Fh(\000)3056 4662 y Fk(L)3113 4632 y Fg(?)3168 4662 y Fp(\()p Fk(\030)3240 4632 y Fg(?)3297 4662 y Fk(;)14 b(\020)3376 4632 y Fg(?)3370 4683 y Ff(0)3432 4662 y Fp(\))3464 4595 y Fh(\001)3525 4662 y Fp(=)23 b Fk(dim)p Fp(\()p Fk(\030)d Fm(\\)15 b Fk(\020)3952 4674 y Ff(0)3990 4662 y Fp(\))31 b Fl(as)0 4851 y(w)m(e)j(use)f(the)g (ob)m(vious)g(form)m(ula)1440 5040 y Fk(dim)p Fp(\()p Fk(a)19 b Fm(\\)g Fk(b)p Fp(\))k Fm(\025)f Fk(dim)p Fp(\()p Fk(a)p Fp(\))d(+)f Fk(dim)p Fp(\()p Fk(b)p Fp(\))g Fm(\000)g Fk(d:)1270 b Fp(\(4.7\))2028 5239 y(54)p eop %%Page: 55 9 55 8 bop 0 83 a Fl(The)34 b(relation)c(\(4.6\))i(is)g(pro)m(v)m(ed.)45 b(No)m(w)33 b(sho)m(w)h(that)860 367 y Fk(Q)p Fp(\()p Fk(\030)t(;)14 b(\021)s Fp(\))24 b(=)e Fk(V)d(ol)1356 333 y Ff(\()p Fj(d)p Fg(\000)p Fj(k)q Ff(+)p Fj(j)s Fg(\000)p Fj(i)p Ff(\))1706 275 y Fh(\020)1756 367 y Fk(\030)1792 379 y Fj(j)s Ff(+1)1911 367 y Fk(;)14 b(:)g(:)g(:)f(;)h(\030)2131 379 y Fj(d)2170 367 y Fk(;)g(b)2243 275 y Fh(\020)2293 367 y Fp(\()p Fk(\030)22 b Fm(\\)d Fk(\020)2493 379 y Ff(0)2531 367 y Fp(\))2563 333 y Fg(?)2563 387 y Fj(\030)2619 275 y Fh(\021\021)2755 325 y Fj(de)p 2755 338 67 4 v 2755 355 V(f)2887 367 y Fp([)p Fk(\030)k Fm(\\)c Fk(\020)3079 379 y Ff(0)3116 367 y Fk(;)14 b(\021)s Fp(])3220 379 y Fj(\030)3257 367 y Fk(:)689 b Fp(\(4.8\))0 651 y Fl(By)33 b(de\014nition)f(of)g Fk(Q)p Fp(\()p Fk(\030)t(;)14 b(\021)s Fp(\))p Fl(,)33 b(o)m(wing)f(to)g(\(4.6\),)1472 935 y Fk(Q)p Fp(\()p Fk(\030)t(;)14 b(\021)s Fp(\))24 b(=)e Fk(V)d(ol)1968 900 y Ff(\()p Fj(j)s Fg(\000)p Fj(i)p Ff(\))2144 867 y Fh(\000)2182 935 y Fk(P)12 b(r)2284 947 y Fj(\030)r Fg(\\)p Fj(\020)2391 955 y Fb(0)2428 935 y Fp(\()p Fk(\021)s Fp(\))2536 900 y Fg(?)2536 955 y Fj(\030)2593 867 y Fh(\001)2645 935 y Fk(:)1301 b Fp(\(4.9\))0 1218 y Fl(The)34 b(follo)m(wing)29 b(form)m(ula)i(holds)h(for)g(an)m(y) i(\015ats)f Fk(a)f Fl(and)h Fk(c)f Fl(in)g(arbitrary)f(am)m(bien)m(t)h (space:)920 1502 y Fp([)p Fk(a)987 1468 y Fg(?)1043 1502 y Fk(;)14 b(c)p Fp(])1161 1460 y Fj(de)p 1161 1473 V 1161 1490 V 1 w(f)1294 1502 y Fk(V)19 b(ol)1441 1435 y Fh(\000)1479 1502 y Fk(b)p Fp(\()p Fk(a)p Fp(\))p Fk(;)14 b(b)p Fp(\()p Fk(c)1764 1468 y Fg(?)1820 1502 y Fp(\))1852 1435 y Fh(\001)1913 1502 y Fp(=)23 b Fk(V)c(ol)r Fp(\()p Fk(P)12 b(r)2269 1514 y Fj(c)2303 1502 y Fp(\()p Fk(a)p Fp(\)\))24 b(=)e Fk(V)d(ol)2702 1410 y Fh(\020)2751 1502 y Fk(P)12 b(r)2853 1519 y Fj(a)2889 1502 y Fc(?)2943 1410 y Fh(\020)2992 1502 y Fk(c)3028 1468 y Fg(?)3084 1410 y Fh(\021\021)3197 1502 y Fk(:)0 1786 y Fl(Hence,)34 b(equalit)m(y)e(\(4.9\))g(yields)849 2070 y Fk(Q)p Fp(\()p Fk(\030)t(;)14 b(\021)s Fp(\))23 b(=)g Fk(V)c(ol)1345 2036 y Ff(\()p Fj(j)s Fg(\000)p Fj(i)p Ff(\))1521 2003 y Fh(\000)1559 2070 y Fk(P)12 b(r)1661 2082 y Fj(\030)r Fg(\\)p Fj(\020)1768 2090 y Fb(0)1805 2070 y Fp(\()p Fk(\021)s Fp(\))1913 2036 y Fg(?)1913 2090 y Fj(\030)1969 2003 y Fh(\001)2031 2070 y Fp(=)22 b Fk(V)d(ol)2266 2003 y Fh(\000)2304 2070 y Fk(P)12 b(r)2406 2082 y Fj(\021)2446 2070 y Fp(\()p Fk(\030)23 b Fm(\\)c Fk(\020)2647 2082 y Ff(0)2685 2070 y Fp(\))2717 2036 y Fg(?)2717 2090 y Fj(\030)2773 2003 y Fh(\001)2834 2070 y Fp(=)k([)p Fk(\030)f Fm(\\)d Fk(\020)3113 2082 y Ff(0)3151 2070 y Fk(;)14 b(\021)s Fp(])3255 2082 y Fj(\030)0 2354 y Fl(\(here)33 b(the)g(am)m(bien)m(t)f(space)i(is)e Fk(\030)t Fl(\).)44 b(Th)m(us,)34 b(relation)c(\(4.8\))i(is)h(pro)m(v)m(ed,)h(and)e(one)h (obtains)f(b)m(y)i(\(4.5\))1410 2638 y Fk(c)1446 2650 y Fj(\013)1493 2638 y Fp(\()p Fk(\030)t Fp(\))24 b(=)1868 2525 y Fh(Z)1691 2767 y Fj(\021)r Fg(2)p Fj(G)p Ff(\()p Fj(i;d)p Ff(\):)p Fj(\021)r Fg(\032)p Fj(\030)2110 2638 y Fp([)p Fk(\030)f Fm(\\)c Fk(\020)2302 2650 y Ff(0)2339 2638 y Fk(;)14 b(\021)s Fp(])2443 2603 y Fj(\013)2443 2658 y(\030)2505 2638 y Fk(\033)s Fp(\()p Fk(d\021)s Fp(\))p Fk(:)0 2967 y Fl(W)-8 b(e)33 b(pro)m(v)m(e)h(that)f Fk(c)679 2979 y Fj(\013)726 2967 y Fp(\()p Fk(\030)t Fp(\))g Fl(do)s(es)g(not)g(dep)s(end)h(on)e Fk(\030)t Fl(.)44 b(According)33 b(to)f(\(4.2\),)g(it)g(is)g(su\016cien)m(t)i(to) e(consider)h(only)g(the)0 3156 y(case)1797 3345 y Fk(\030)1837 3311 y Fg(?)1912 3345 y Fm(\\)19 b Fk(\020)2028 3311 y Fg(?)2022 3366 y Ff(0)2107 3345 y Fp(=)k Fm(f)p Fp(0)p Fm(g)p Fk(:)1584 b Fp(\(4.10\))0 3590 y Fl(F)-8 b(or)32 b(an)m(y)h Fk(\030)27 b Fm(2)d Fk(G)p Fp(\()p Fk(j;)14 b(d)p Fp(\))33 b Fl(there)h(exists)f(a)f(rotation)f Fk(\015)d Fm(2)23 b Fk(S)5 b(O)r Fp(\()p Fk(d)p Fp(\))p Fl(,)34 b(suc)m(h)g(that)e Fk(\030)c Fp(=)22 b Fk(\015)5 b(\030)2855 3602 y Ff(0)2892 3590 y Fl(,)33 b Fk(\030)2988 3602 y Ff(0)3049 3590 y Fp(=)22 b Fk(L)p Fm(f)p Fk(e)3274 3602 y Ff(1)3310 3590 y Fk(;)14 b(:)g(:)g(:)g(;)g(e)3534 3602 y Fj(j)3568 3590 y Fm(g)p Fl(.)43 b(Then)637 3874 y Fk(c)673 3886 y Fj(\013)720 3874 y Fp(\()p Fk(\030)t Fp(\))24 b(=)1110 3761 y Fh(Z)918 4003 y Fj(\021)r Fg(2)p Fj(G)p Ff(\()p Fj(i;d)p Ff(\):)p Fj(\021)r Fg(\032)p Fj(\030)1318 4011 y Fb(0)1368 3874 y Fp([)p Fk(\015)5 b(\030)1475 3886 y Ff(0)1530 3874 y Fm(\\)19 b Fk(\020)1640 3886 y Ff(0)1678 3874 y Fk(;)14 b(\015)5 b(\021)s Fp(])1830 3840 y Fj(\013)1830 3895 y(\015)t(\030)1899 3903 y Fb(0)1948 3874 y Fk(\033)s Fp(\()p Fk(d\021)s Fp(\))25 b(=)2436 3761 y Fh(Z)2244 4003 y Fj(\021)r Fg(2)p Fj(G)p Ff(\()p Fj(i;d)p Ff(\):)p Fj(\021)r Fg(\032)p Fj(\030)2644 4011 y Fb(0)2694 3874 y Fp([)p Fk(\030)2753 3886 y Ff(0)2809 3874 y Fm(\\)18 b Fk(\015)2930 3840 y Fg(\000)p Ff(1)3019 3874 y Fk(\020)3055 3886 y Ff(0)3093 3874 y Fk(;)c(\021)s Fp(])3197 3840 y Fj(\013)3197 3895 y(\030)3227 3903 y Fb(0)3277 3874 y Fk(\033)s Fp(\()p Fk(d\021)s Fp(\))p Fk(;)0 4204 y Fl(and)33 b(\(4.10\))e(has)i(the)g(form)1235 4393 y Fp(\()p Fk(\015)5 b(\030)1351 4405 y Ff(0)1388 4393 y Fp(\))1420 4358 y Fg(?)1495 4393 y Fm(\\)19 b Fk(\020)1611 4358 y Fg(?)1605 4413 y Ff(0)1690 4393 y Fp(=)k Fk(\015)5 b(\030)1866 4358 y Fg(?)1862 4413 y Ff(0)1940 4393 y Fm(\\)19 b Fk(\020)2056 4358 y Fg(?)2050 4413 y Ff(0)2136 4393 y Fp(=)j Fk(\030)2263 4358 y Fg(?)2259 4413 y Ff(0)2338 4393 y Fm(\\)d Fk(\015)2460 4358 y Fg(\000)p Ff(1)2548 4393 y Fk(\020)2590 4358 y Fg(?)2584 4413 y Ff(0)2670 4393 y Fp(=)k Fm(f)p Fp(0)p Fm(g)p Fk(:)0 4638 y Fl(Without)32 b(loss)g(of)g(generalit)m(y)-8 b(,)32 b(one)h(can)g(substitute)g Fk(\015)k Fl(for)32 b Fk(\015)2249 4607 y Fg(\000)p Ff(1)2338 4638 y Fl(.)43 b(Th)m(us,)35 b(w)m(e)e(ha)m(v)m(e)h(to)e(sho)m(w)i(that)1325 4921 y Fp(~)-44 b Fk(c)1359 4933 y Fj(\013)1406 4921 y Fp(\()p Fk(\015)5 b Fp(\))1542 4879 y Fj(de)p 1542 4892 V 1542 4909 V(f)1862 4808 y Fh(Z)1669 5050 y Fj(\021)r Fg(2)p Fj(G)p Ff(\()p Fj(i;d)p Ff(\):)p Fj(\021)r Fg(\032)p Fj(\030)2069 5058 y Fb(0)2120 4921 y Fp([)p Fk(\030)2179 4933 y Ff(0)2235 4921 y Fm(\\)18 b Fk(\015)5 b(\020)2392 4933 y Ff(0)2430 4921 y Fk(;)14 b(\021)s Fp(])2534 4887 y Fj(\013)2534 4942 y(\030)2564 4950 y Fb(0)2614 4921 y Fk(\033)s Fp(\()p Fk(d\021)s Fp(\))1113 b(\(4.11\))2028 5239 y(55)p eop %%Page: 56 10 56 9 bop 0 83 a Fl(is)32 b(constan)m(t)i(on)e(the)h(set)1371 273 y Fk(G)1436 285 y Fj(j)s(k)1531 273 y Fp(=)1618 206 y Fh(\010)1667 273 y Fk(\015)27 b Fm(2)d Fk(S)5 b(O)r Fp(\()p Fk(d)p Fp(\))24 b(:)f Fk(\030)2154 239 y Fg(?)2150 294 y Ff(0)2229 273 y Fm(\\)c Fk(\015)5 b(\020)2393 239 y Fg(?)2387 294 y Ff(0)2472 273 y Fp(=)22 b Fm(f)p Fp(0)p Fm(g)2685 206 y Fh(\011)2746 273 y Fk(:)1159 b Fp(\(4.12\))0 521 y Fl(First,)32 b(let)g(us)h(\014rst)g(pro)m(v)m(e)h(that)1379 711 y Fk(dim)p Fp(\()p Fk(\030)1592 723 y Ff(0)1648 711 y Fm(\\)19 b Fk(\015)5 b(\020)1806 723 y Ff(0)1843 711 y Fp(\))24 b(=)e Fk(j)i Fp(+)18 b Fk(k)j Fm(\000)d Fk(d;)112 b(\015)27 b Fm(2)d Fk(G)2666 723 y Fj(j)s(k)2738 711 y Fk(:)0 959 y Fl(By)33 b(\(4.7\),)e(the)i(dimension)e(of)g Fk(\030)1187 971 y Ff(0)1242 959 y Fm(\\)18 b Fk(\015)5 b(\020)1399 971 y Ff(0)1468 959 y Fl(can)33 b(not)f(b)s(e)g(less)g (than)h Fk(j)22 b Fp(+)17 b Fk(k)k Fm(\000)c Fk(d)p Fl(.)43 b(Let)33 b(us)f(pro)m(v)m(e)i(that)e(it)f(also)g(can)h(not)0 1148 y(b)s(e)j(greater)g(than)g Fk(j)25 b Fp(+)19 b Fk(k)k Fm(\000)c Fk(d)p Fl(.)51 b(Supp)s(ose,)36 b(ex)g(adv)m(erso,)h(that)d Fk(dim)p Fp(\()p Fk(\030)2463 1160 y Ff(0)2521 1148 y Fm(\\)20 b Fk(\015)5 b(\020)2680 1160 y Ff(0)2717 1148 y Fp(\))27 b(=)f Fk(m)g(>)g(j)e Fp(+)c Fk(k)i Fm(\000)e Fk(d)p Fl(.)50 b(Let)36 b Fk(\034)3684 1160 y Ff(1)3721 1148 y Fk(;)14 b(:)g(:)g(:)g(;)g(\034)3942 1160 y Fj(m)4040 1148 y Fl(b)s(e)0 1337 y(the)33 b(basis)g(in)e Fk(\030)557 1349 y Ff(0)613 1337 y Fm(\\)19 b Fk(\015)5 b(\020)771 1349 y Ff(0)808 1337 y Fl(.)44 b(Amplify)30 b(it)i(to)g(the)h(bases)h (in)e Fk(\030)2046 1349 y Ff(0)2116 1337 y Fl(and)h Fk(\015)5 b(\020)2390 1349 y Ff(0)2427 1337 y Fl(:)807 1627 y Fk(\030)843 1639 y Ff(0)904 1627 y Fp(=)23 b Fk(L)p Fm(f)p Fk(\034)1127 1639 y Ff(1)1163 1627 y Fk(;)14 b(:)g(:)g(:)g(;)g(\034)1384 1639 y Fj(m)1447 1627 y Fk(;)g(\034)1520 1639 y Fj(m)p Ff(+1)1667 1627 y Fk(;)g(:)g(:)g(:)g(;)g(\034)1888 1639 y Fj(j)1923 1627 y Fm(g)p Fk(;)111 b(\015)5 b(\020)2183 1639 y Ff(0)2243 1627 y Fp(=)23 b Fk(L)p Fm(f)p Fk(\034)2466 1639 y Ff(1)2502 1627 y Fk(;)14 b(:)g(:)g(:)g(;)g(\034)2723 1639 y Fj(m)2786 1627 y Fk(;)19 b Fp(~)-47 b Fk(\034)2859 1639 y Fj(m)p Ff(+1)3007 1627 y Fk(;)14 b(:)g(:)g(:)f(;)18 b Fp(~)-46 b Fk(\034)3227 1639 y Fj(k)3268 1627 y Fm(g)p Fk(:)0 1917 y Fl(The)34 b(n)m(um)m(b)s(er)g(of)f(distinct)g(unit)g(v)m (ectors)i(in)e Fk(\030)1713 1929 y Ff(0)1751 1917 y Fl(,)g Fk(\015)5 b(\020)1895 1929 y Ff(0)1966 1917 y Fl(is)33 b(equal)h(to)f Fk(j)24 b Fp(+)18 b Fk(k)k Fm(\000)d Fk(m)p Fl(.)46 b(As)34 b Fk(m)24 b(>)g(j)g Fp(+)19 b Fk(k)j Fm(\000)c Fk(d)p Fl(,)34 b(that)g(n)m(um)m(b)s(er)0 2106 y(is)h(less)g(than)g Fk(d)p Fl(.)51 b(So)35 b(there)g(exists)h(at)f (least)f(one)i(unit)e(v)m(ector)i Fk(x)27 b Fm(2)g Fn(I)-14 b(R)2582 2069 y Fj(d)2621 2106 y Fl(,)35 b(that)g(do)s(es)h(not)e(b)s (elong)g(to)h(the)h(linear)0 2295 y(h)m(ull)31 b(of)h(the)h(bases)h(in) e Fk(\030)877 2307 y Ff(0)947 2295 y Fl(and)h Fk(\015)5 b(\020)1221 2307 y Ff(0)1258 2295 y Fl(.)43 b(Then)34 b Fk(x)24 b Fm(2)f Fk(\030)1772 2264 y Fg(?)1768 2315 y Ff(0)1847 2295 y Fm(\\)c Fp(\()p Fk(\015)5 b(\020)2037 2307 y Ff(0)2074 2295 y Fp(\))2106 2264 y Fg(?)2162 2295 y Fl(.)44 b(W)-8 b(e)33 b(arriv)m(ed)f(at)h(the)g(con)m(tradiction)e (with)h(\(4.12\).)166 2484 y(Th)m(us,)47 b(w)m(e)d(pro)m(v)m(ed)g(that) f(an)m(y)g(transform)f Fk(\015)g Fm(2)c Fk(G)2051 2496 y Fj(j)s(k)2166 2484 y Fl(preserv)m(es)45 b(the)f(dimension)d(of)h Fk(\014)3454 2442 y Fj(de)p 3454 2455 67 4 v 3454 2472 V(f)3601 2484 y Fk(\030)3637 2496 y Ff(0)3699 2484 y Fm(\\)24 b Fk(\015)5 b(\020)3862 2496 y Ff(0)3937 2484 y Fm(\032)37 b Fk(\030)4075 2496 y Ff(0)4113 2484 y Fl(.)0 2673 y(Iden)m(tifying)32 b Fk(\030)529 2685 y Ff(0)599 2673 y Fl(with)g Fn(I)-14 b(R)915 2636 y Fj(j)950 2673 y Fl(,)33 b(w)m(e)h(can)e(rewrite)h(the)g(relation)e(\(4.11\))h(as)g (follo)m(ws:)1080 2962 y Fp(\026)-44 b Fk(c)1114 2974 y Fj(\013)1162 2962 y Fp(\()p Fk(\014)t Fp(\))23 b(=)1583 2849 y Fh(Z)1370 3095 y Fj(\021)r Fg(2)p Fj(G)p Ff(\()p Fj(i;d)p Ff(\):)p Fj(\021)r Fg(\032)p Fa(I)-11 b(R)1811 3075 y Fd(j)1861 2962 y Fp([)p Fk(\014)t(;)14 b(\021)s Fp(])2039 2928 y Fj(\013)2039 2988 y Fa(I)-11 b(R)2112 2969 y Fd(j)2161 2962 y Fk(\033)s Fp(\()p Fk(d\021)s Fp(\))24 b(=)2521 2849 y Fh(Z)2456 3091 y Fj(G)p Ff(\()p Fj(i;j)s Ff(\))2651 2962 y Fp([)p Fk(\014)t(;)14 b(\021)s Fp(])2829 2928 y Fj(\013)2829 2988 y Fa(I)-11 b(R)2902 2969 y Fd(j)2951 2962 y Fk(d\021)s(;)0 3296 y(\014)31 b Fm(\032)26 b Fk(\030)205 3308 y Ff(0)242 3296 y Fl(,)36 b Fk(dim)p Fp(\()p Fk(\014)t Fp(\))27 b(=)f Fk(j)f Fp(+)19 b Fk(k)k Fm(\000)c Fk(d)p Fl(.)51 b(No)m(w)35 b(pro)m(v)m(e)h(that)h Fp(\026)-44 b Fk(c)1836 3308 y Fj(\013)1883 3296 y Fp(\()p Fk(\014)t Fp(\))36 b Fl(do)s(es)f(not)g(dep)s(end)h(on)f Fk(\014)c Fm(2)c Fk(G)p Fp(\()p Fk(j)e Fp(+)19 b Fk(k)k Fm(\000)c Fk(d;)14 b(j)5 b Fp(\))p Fl(.)51 b(Indeed,)37 b(b)m(y)0 3485 y(rotation)31 b(in)m(v)-5 b(ariance)32 b(\(since)h Fk(dim)p Fp(\()p Fk(\014)t Fp(\))g Fl(do)s(es)g(not)f(dep)s (end)i(on)e Fk(\015)c Fm(\032)23 b Fk(G)2496 3497 y Fj(j)s(k)2568 3485 y Fl(\),)1301 3774 y Fp(\026)-44 b Fk(c)1335 3786 y Fj(\013)1382 3774 y Fp(\()p Fk(\014)t Fp(\))24 b(=)h(\026)-44 b Fk(c)1645 3786 y Fj(\013)1692 3774 y Fp(\()p Fk(\014)1771 3786 y Ff(0)1809 3774 y Fp(\))p Fk(;)111 b(\014)2022 3786 y Ff(0)2083 3774 y Fp(=)22 b Fk(L)p Fm(f)p Fk(e)2308 3786 y Ff(1)2344 3774 y Fk(;)14 b(:)g(:)g(:)g(;)g(e)2568 3786 y Fj(j)s Ff(+)p Fj(k)q Fg(\000)p Fj(d)2776 3774 y Fm(g)p Fk(:)0 4064 y Fl(Th)m(us,)34 b(w)m(e)g(ha)m(v)m(e)g(pro)m(v)m (ed)g(that)1639 4254 y Fk(c)p Fp(\()p Fk(\013)p Fp(\))24 b(=)1951 4141 y Fh(Z)1886 4383 y Fj(G)p Ff(\()p Fj(i;j)s Ff(\))2081 4254 y Fp([)p Fk(\014)2151 4266 y Ff(0)2188 4254 y Fk(;)14 b(\021)s Fp(])2292 4220 y Fj(\013)2292 4280 y Fa(I)-11 b(R)2365 4261 y Fd(j)2414 4254 y Fk(d\021)0 4542 y Fl(is)32 b(a)g(constan)m(t,)h Fk(\014)646 4554 y Ff(0)706 4542 y Fp(=)23 b Fk(L)p Fm(f)p Fk(e)932 4554 y Ff(1)968 4542 y Fk(;)14 b(:)g(:)g(:)g(;)g(e)1192 4554 y Fj(j)s Ff(+)p Fj(k)q Fg(\000)p Fj(d)1400 4542 y Fm(g)23 b(2)g Fk(G)p Fp(\()p Fk(j)h Fp(+)17 b Fk(k)k Fm(\000)d Fk(d;)c(j)5 b Fp(\))p Fk(:)32 b Fl(No)m(w)h(our)f(aim)f(is)g(to)h (calculate)f Fk(c)p Fp(\()p Fk(\013)p Fp(\))p Fl(.)45 b(F)-8 b(or)31 b Fk(n)18 b Fp(+)g Fk(r)25 b Fm(\025)e Fk(m)p Fl(,)0 4731 y Fk(\014)47 4743 y Ff(0)107 4731 y Fp(=)g Fk(L)p Fm(f)p Fk(e)333 4743 y Ff(1)369 4731 y Fk(;)14 b(:)g(:)g(:)f(;)h(e)592 4743 y Fj(r)629 4731 y Fm(g)32 b Fl(w)m(e)h(put)1477 4921 y Fk(b)1513 4933 y Fj(\013)1560 4921 y Fp(\()p Fk(n;)14 b(m;)g(r)r Fp(\))24 b(=)2042 4808 y Fh(Z)1954 5050 y Fj(G)p Ff(\()p Fj(n;m)p Ff(\))2195 4921 y Fp([)p Fk(\014)2265 4933 y Ff(0)2302 4921 y Fk(;)14 b(\021)s Fp(])2406 4887 y Fj(\013)2406 4942 y Fa(I)-11 b(R)2479 4923 y Fd(m)2552 4921 y Fk(d\021)s(:)2028 5239 y Fp(56)p eop %%Page: 57 11 57 10 bop 0 83 a Fl(Then)36 b Fk(c)p Fp(\()p Fk(\013)p Fp(\))27 b(=)e Fk(b)563 95 y Fj(\013)610 83 y Fp(\()p Fk(i;)14 b(j;)g(j)25 b Fp(+)19 b Fk(k)k Fm(\000)c Fk(d)p Fp(\))p Fl(.)50 b(W)-8 b(e)35 b(calculate)f Fk(b)1839 95 y Fj(\013)1886 83 y Fp(\()p Fk(n;)14 b(m;)g(r)r Fp(\))35 b Fl(for)f(all)f Fk(n;)14 b(m;)g(r)r Fl(,)35 b(suc)m(h)h(that)f Fk(n)19 b Fp(+)g Fk(r)29 b Fm(\025)d Fk(m)g Fm(\025)f Fp(2)p Fl(.)50 b(A)m(t)35 b(the)0 272 y(\014rst)e(step,)h(w)m(e)f(pro)m (v)m(e)h(that)1342 461 y Fk(b)1378 473 y Fj(\013)1425 461 y Fp(\()p Fk(k)s(;)14 b(d;)g(i)p Fp(\))23 b(=)f Fk(b)1827 473 y Fj(\013)1874 461 y Fp(\()p Fk(i;)14 b(j;)g(i)k Fp(+)g Fk(j)24 b Fm(\000)18 b Fk(d)p Fp(\))h Fm(\001)f Fk(b)2485 473 y Fj(\013)2532 461 y Fp(\()p Fk(k)s(;)c(d;)g(j)5 b Fp(\))1130 b(\(4.13\))0 702 y Fl(for)32 b Fk(i)18 b Fp(+)g Fk(k)26 b Fm(\025)d Fk(d)p Fl(,)32 b Fk(i)23 b(<)g(j)5 b Fl(,)32 b Fk(i)23 b Fm(\025)g Fk(d=)p Fp(2)p Fl(,)31 b Fk(d)24 b Fm(\025)e Fp(2)p Fl(.)43 b(In)m(tegrate)34 b(the)f(equalit)m(y)1573 864 y Fh(Z)1395 1106 y Fj(\021)r Fg(2)p Fj(G)p Ff(\()p Fj(i;d)p Ff(\):)p Fj(\021)r Fg(\032)p Fj(\030)1815 977 y Fp([)p Fk(\021)s(;)14 b(\020)6 b Fp(])1984 943 y Fj(\013)2046 977 y Fk(\033)s Fp(\()p Fk(d\021)s Fp(\))25 b(=)d Fk(c)p Fp(\()p Fk(\013)p Fp(\)[)p Fk(\030)t(;)14 b(\020)6 b Fp(])2677 943 y Fj(\013)3928 977 y Fp(\(4.14\))0 1298 y Fl(with)27 b(resp)s(ect)j(to)d Fk(\020)6 b Fl(,)29 b(where)g Fk(\020)h Fm(2)23 b Fk(G)p Fp(\()p Fk(k)s(;)14 b(d)p Fp(\))29 b Fl(and)f Fk(\030)f Fm(2)c Fk(G)p Fp(\()p Fk(j;)14 b(d)p Fp(\))29 b Fl(\(\(4.14\))e(follo)m(ws)g(from)g(\(1.1\))g (b)m(y)i(rotation)d(in)m(v)-5 b(ariance\).)0 1487 y(By)33 b(F)-8 b(ubini's)31 b(theorem,)1181 1682 y Fh(Z)1003 1924 y Fj(\021)r Fg(2)p Fj(G)p Ff(\()p Fj(i;d)p Ff(\):)p Fj(\021)r Fg(\032)p Fj(\030)1437 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2576 y Fk(\033)s Fp(\()p Fk(d\021)s Fp(\))25 b(=)d Fk(c)p Fp(\()p Fk(\013)p Fp(\))p Fk(b)2536 2588 y Fj(\013)2584 2576 y Fp(\()p Fk(k)s(;)14 b(d;)g(j)5 b Fp(\))p Fk(:)0 2897 y Fl(As)33 b(the)g(total)e(mass)i(of)f(the)h (measure)g Fk(\033)j Fl(is)c(one,)h(the)g(ab)s(o)m(v)m(e)g(relation)e (completes)h(the)h(pro)s(of)f(of)g(\(4.13\).)166 3086 y(By)h(lemma)d(2.2)j(\(a\))f(of)g([13])g(\(where)i(the)f(op)s(erators)f Fk(A)h Fl(and)g Fk(A)2509 3056 y Fg(\003)2580 3086 y Fl(are)f(applied)g(to)g(a)g(constan)m(t)i(function\),)648 3381 y Fk(b)684 3393 y Fj(\013)731 3381 y Fp(\()p Fk(d)19 b Fm(\000)f Fp(1)p Fk(;)c(d;)g(d)k Fm(\000)g Fk(k)s Fp(\))23 b(=)g Fk(b)1436 3393 y Fj(\013)1483 3381 y Fp(\()p Fk(d)c Fm(\000)f Fk(k)s(;)c(d;)g(d)k Fm(\000)g Fp(1\))23 b(=)2162 3324 y Fk(!)2214 3336 y Fj(d)p Fg(\000)p Fj(k)2341 3324 y Fk(!)2393 3336 y Fj(k)p 2162 3362 272 4 v 2252 3438 a Fk(!)2304 3450 y Fj(d)2500 3251 y Ff(1)2457 3268 y Fh(Z)2463 3503 y Ff(0)2540 3381 y Fp(\(1)18 b Fm(\000)g Fk(t)2745 3346 y Ff(2)2782 3381 y Fp(\))2814 3346 y Ff(\()p Fj(k)q Fg(\000)p Ff(2\))p Fj(=)p Ff(2)3060 3381 y Fk(t)3090 3346 y Fj(d)p Fg(\000)p Fj(k)q Fg(\000)p Ff(1+)p Fj(\013)3396 3381 y Fk(dt:)436 b Fp(\(4.15\))0 3681 y Fl(The)34 b(in)m(tegral)c(in)i (the)h(righ)m(t-hand)f(side)h(of)f(\(4.15\))f(is)i(equal)f(to)1156 3919 y Fp(1)p 1156 3956 42 4 v 1156 4032 a(2)1264 3846 y Ff(1)1221 3862 y Fh(Z)1228 4098 y Ff(0)1304 3975 y Fp(\(1)18 b Fm(\000)g Fk(u)p Fp(\))1559 3941 y Fj(k)q(=)p Ff(2)p Fg(\000)p Ff(1)1752 3975 y Fk(u)1810 3915 y Fd(d)p Fc(\000)p Fd(k)q Fb(+)p Fd(\013)p 1810 3929 189 4 v 1890 3961 a Fb(2)2008 3941 y Fg(\000)p Ff(1)2097 3975 y Fk(du)23 b Fp(=)2308 3919 y(1)p 2308 3956 42 4 v 2308 4032 a(2)2370 3914 y(\000)2436 3847 y Fh(\000)2484 3881 y Fj(k)p 2484 3895 37 4 v 2486 3942 a Ff(2)2530 3847 y Fh(\001)2582 3914 y Fp(\000)2648 3847 y Fh(\000)2696 3881 y Fj(d)p Ff(+)p Fj(\013)p Fg(\000)p Fj(k)p 2696 3895 218 4 v 2788 3942 a Ff(2)2923 3847 y Fh(\001)p 2370 3956 592 4 v 2520 4039 a 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