%!PS-Adobe-2.0 %%Creator: dvips 5.83 Copyright 1998 Radical Eye Software %%Title: Q-spl-cl.dvi %%CreationDate: Mon Nov 4 12:58:16 2002 %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\EMTEX\BIN\dvips32.EXE -Pljh_a4 Q-spl-cl %+ -pj:C:\WINNT\TEMP\dv1.mfj %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.11.02:1123 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true 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b Fo(\000)g Fn(t)2075 918 y Fm(i)2102 906 y Fp(\))h(+)e Fn(q)2278 918 y Fm(i)2306 906 y Fp(,)32 b(where)e Fn(h)2652 918 y Fm(i)2709 906 y Fp(=)e Fn(t)2832 918 y Fm(i)p Fy(+1)2965 906 y Fo(\000)20 b Fn(t)3080 918 y Fm(i)3108 906 y Fn(;)45 b(q)3213 918 y Fm(i)3269 906 y Fn(>)29 b Fp(0.)333 1005 y(W)-7 b(e)28 b(consider)f(no)n(w)g(the)h(CCT-system)e Fo(f)p Fn(u)1688 1017 y Fy(1)1725 1005 y Fn(;)14 b(u)1810 1017 y Fy(2)1846 1005 y Fn(;)g(u)1931 1017 y Fy(3)1968 1005 y Fn(;)g(u)2053 1017 y Fy(4)2090 1005 y Fo(g)p Fp(:)1044 1207 y Fn(u)1092 1219 y Fy(1)1129 1207 y Fp(\()p Fn(x)p Fp(\))84 b(=)f(1)1044 1379 y Fn(u)1092 1391 y Fy(2)1129 1379 y Fp(\()p Fn(x)p Fp(\))h(=)1472 1266 y Ff(Z)1555 1286 y Fm(x)1518 1454 y(a)1624 1379 y Fn(dt)1697 1391 y Fy(2)3321 1379 y Fp(\(1\))1044 1616 y Fn(u)1092 1628 y Fy(3)1129 1616 y Fp(\()p Fn(x)p Fp(\))g(=)1472 1503 y Ff(Z)1555 1523 y Fm(x)1518 1691 y(a)1624 1616 y Fn(dt)1697 1628 y Fy(2)1762 1503 y Ff(Z)1845 1523 y Fm(t)1870 1531 y Fg(2)1808 1691 y Fm(a)1921 1616 y Fn(q)s Fp(\()p Fn(t)2023 1628 y Fy(3)2060 1616 y Fp(\))14 b Fn(dt)2179 1628 y Fy(3)1044 1853 y Fn(u)1092 1865 y Fy(4)1129 1853 y Fp(\()p Fn(x)p Fp(\))84 b(=)1472 1740 y Ff(Z)1555 1760 y Fm(x)1518 1928 y(a)1624 1853 y Fn(dt)1697 1865 y Fy(2)1762 1740 y Ff(Z)1845 1760 y Fm(t)1870 1768 y Fg(2)1808 1928 y Fm(a)1921 1853 y Fn(q)s Fp(\()p Fn(t)2023 1865 y Fy(3)2060 1853 y Fp(\))14 b Fn(dt)2179 1865 y Fy(3)2245 1740 y Ff(Z)2328 1760 y Fm(t)2353 1768 y Fg(3)2291 1928 y Fm(a)2417 1853 y Fn(dt)2490 1865 y Fy(4)2693 1853 y Fn(;)333 2097 y Fp(with)28 b(generalized)e(deriv)-5 b(ativ)n(es)1568 2298 y Fn(L)1625 2310 y Fy(1)1744 2298 y Fp(=)83 b Fn(D)1568 2454 y(L)1625 2466 y Fy(2)1744 2454 y Fp(=)1902 2397 y(1)p 1902 2435 42 4 v 1903 2511 a Fn(q)1953 2454 y(D)2024 2419 y Fy(2)1568 2670 y Fn(L)1625 2682 y Fy(3)1744 2670 y Fp(=)g Fn(D)1973 2613 y Fp(1)p 1973 2651 V 1974 2727 a Fn(q)2024 2670 y(D)2095 2635 y Fy(2)1568 2886 y Fn(L)1625 2898 y Fy(4)1744 2886 y Fp(=)g Fn(D)1963 2851 y Fy(2)2010 2829 y Fp(1)p 2010 2866 V 2011 2943 a Fn(q)2062 2886 y(D)2133 2851 y Fy(2)2170 2886 y Fn(:)333 3127 y Fp(W)-7 b(e)31 b(wish)f(to)h(construct)f(a)g(lo) r(cal)f(basis)h(for)g(the)h(space)f(spanned)g(b)n(y)g(these)g (functions,)i(that)f(is,)333 3226 y(B-splines)j(in)h Fo(S)6 b Fp(\(4)p Fn(;)14 b Fe(m)p Fn(;)g(d)p Fd(\033)r Fn(;)g Fp(\001\),)37 b(where)d Fe(m)f Fp(is)i(m)n(ultiplicit)n(y)g(v)n (ector)e Fe(m)h Fp(=)g(\(1)p Fn(;)14 b(:)g(:)g(:)f(;)h Fp(1\))3090 3196 y Fm(T)3142 3226 y Fp(,)37 b Fn(d)p Fd(\033)h Fp(:=)333 3326 y(\()p Fn(dt)438 3338 y Fy(2)476 3326 y Fn(;)14 b(q)s Fp(\()p Fn(t)615 3338 y Fy(3)652 3326 y Fp(\))g Fn(dt)771 3338 y Fy(3)809 3326 y Fn(;)g(dt)919 3338 y Fy(4)956 3326 y Fp(\))988 3296 y Fm(T)1041 3326 y Fp(,)27 b(and)h(\001)23 b(=)g Fo(f)p Fn(t)1505 3338 y Fm(i)1532 3326 y Fo(g)1574 3290 y Fm(n)p Fy(+4)1574 3349 y Fm(i)p Fy(=)p Fl(\000)p Fy(2)1737 3326 y Fp(.)499 3473 y(First,)k(w)n(e)g(will)h(fo)r(cus)f(on)g Fk(r)l(e)l(duc)l(e)l(d)j (system)d Fo(f)p Fn(u)1981 3485 y Fy(1)p Fm(;)p Fy(1)2070 3473 y Fn(;)14 b(u)2155 3485 y Fy(1)p Fm(;)p Fy(2)2245 3473 y Fn(;)g(u)2330 3485 y Fy(1)p Fm(;)p Fy(3)2419 3473 y Fo(g)p Fp(,)27 b(whic)n(h)h(is)f(in)g Fo(S)6 b Fp(\(3)p Fn(;)14 b Fe(m)p Fn(;)g(d)p Fd(\033)3315 3442 y Fy(\(1\))3404 3473 y Fn(;)333 3572 y Fp(\001\))p Fn(;)42 b(d)p Fd(\033)602 3542 y Fy(\(1\))715 3572 y Fp(=)22 b(\()p Fn(q)s Fp(\()p Fn(t)936 3584 y Fy(3)974 3572 y Fp(\))14 b Fn(dt)1093 3584 y Fy(3)1131 3572 y Fn(;)g(dt)1241 3584 y Fy(4)1278 3572 y Fp(\).)37 b(CCT-system)27 b(is)h(in)f(this)h(case:)1256 3774 y Fn(u)1304 3786 y Fy(1)p Fm(;)p Fy(1)1394 3774 y Fp(\()p Fn(x)p Fp(\))84 b(=)e(1)1256 3948 y Fn(u)1304 3960 y Fy(1)p Fm(;)p Fy(2)1394 3948 y Fp(\()p Fn(x)p Fp(\))i(=)1736 3835 y Ff(Z)1819 3856 y Fm(x)1782 4024 y(a)1875 3948 y Fn(q)s Fp(\()p Fn(t)1977 3960 y Fy(3)2015 3948 y Fp(\))14 b Fn(dt)2134 3960 y Fy(3)3321 3948 y Fp(\(2\))1256 4185 y Fn(u)1304 4197 y Fy(1)p Fm(;)p Fy(3)1394 4185 y Fp(\()p Fn(x)p Fp(\))84 b(=)1736 4072 y Ff(Z)1819 4093 y Fm(x)1782 4261 y(a)1875 4185 y Fn(q)s Fp(\()p Fn(t)1977 4197 y Fy(3)2015 4185 y Fp(\))14 b Fn(dt)2134 4197 y Fy(3)2199 4072 y Ff(Z)2282 4093 y Fm(t)2307 4101 y Fg(3)2245 4261 y Fm(a)2371 4185 y Fn(dt)2444 4197 y Fy(4)2481 4185 y Fn(:)499 4487 y Fp(Let)33 b(us)f(consider)f(no)n(w)h (less)g(smo)r(oth)g(B-splines)2099 4466 y Ff(e)2085 4487 y Fn(T)2146 4457 y Fy(3)2134 4509 y Fm(j)2215 4487 y Fp(from)g(the)h(space)e Fo(S)6 b Fp(\(3)p Fn(;)30 b Fe(~)-64 b(m)o Fn(;)14 b(d)p Fd(\033)3177 4457 y Fy(\(1\))3266 4487 y Fn(;)g Fp(\001\),)333 4598 y(where)29 b(the)h(m)n(ultiplicit)n (y)g(v)n(ector)44 b Fe(~)-64 b(m)25 b Fp(=)h(\(2)p Fn(;)14 b(:)g(:)g(:)f(;)h Fp(2\))1947 4568 y Fm(T)2029 4598 y Fp(is)29 b(on)g(the)h(same)f(knot)g(sequence.)42 b(F)-7 b(or)29 b(the)333 4698 y(\014xed)f(index)h Fn(i)p Fp(,)f(w)n(e)f (denote)h(the)h(p)r(oin)n(ts)f(in)g(the)h(new)f(extended)h(partition)e (as)h Fn(t)2868 4710 y Fm(i)2919 4698 y Fp(=)3009 4682 y(~)3008 4698 y Fn(t)3038 4710 y Fm(r)r Fl(\000)p Fy(1)3184 4698 y Fp(=)3273 4682 y(~)3272 4698 y Fn(t)3302 4710 y Fm(r)3363 4698 y Fn(<)334 4791 y Fp(~)333 4806 y Fn(t)363 4818 y Fm(r)r Fy(+1)484 4806 y Fp(,)i(and)g(p)r(olynomial)f(B-splines)g (on)g(this)h(partition)f(simply)h(as)2507 4785 y Ff(e)2490 4806 y Fn(B)2557 4776 y Fm(n)2553 4828 y(j)2602 4806 y Fp(.)43 b(It)30 b(is)g(easily)f(seen)g(from)p eop %%Page: 3 3 3 2 bop 333 274 a Fi(Q-splines)2775 b(3)p 333 324 3095 4 v 333 515 a Fp(the)28 b(de\014nition)g(of)g(the)g(basis)e(\(2\))i (that)g(w)n(e)f(can)h(write)2101 494 y Ff(e)2087 515 y Fn(T)2148 485 y Fy(3)2136 536 y Fm(j)2212 515 y Fp(as)1373 758 y Ff(e)1359 779 y Fn(T)1420 745 y Fy(3)1408 800 y Fm(r)r Fl(\000)p Fy(1)1529 779 y Fp(\()p Fn(x)p Fp(\))84 b(=)1955 676 y Fm(s)1911 701 y Ff(X)1872 877 y Fm(j)s Fy(=)p Fm(s)p Fl(\000)p Fy(3)2084 779 y Fn(a)2128 791 y Fm(j)2180 758 y Ff(b)2162 779 y Fn(B)2229 745 y Fy(4)2225 800 y Fm(j)2267 779 y Fp(\()p Fn(x)p Fp(\))943 b(\(3\))1446 1059 y Ff(e)1432 1080 y Fn(T)1493 1045 y Fy(3)1481 1100 y Fm(r)1529 1080 y Fp(\()p Fn(x)p Fp(\))84 b(=)1913 976 y Fm(s)p Fy(+3)1911 1001 y Ff(X)1872 1178 y Fm(j)s Fy(=)p Fm(s)p Fl(\000)p Fy(3)2084 1080 y Fn(a)2128 1092 y Fm(j)2180 1059 y Ff(b)2162 1080 y Fn(B)2229 1045 y Fy(4)2225 1100 y Fm(j)2267 1080 y Fp(\()p Fn(x)p Fp(\))p Fn(:)920 b Fp(\(4\))350 1352 y Ff(b)333 1373 y Fn(B)400 1343 y Fy(4)396 1395 y Fm(j)472 1373 y Fo(2)35 b(S)6 b Fp(\(4)p Fn(;)30 b Fe(^)-64 b(m)n Fn(;)14 b(d)p Fd(\025)p Fn(;)g Fp(\001\),)37 b(where)50 b Fe(^)-64 b(m)33 b Fp(=)h(\(3)p Fn(;)14 b(:)g(:)g(:)f(;)h Fp(3\))1934 1343 y Fm(T)2021 1373 y Fp(on)34 b(the)h(same)f(knot)g (sequence)g(\001)g(and)h Fn(d)p Fd(\025)333 1473 y Fp(is)29 b(densit)n(y)g(v)n(ector)f(made)h(only)g(from)g(Leb)r(esgue)g (measures.)41 b(P)n(oin)n(ts)28 b(in)h(this)h(partition)f(will)g(b)r(e) 333 1572 y Fn(t)363 1584 y Fm(i)414 1572 y Fp(=)503 1557 y(^)501 1572 y Fn(t)531 1584 y Fm(s)p Fl(\000)p Fy(2)675 1572 y Fp(=)764 1557 y(^)763 1572 y Fn(t)793 1584 y Fm(s)p Fl(\000)p Fy(1)936 1572 y Fp(=)1025 1557 y(^)1024 1572 y Fn(t)1054 1584 y Fm(s)1112 1572 y Fn(<)1201 1557 y Fp(^)1200 1572 y Fn(t)1230 1584 y Fm(s)p Fy(+1)1349 1572 y Fp(.)499 1698 y(W)-7 b(e)42 b(will)g(use)f(this)h(general)e(theorem)g (whic)n(h)i(is)f(a)g(generalization)f(to)h(Tc)n(heb)n(yc)n(he\016an)333 1798 y(splines)28 b(of)f(the)h(w)n(ell)f(kno)n(wn)g(deriv)-5 b(ativ)n(e)27 b(form)n(ula)g(for)g(p)r(olynomial)g(B-splines:)333 1959 y Fe(Theorem)j(1)42 b Fk(L)l(et)29 b Fn(L)1028 1971 y Fy(1)p Fm(;d)p Fd(\033)1210 1959 y Fk(b)l(e)h(the)g(\014rst)f(gener)l (alize)l(d)j(derivative)g(with)f(r)l(esp)l(e)l(ct)f(to)g(CCT-system)333 2059 y Fo(S)6 b Fp(\()p Fn(n;)14 b(d)p Fd(\033)t Fp(\))p Fk(,)41 b(and)e(let)g(the)g(multiplicity)h(ve)l(ctor)f Fe(m)e Fp(=)i(\()p Fn(n)2157 2071 y Fy(1)2194 2059 y Fn(;)14 b(:)g(:)g(:)g(;)g(n)2429 2071 y Fm(k)2469 2059 y Fp(\))2501 2029 y Fm(T)2593 2059 y Fk(satisfy)39 b Fn(n)2910 2071 y Fm(i)2977 2059 y Fn(<)f(n)25 b Fo(\000)g Fp(1)38 b Fk(for)333 2240 y Fn(i)e Fp(=)h(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(k)s Fk(.)61 b(Then)38 b(for)g Fn(x)g Fo(2)f Fp([)p Fn(a;)14 b(b)p Fp(])37 b Fk(and)g Fn(i)g Fp(=)f(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(n)24 b Fp(+)2364 2136 y Fm(k)2323 2161 y Ff(X)2329 2338 y Fm(i)p Fy(=1)2456 2240 y Fn(n)2506 2252 y Fm(i)2571 2240 y Fk(the)38 b(fol)t(lowing)h (derivative)333 2412 y(formula)31 b(holds:)991 2669 y Fn(L)1048 2681 y Fy(1)p Fm(;d)p Fd(\033)1199 2669 y Fn(T)1260 2635 y Fm(n)1248 2689 y(i;d)p Fd(\033)1389 2669 y Fp(\()p Fn(x)p Fp(\))24 b(=)1622 2596 y Fn(T)1683 2561 y Fm(n)p Fl(\000)p Fy(1)1671 2621 y Fm(i;R)1764 2629 y Fh(n)p Fc(\000)p Fg(1)1877 2621 y Fm(d)p Fd(\033)1976 2596 y Fp(\()p Fn(x)p Fp(\))p 1622 2650 467 4 v 1714 2726 a Fn(C)1773 2738 y Fm(n)p Fl(\000)p Fy(1)1903 2726 y Fp(\()p Fn(i)p Fp(\))2117 2669 y Fo(\000)2210 2596 y Fn(T)2271 2561 y Fm(n)p Fl(\000)p Fy(1)2259 2621 y Fm(i)p Fy(+1)p Fm(;R)2436 2629 y Fh(n)p Fc(\000)p Fg(1)2549 2621 y Fm(d)p Fd(\033)2648 2596 y Fp(\()p Fn(x)p Fp(\))p 2210 2650 551 4 v 2272 2726 a Fn(C)2331 2738 y Fm(n)p Fl(\000)p Fy(1)2462 2726 y Fp(\()p Fn(i)18 b Fp(+)g(1\))3321 2669 y(\(5\))333 2899 y Fk(wher)l(e)1260 3041 y Fn(C)1319 3053 y Fm(n)p Fl(\000)p Fy(1)1450 3041 y Fp(\()p Fn(i)p Fp(\))23 b(:=)1677 2928 y Ff(Z)1760 2948 y Fm(t)1785 2956 y Fh(i)p Fg(+)p Fh(n)p Fc(\000)p Fg(1)1723 3116 y Fm(t)1748 3124 y Fh(i)1981 3041 y Fn(T)2042 3005 y Fm(n)p Fl(\000)p Fy(1)2030 3066 y Fm(i;R)2123 3074 y Fh(n)p Fc(\000)p Fg(1)2237 3066 y Fm(d)p Fd(\033)2350 3041 y Fn(d\033)2440 3053 y Fy(2)2477 3041 y Fn(;)821 b Fp(\(6\))333 3258 y Fk(and)30 b(density)h(ve)l(ctors)f Fn(d)p Fd(\033)c Fp(=)d(\()p Fn(d\033)1394 3270 y Fy(2)1432 3258 y Fp(\()p Fn(t)p Fp(\))p Fn(;)14 b(:)g(:)g(:)g(;)g(d\033)1801 3270 y Fm(n)1847 3258 y Fp(\()p Fn(t)p Fp(\)\))1973 3227 y Fm(T)2049 3258 y Fo(2)23 b Fe(R)2199 3227 y Fm(n)p Fl(\000)p Fy(1)2359 3258 y Fk(and)30 b Fn(R)2583 3270 y Fm(n)p Fl(\000)p Fy(1)2713 3258 y Fn(d)p Fd(\033)c Fp(=)d(\()p Fn(d\033)3049 3270 y Fy(3)3087 3258 y Fp(\()p Fn(t)p Fp(\))p Fn(;)14 b(:)g(:)g(:)g(;)333 3357 y(d\033)423 3369 y Fm(n)469 3357 y Fp(\()p Fn(t)p Fp(\)\))595 3327 y Fm(T)671 3357 y Fo(2)24 b Fe(R)822 3327 y Fm(n)p Fl(\000)p Fy(2)951 3357 y Fk(.)499 3518 y Fp(F)-7 b(rom)27 b(\(2\))h(and)f (Theorem)g(1)g(it)h(is)g(ob)n(vious)e(that)1284 3754 y(\026)1275 3775 y Fn(L)1332 3787 y Fy(1)1382 3754 y Ff(e)1368 3775 y Fn(T)1429 3740 y Fy(3)1417 3795 y Fm(r)r Fl(\000)p Fy(1)1539 3775 y Fp(\()p Fn(x)p Fp(\))e(=)1812 3694 y Ff(e)1795 3715 y Fn(B)1862 3685 y Fy(2)1858 3735 y Fm(r)r Fl(\000)p Fy(1)1980 3715 y Fp(\()p Fn(x)p Fp(\))p 1771 3755 344 4 v 1788 3825 a Ff(e)1771 3846 y Fn(C)1830 3858 y Fy(2)1868 3846 y Fp(\()p Fn(r)d Fo(\000)d Fp(1\))2144 3775 y Fo(\000)2254 3697 y Ff(e)2237 3718 y Fn(B)2304 3688 y Fy(2)2300 3739 y Fm(r)2341 3718 y Fp(\()p Fn(x)p Fp(\))p 2237 3755 217 4 v 2261 3825 a Ff(e)2245 3846 y Fn(C)2304 3858 y Fy(2)2341 3846 y Fp(\()p Fn(r)r Fp(\))2463 3775 y Fn(;)333 4032 y Fp(where)583 4011 y(\026)573 4032 y Fn(L)630 4044 y Fy(1)690 4032 y Fp(=)788 3999 y Fy(1)p 788 4013 34 4 v 788 4061 a Fm(q)831 4032 y Fn(D)29 b Fp(is)f(generalized)e(deriv)-5 b(ativ)n(e)27 b(for)g(the)h(reduced)f (CCT-system,)g(and)1399 4288 y Ff(e)1382 4309 y Fn(C)1441 4321 y Fy(2)1479 4309 y Fp(\()p Fn(j)5 b Fp(\))23 b(=)1693 4196 y Ff(Z)1777 4205 y Fy(~)1776 4216 y Fm(t)1801 4224 y Fh(j)r Fg(+2)1740 4374 y Fy(~)1739 4385 y Fm(t)1764 4393 y Fh(j)1938 4288 y Ff(e)1921 4309 y Fn(B)1988 4275 y Fy(2)1984 4329 y Fm(j)2025 4309 y Fp(\()p Fn(t)p Fp(\))14 b Fn(q)s Fp(\()p Fn(t)p Fp(\))g Fn(dt:)333 4553 y Fp(In)28 b(particular)926 4762 y Ff(e)909 4783 y Fn(C)968 4795 y Fy(2)1006 4783 y Fp(\()p Fn(r)21 b Fo(\000)d Fp(1\))23 b(=)1374 4726 y(\(2)p Fn(q)1485 4738 y Fm(i)1531 4726 y Fp(+)18 b Fn(q)1651 4738 y Fm(i)p Fy(+1)1763 4726 y Fp(\))p Fn(h)1843 4738 y Fm(i)p 1374 4763 497 4 v 1601 4839 a Fp(6)1880 4783 y Fn(;)2017 4762 y Ff(e)2000 4783 y Fn(C)2059 4795 y Fy(2)2097 4783 y Fp(\()p Fn(r)r Fp(\))24 b(=)2322 4726 y(\()p Fn(q)2391 4738 y Fm(i)2437 4726 y Fp(+)18 b(2)p Fn(q)2599 4738 y Fm(i)p Fy(+1)2711 4726 y Fp(\))p Fn(h)2791 4738 y Fm(i)p 2322 4763 V 2549 4839 a Fp(6)2828 4783 y Fn(:)p eop %%Page: 4 4 4 3 bop 333 274 a Fi(4)2269 b(T.)23 b(Bosner)h(and)g(M.)f(Rogina)p 333 324 3095 4 v 333 515 a Fp(F)-7 b(rom)27 b(the)h(simple)g(prop)r (erties)f(of)g(B-splines)433 719 y Ff(e)419 740 y Fn(T)480 705 y Fy(3)468 760 y Fm(r)r Fl(\000)p Fy(1)590 740 y Fp(\()p Fn(t)652 752 y Fm(i)680 740 y Fp(\))c(=)g(0)p Fn(;)939 719 y Fp(\026)929 740 y Fn(L)986 752 y Fy(1)1036 719 y Ff(e)1022 740 y Fn(T)1083 705 y Fy(3)1071 760 y Fm(r)r Fl(\000)p Fy(1)1193 740 y Fp(\()p Fn(t)1255 704 y Fy(+)1255 763 y Fm(i)1310 740 y Fp(\))g(=)1614 684 y(1)p 1463 721 344 4 v 1480 791 a Ff(e)1463 812 y Fn(C)1522 824 y Fy(2)1560 812 y Fp(\()p Fn(r)e Fo(\000)d Fp(1\))1817 740 y Fn(;)1895 719 y Ff(e)1881 740 y Fn(T)1942 705 y Fy(3)1930 760 y Fm(r)r Fl(\000)p Fy(1)2052 740 y Fp(\()p Fn(t)2114 752 y Fm(i)p Fy(+1)2226 740 y Fp(\))23 b(=)g(0)p Fn(;)2485 719 y Fp(\026)2475 740 y Fn(L)2532 752 y Fy(1)2582 719 y Ff(e)2569 740 y Fn(T)2630 705 y Fy(3)2618 760 y Fm(r)r Fl(\000)p Fy(1)2739 740 y Fp(\()p Fn(t)2801 704 y Fl(\000)2801 763 y Fm(i)p Fy(+1)2913 740 y Fp(\))g(=)g Fo(\000)3210 684 y Fp(1)p 3131 721 201 4 v 3147 791 a Ff(e)3131 812 y Fn(C)3190 824 y Fy(2)3227 812 y Fp(\()p Fn(r)r Fp(\))333 994 y(w)n(e)k(get)h(the)g(co)r(e\016cien)n(ts)f(for)g (\(3\):)899 1223 y Fn(a)943 1235 y Fm(s)p Fl(\000)p Fy(3)1087 1223 y Fp(=)c Fn(a)1219 1235 y Fm(s)1277 1223 y Fp(=)g(0)p Fn(;)41 b(a)1515 1235 y Fm(s)p Fl(\000)p Fy(2)1658 1223 y Fp(=)1881 1167 y(2)p Fn(q)1960 1179 y Fm(i)p 1756 1204 357 4 v 1756 1280 a Fp(2)p Fn(q)1835 1292 y Fm(i)1880 1280 y Fp(+)18 b Fn(q)2000 1292 y Fm(i)p Fy(+1)2122 1223 y Fn(;)42 b(a)2231 1235 y Fm(s)p Fl(\000)p Fy(1)2374 1223 y Fp(=)2555 1167 y(2)p Fn(q)2634 1179 y Fm(i)p Fy(+1)p 2472 1204 V 2472 1280 a Fn(q)2509 1292 y Fm(i)2555 1280 y Fp(+)18 b(2)p Fn(q)2717 1292 y Fm(i)p Fy(+1)2838 1223 y Fn(:)333 1480 y Fp(So,)485 1459 y Ff(e)471 1480 y Fn(T)532 1450 y Fy(3)520 1500 y Fm(r)r Fl(\000)p Fy(1)669 1480 y Fp(is)28 b(\014nally)f(lo)r(oking)g(as)964 1695 y Ff(e)950 1716 y Fn(T)1011 1682 y Fy(3)999 1737 y Fm(r)r Fl(\000)p Fy(1)1120 1716 y Fp(\()p Fn(x)p Fp(\))d(=)1478 1660 y(2)p Fn(q)1557 1672 y Fm(i)p 1353 1697 V 1353 1773 a Fp(2)p Fn(q)1432 1785 y Fm(i)1477 1773 y Fp(+)18 b Fn(q)1597 1785 y Fm(i)p Fy(+1)1737 1695 y Ff(b)1719 1716 y Fn(B)1786 1682 y Fy(4)1782 1737 y Fm(s)p Fl(\000)p Fy(2)1903 1716 y Fp(\()p Fn(x)p Fp(\))h(+)2209 1660 y(2)p Fn(q)2288 1672 y Fm(i)p Fy(+1)p 2126 1697 V 2126 1773 a Fn(q)2163 1785 y Fm(i)2209 1773 y Fp(+)f(2)p Fn(q)2371 1785 y Fm(i)p Fy(+1)2510 1695 y Ff(b)2492 1716 y Fn(B)2559 1682 y Fy(4)2555 1737 y Fm(s)p Fl(\000)p Fy(1)2676 1716 y Fp(\()p Fn(x)p Fp(\))p Fn(:)511 b Fp(\(7\))333 1973 y(Again,)27 b(for)g(calculating) 1159 1952 y Ff(e)1145 1973 y Fn(T)1206 1943 y Fy(3)1194 1993 y Fm(r)1242 1973 y Fp(,)h(w)n(e)f(use)h(the)g(equations)1024 2152 y Ff(e)1010 2173 y Fn(T)1071 2139 y Fy(3)1059 2194 y Fm(r)1107 2173 y Fp(\()p Fn(t)1169 2185 y Fm(i)1197 2173 y Fp(\))23 b(=)1350 2152 y(\026)1340 2173 y Fn(L)1397 2185 y Fy(1)1448 2152 y Ff(e)1434 2173 y Fn(T)1495 2139 y Fy(3)1483 2194 y Fm(r)1531 2173 y Fp(\()p Fn(t)1593 2185 y Fm(i)1621 2173 y Fp(\))g(=)1778 2152 y Ff(e)1764 2173 y Fn(T)1825 2139 y Fy(3)1813 2194 y Fm(r)1861 2173 y Fp(\()p Fn(t)1923 2185 y Fm(i)p Fy(+2)2035 2173 y Fp(\))h(=)2188 2152 y(\026)2178 2173 y Fn(L)2235 2185 y Fy(1)2286 2152 y Ff(e)2272 2173 y Fn(T)2333 2139 y Fy(3)2321 2194 y Fm(r)2369 2173 y Fp(\()p Fn(t)2431 2185 y Fm(i)p Fy(+2)2543 2173 y Fp(\))g(=)e(0)p Fn(;)769 2394 y Ff(e)755 2415 y Fn(T)816 2381 y Fy(3)804 2435 y Fm(r)853 2415 y Fp(\()p Fn(t)915 2427 y Fm(i)p Fy(+1)1027 2415 y Fp(\))h(=)g(1)p Fn(;)1285 2394 y Fp(\026)1276 2415 y Fn(L)1333 2427 y Fy(1)1383 2394 y Ff(e)1369 2415 y Fn(T)1430 2381 y Fy(3)1418 2435 y Fm(r)1467 2415 y Fp(\()p Fn(t)1529 2379 y Fl(\000)1529 2438 y Fm(i)p Fy(+1)1641 2415 y Fp(\))g(=)1873 2359 y(1)p 1794 2396 201 4 v 1810 2466 a Ff(e)1794 2487 y Fn(C)1853 2499 y Fy(2)1890 2487 y Fp(\()p Fn(r)r Fp(\))2004 2415 y Fn(;)2079 2394 y Fp(\026)2069 2415 y Fn(L)2126 2427 y Fy(1)2177 2394 y Ff(e)2163 2415 y Fn(T)2224 2381 y Fy(3)2212 2435 y Fm(r)2260 2415 y Fp(\()p Fn(t)2322 2379 y Fy(+)2322 2438 y Fm(i)p Fy(+1)2434 2415 y Fp(\))g(=)g Fo(\000)2803 2359 y Fp(1)p 2652 2396 344 4 v 2668 2466 a Ff(e)2652 2487 y Fn(C)2711 2499 y Fy(2)2748 2487 y Fp(\()p Fn(r)f Fp(+)c(1\))333 2640 y(to)28 b(get)f(the)h(co)r (e\016cien)n(ts)f(for)g(\(4\))1151 2830 y Fn(a)1195 2842 y Fm(s)p Fl(\000)p Fy(3)1338 2830 y Fp(=)c Fn(a)1470 2842 y Fm(s)p Fl(\000)p Fy(2)1613 2830 y Fp(=)g Fn(a)1745 2842 y Fm(s)p Fy(+2)1887 2830 y Fp(=)g Fn(a)2019 2842 y Fm(s)p Fy(+3)2161 2830 y Fp(=)g(0)p Fn(;)41 b(a)2399 2842 y Fm(s)2458 2830 y Fp(=)22 b(1)p Fn(;)1116 3052 y(a)1160 3064 y Fm(s)p Fl(\000)p Fy(1)1303 3052 y Fp(=)1547 2996 y Fn(q)1584 3008 y Fm(i)p 1401 3033 357 4 v 1401 3109 a Fp(2)p Fn(q)1480 3121 y Fm(i)p Fy(+1)1610 3109 y Fp(+)c Fn(q)1730 3121 y Fm(i)1767 3052 y Fn(;)97 b(a)1931 3064 y Fm(s)p Fy(+1)2074 3052 y Fp(=)2317 2996 y Fn(q)2354 3008 y Fm(i)p Fy(+2)p 2171 3033 441 4 v 2171 3109 a Fp(2)p Fn(q)2250 3121 y Fm(i)p Fy(+1)2380 3109 y Fp(+)18 b Fn(q)2500 3121 y Fm(i)p Fy(+2)2622 3052 y Fn(;)333 3256 y Fp(and,)28 b(\014nally)800 3443 y Ff(e)786 3464 y Fn(T)847 3429 y Fy(3)835 3484 y Fm(r)883 3464 y Fp(\()p Fn(x)p Fp(\))c(=)1262 3407 y Fn(q)1299 3419 y Fm(i)p 1116 3444 357 4 v 1116 3520 a Fn(q)1153 3532 y Fm(i)1199 3520 y Fp(+)18 b(2)p Fn(q)1361 3532 y Fm(i)p Fy(+1)1500 3443 y Ff(b)1482 3464 y Fn(B)1549 3429 y Fy(4)1545 3484 y Fm(s)p Fl(\000)p Fy(1)1666 3464 y Fp(\()p Fn(x)p Fp(\))h(+)1896 3443 y Ff(b)1879 3464 y Fn(B)1946 3429 y Fy(4)1942 3484 y Fm(s)1983 3464 y Fp(\()p Fn(x)p Fp(\))h(+)2353 3407 y Fn(q)2390 3419 y Fm(i)p Fy(+2)p 2207 3444 441 4 v 2207 3520 a Fp(2)p Fn(q)2286 3532 y Fm(i)p Fy(+1)2416 3520 y Fp(+)e Fn(q)2536 3532 y Fm(i)p Fy(+2)2675 3443 y Ff(b)2657 3464 y Fn(B)2724 3429 y Fy(4)2720 3484 y Fm(s)p Fy(+1)2840 3464 y Fp(\()p Fn(x)p Fp(\))p Fn(:)347 b Fp(\(8\))333 3701 y(In)n(tegrating)40 b(\(5\))h(from)f(Theorem)g(1,)k(w)n(e)d(can)f(further)h(calculate)f (splines)h(of)g(higher)f(order.)333 3801 y(Therefore)26 b(w)n(e)i(will)f(start)h(with)g(the)g(equation)847 4013 y Ff(e)833 4034 y Fn(T)894 3999 y Fy(4)882 4054 y Fm(r)r Fl(\000)p Fy(1)1003 4034 y Fp(\()p Fn(x)p Fp(\))c(=)1387 3978 y(1)p 1236 4015 344 4 v 1252 4085 a Ff(e)1236 4106 y Fn(C)1295 4118 y Fy(3)1332 4106 y Fp(\()p Fn(r)e Fo(\000)c Fp(1\))1603 3921 y Ff(Z)1686 3941 y Fm(x)1651 4098 y Fy(~)1650 4109 y Fm(t)1675 4117 y Fh(r)q Fc(\000)p Fg(1)1813 4013 y Ff(e)1799 4034 y Fn(T)1860 3999 y Fy(3)1848 4054 y Fm(r)r Fl(\000)p Fy(1)1970 4034 y Fp(\()p Fn(t)p Fp(\))c Fn(dt)19 b Fo(\000)2342 3978 y Fp(1)p 2263 4015 201 4 v 2279 4085 a Ff(e)2263 4106 y Fn(C)2322 4118 y Fy(3)2359 4106 y Fp(\()p Fn(r)r Fp(\))2487 3921 y Ff(Z)2570 3941 y Fm(x)2535 4098 y Fy(~)2533 4109 y Fm(t)2558 4117 y Fh(r)2640 4013 y Ff(e)2626 4034 y Fn(T)2687 3999 y Fy(3)2675 4054 y Fm(r)2723 4034 y Fp(\()p Fn(t)p Fp(\))14 b Fn(dt;)394 b Fp(\(9\))333 4288 y(where)27 b(it)h(is)g(easy)e(to)i(see)f(from)g (\(7\))h(and)f(\(8\))h(that)822 4507 y Ff(e)805 4528 y Fn(C)864 4540 y Fy(3)902 4528 y Fp(\()p Fn(r)21 b Fo(\000)d Fp(1\))83 b(=)1390 4472 y Fn(h)1438 4484 y Fm(i)p 1390 4509 76 4 v 1407 4585 a Fp(4)1489 4411 y Ff(\024)1668 4472 y Fp(2)p Fn(q)1747 4484 y Fm(i)p 1543 4509 357 4 v 1543 4585 a Fp(2)p Fn(q)1622 4597 y Fm(i)1667 4585 y Fp(+)18 b Fn(q)1787 4597 y Fm(i)p Fy(+1)1927 4528 y Fp(+)2104 4472 y(2)p Fn(q)2183 4484 y Fm(i)p Fy(+1)p 2020 4509 V 2020 4585 a Fn(q)2057 4597 y Fm(i)2104 4585 y Fp(+)g(2)p Fn(q)2266 4597 y Fm(i)p Fy(+1)2387 4411 y Ff(\025)3280 4528 y Fp(\(10\))965 4740 y Ff(e)948 4761 y Fn(C)1007 4773 y Fy(3)1045 4761 y Fp(\()p Fn(r)r Fp(\))84 b(=)1390 4704 y(1)p 1390 4741 42 4 v 1390 4818 a(4)1455 4644 y Ff(\024)1617 4704 y Fn(q)1654 4716 y Fm(i)1681 4704 y Fn(h)1729 4716 y Fm(i)p 1509 4741 357 4 v 1509 4818 a Fn(q)1546 4830 y Fm(i)1592 4818 y Fp(+)18 b(2)p Fn(q)1754 4830 y Fm(i)p Fy(+1)1893 4761 y Fp(+)g Fn(h)2024 4773 y Fm(i)2070 4761 y Fp(+)g Fn(h)2201 4773 y Fm(i)p Fy(+1)2331 4761 y Fp(+)2491 4704 y Fn(q)2528 4716 y Fm(i)p Fy(+2)2639 4704 y Fn(h)2687 4716 y Fm(i)p Fy(+1)p 2424 4741 441 4 v 2424 4818 a Fp(2)p Fn(q)2503 4830 y Fm(i)p Fy(+1)2633 4818 y Fp(+)g Fn(q)2753 4830 y Fm(i)p Fy(+2)2875 4644 y Ff(\025)2933 4761 y Fn(:)324 b Fp(\(11\))p eop %%Page: 5 5 5 4 bop 333 274 a Fi(Q-splines)2775 b(5)p 333 324 3095 4 v 333 515 a Fp(F)-7 b(rom)27 b(\(9\))g(using)g(\(7\),)h(\(8\))f(and)h (the)f(w)n(ell)g(kno)n(wn)g(recurrence)f(for)g(in)n(tegrals)g(of)i(the) f(p)r(olynomial)333 614 y(B-splines)1151 678 y Ff(Z)1234 699 y Fm(x)1198 867 y Fl(\0001)1334 791 y Fn(B)1401 757 y Fm(k)1397 812 y(i)1442 791 y Fp(\()p Fn(t)p Fp(\))14 b Fn(dt)23 b Fp(=)1744 735 y Fn(t)1774 747 y Fm(i)p Fy(+)p Fm(k)1908 735 y Fo(\000)18 b Fn(t)2021 747 y Fm(i)p 1744 772 305 4 v 1873 848 a Fn(k)2072 687 y Fm(i)p Fy(+)p Fm(k)q Fl(\000)p Fy(1)2110 712 y Ff(X)2117 889 y Fm(j)s Fy(=)p Fm(i)2282 791 y Fn(B)2349 756 y Fm(k)q Fy(+1)2345 814 y Fm(j)2474 791 y Fp(\()p Fn(x)p Fp(\))p Fn(;)333 1017 y Fp(where)k Fo(f)p Fn(t)640 1029 y Fm(i)667 1017 y Fo(g)709 1029 y Fm(i)758 1017 y Fp(is)g(kno)n(w)g(an)n(y)f(extended)i (partition,)g(w)n(e)f(can)f(get)h(b)n(y)g(lo)r(oking)f(separately)g(at) h Fn(x)h Fp(from)333 1116 y(eac)n(h)k(of)g(the)h(subin)n(terv)-5 b(als)27 b([)p Fn(t)1268 1128 y Fm(i)1296 1116 y Fn(;)14 b(t)1363 1128 y Fm(i)p Fy(+1)1474 1116 y Fp(])28 b(and)g([)p Fn(t)1740 1128 y Fm(i)p Fy(+1)1851 1116 y Fn(;)14 b(t)1918 1128 y Fm(i)p Fy(+2)2030 1116 y Fp(],)28 b(that)347 1307 y Ff(e)333 1328 y Fn(T)394 1294 y Fy(4)382 1349 y Fm(r)r Fl(\000)p Fy(1)503 1328 y Fp(\()p Fn(x)p Fp(\))84 b(=)1007 1272 y(1)p 856 1309 344 4 v 872 1379 a Ff(e)856 1400 y Fn(C)915 1412 y Fy(3)953 1400 y Fp(\()p Fn(r)21 b Fo(\000)d Fp(1\))1345 1272 y(2)p Fn(q)1424 1284 y Fm(i)p 1220 1309 357 4 v 1220 1385 a Fp(2)p Fn(q)1299 1397 y Fm(i)1344 1385 y Fp(+)g Fn(q)1464 1397 y Fm(i)p Fy(+1)1596 1272 y Fn(h)1644 1284 y Fm(i)p 1596 1309 76 4 v 1613 1385 a Fp(4)1699 1307 y Ff(b)1682 1328 y Fn(B)1749 1294 y Fy(5)1745 1349 y Fm(s)p Fl(\000)p Fy(2)1865 1328 y Fp(\()p Fn(x)p Fp(\))h(+)2168 1272 y(1)p 2088 1309 201 4 v 2105 1379 a Ff(e)2088 1400 y Fn(C)2147 1412 y Fy(3)2185 1400 y Fp(\()p Fn(r)r Fp(\))2313 1211 y Ff(\022)2384 1272 y Fn(h)2432 1284 y Fm(i)2478 1272 y Fp(+)f Fn(h)2609 1284 y Fm(i)p Fy(+1)p 2384 1309 337 4 v 2532 1385 a Fp(4)2749 1328 y(+)2988 1272 y Fn(q)3025 1284 y Fm(i)p Fy(+2)p 2842 1309 441 4 v 2842 1385 a Fp(2)p Fn(q)2921 1397 y Fm(i)p Fy(+1)3051 1385 y Fp(+)g Fn(q)3171 1397 y Fm(i)p Fy(+2)3303 1272 y Fn(h)3351 1284 y Fm(i)p Fy(+1)p 3303 1309 160 4 v 3362 1385 a Fp(4)3472 1211 y Ff(\023)863 1547 y(b)846 1568 y Fn(B)913 1534 y Fy(5)909 1588 y Fm(s)p Fl(\000)p Fy(1)1029 1568 y Fp(\()p Fn(x)p Fp(\))i(+)1333 1512 y(1)p 1253 1549 201 4 v 1269 1619 a Ff(e)1253 1640 y Fn(C)1312 1652 y Fy(3)1349 1640 y Fp(\()p Fn(r)r Fp(\))1620 1512 y Fn(q)1657 1524 y Fm(i)p Fy(+2)p 1474 1549 441 4 v 1474 1625 a Fp(2)p Fn(q)1553 1637 y Fm(i)p Fy(+1)1682 1625 y Fp(+)e Fn(q)1802 1637 y Fm(i)p Fy(+2)1934 1512 y Fn(h)1982 1524 y Fm(i)p Fy(+1)p 1934 1549 160 4 v 1993 1625 a Fp(4)2121 1547 y Ff(b)2104 1568 y Fn(B)2171 1534 y Fy(5)2167 1588 y Fm(s)2208 1568 y Fp(\()p Fn(x)p Fp(\))p Fn(:)1057 b Fp(\(12\))333 1789 y(In)28 b(the)g(same)f(w)n(a)n(y)-7 b(,)479 1975 y Ff(e)465 1996 y Fn(T)526 1962 y Fy(4)514 2017 y Fm(r)563 1996 y Fp(\()p Fn(x)p Fp(\))84 b(=)995 1940 y(1)p 915 1977 201 4 v 932 2047 a Ff(e)915 2068 y Fn(C)974 2080 y Fy(3)1012 2068 y Fp(\()p Fn(r)r Fp(\))1282 1940 y Fn(q)1319 1952 y Fm(i)p 1136 1977 357 4 v 1136 2053 a Fn(q)1173 2065 y Fm(i)1219 2053 y Fp(+)18 b(2)p Fn(q)1381 2065 y Fm(i)p Fy(+1)1512 1940 y Fn(h)1560 1952 y Fm(i)p 1512 1977 76 4 v 1529 2053 a Fp(4)1615 1975 y Ff(b)1598 1996 y Fn(B)1665 1962 y Fy(5)1661 2017 y Fm(s)p Fl(\000)p Fy(1)1781 1996 y Fp(\()p Fn(x)p Fp(\))i(+)2085 1940 y(1)p 2005 1977 201 4 v 2021 2047 a Ff(e)2005 2068 y Fn(C)2064 2080 y Fy(3)2101 2068 y Fp(\()p Fn(r)r Fp(\))2230 1879 y Ff(\022)2446 1940 y Fn(q)2483 1952 y Fm(i)p 2301 1977 357 4 v 2301 2053 a Fn(q)2338 2065 y Fm(i)2384 2053 y Fp(+)e(2)p Fn(q)2546 2065 y Fm(i)p Fy(+1)2677 1940 y Fn(h)2725 1952 y Fm(i)p 2677 1977 76 4 v 2694 2053 a Fp(4)2781 1996 y(+)2874 1940 y Fn(h)2922 1952 y Fm(i)2968 1940 y Fp(+)g Fn(h)3099 1952 y Fm(i)p Fy(+1)p 2874 1977 337 4 v 3021 2053 a Fp(4)3220 1879 y Ff(\023)923 2215 y(b)905 2236 y Fn(B)972 2201 y Fy(5)968 2256 y Fm(s)1010 2236 y Fp(\()p Fn(x)p Fp(\))h(+)1384 2180 y(1)p 1233 2217 344 4 v 1250 2287 a Ff(e)1233 2308 y Fn(C)1292 2320 y Fy(3)1330 2308 y Fp(\()p Fn(r)i Fp(+)d(1\))1722 2180 y(2)p Fn(q)1801 2192 y Fm(i)p Fy(+2)p 1597 2217 441 4 v 1597 2293 a Fn(q)1634 2305 y Fm(i)p Fy(+1)1764 2293 y Fp(+)g(2)p Fn(q)1926 2305 y Fm(i)p Fy(+2)2057 2180 y Fn(h)2105 2192 y Fm(i)p Fy(+1)p 2057 2217 160 4 v 2116 2293 a Fp(4)2244 2215 y Ff(b)2227 2236 y Fn(B)2294 2201 y Fy(5)2290 2256 y Fm(s)p Fy(+1)2409 2236 y Fp(\()p Fn(x)p Fp(\))p Fn(:)737 b Fp(\(13\))499 2482 y(Next)39 b(Lemma)g(and)g (Theorem)f(are)g(connecting)h(general)e(B-splines)i(with)g(less)g(smo)r (oth)333 2581 y(ones,)27 b(whic)n(h)h(are)e(simpler)h(to)h(calculate,)f (and)g(whic)n(h)h(w)n(e,)f(in)h(our)f(case)g(already)f(ha)n(v)n(e:)333 2731 y Fe(Lemma)k(2)41 b Fk(L)l(et)28 b Fn(T)964 2701 y Fy(3)952 2762 y Fm(i;d)p Fd(\033)1089 2745 y Fg(\(1\))1193 2731 y Fo(2)c(S)6 b Fp(\(3)p Fn(;)14 b Fe(m)p Fn(;)g(d)p Fd(\033)1659 2701 y Fy(\(1\))1748 2731 y Fn(;)g Fp(\001\))28 b Fk(b)l(e)h(a)f(Tchebyche\016an)j Fp(3)2693 2701 y Fm(r)r(d)2792 2731 y Fk(or)l(der)e(spline)g(asso-)333 2862 y(ciate)l(d)k(with)f(the)g (multiplicity)h(ve)l(ctor)f Fe(m)25 b Fp(=)h(\(1)p Fn(;)14 b(:)g(:)g(:)f(;)h Fp(1\))2112 2832 y Fm(T)2164 2862 y Fk(,)33 b(and)f(let)g(us)f(assume)g(that)3092 2841 y Ff(e)3078 2862 y Fn(T)3139 2832 y Fy(3)3127 2893 y Fm(i;d)p Fd(\033)3264 2877 y Fg(\(1\))3372 2862 y Fo(2)333 2986 y(S)6 b Fp(\(3)p Fn(;)30 b Fe(~)-64 b(m)o Fn(;)14 b(d)p Fd(\033)720 2956 y Fy(\(1\))809 2986 y Fn(;)g Fp(\001\))31 b Fk(ar)l(e)f(B-splines)h(asso)l(ciate)l(d)h(with)f(multiplicity)g(ve)l (ctor)47 b Fe(~)-64 b(m)22 b Fp(=)i(\(2)p Fn(;)14 b(:)g(:)g(:)f(;)h Fp(2\))3256 2956 y Fm(T)3338 2986 y Fk(on)333 3086 y(the)37 b(same)h(knot)f(se)l(quenc)l(e.)60 b(If)37 b Fo(f)p Fn(t)1449 3098 y Fy(1)1486 3086 y Fn(;)14 b(:)g(:)g(:)g(;)g(t)1701 3098 y Fm(k)q Fy(+6)1825 3086 y Fo(g)37 b Fk(and)g Fo(f)2115 3071 y Fp(~)2114 3086 y Fn(t)2144 3098 y Fy(1)2181 3086 y Fn(;)14 b(:)g(:)g(:)g(;)2367 3071 y Fp(~)2366 3086 y Fn(t)2396 3098 y Fy(2)p Fm(k)q Fy(+6)2554 3086 y Fo(g)36 b Fk(ar)l(e)h(the)h(asso)l(ciate)l(d)g(ex-)333 3186 y(tende)l(d)30 b(p)l(artitions,)h(and)f Fn(r)j Fk(an)c(index)h(such)g(that)g Fn(t)1953 3198 y Fm(i)2004 3186 y Fp(=)2092 3170 y(~)2091 3186 y Fn(t)2121 3198 y Fm(r)2181 3186 y Fn(<)2270 3170 y Fp(~)2269 3186 y Fn(t)2299 3198 y Fm(r)r Fy(+1)2420 3186 y Fk(,)g(then)f(for)i Fn(i)23 b Fp(=)f(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(k)22 b Fp(+)c(3)p Fk(:)606 3357 y Fn(T)667 3323 y Fy(3)655 3384 y Fm(i;d)p Fd(\033)793 3367 y Fg(\(1\))897 3357 y Fp(=)23 b Fn(T)1046 3323 y Fy(3)1034 3384 y Fm(i;d)p Fd(\033)1171 3367 y Fg(\(1\))1253 3357 y Fp(\()p Fn(t)1315 3369 y Fm(i)p Fy(+1)1427 3357 y Fp(\))1473 3336 y Ff(e)1459 3357 y Fn(T)1520 3323 y Fy(3)1508 3384 y Fm(r)n(;d)p Fd(\033)1651 3367 y Fg(\(1\))1751 3357 y Fp(+)1848 3336 y Ff(e)1834 3357 y Fn(T)1895 3323 y Fy(3)1883 3384 y Fm(r)r Fy(+1)p Fm(;d)p Fd(\033)2113 3367 y Fg(\(1\))2213 3357 y Fp(+)18 b Fn(T)2357 3323 y Fy(3)2345 3384 y Fm(i;d)p Fd(\033)2483 3367 y Fg(\(1\))2564 3357 y Fp(\()p Fn(t)2626 3369 y Fm(i)p Fy(+2)2738 3357 y Fp(\))2784 3336 y Ff(e)2770 3357 y Fn(T)2831 3323 y Fy(3)2819 3384 y Fm(r)r Fy(+2)p Fm(;d)p Fd(\033)3050 3367 y Fg(\(1\))3131 3357 y Fn(:)333 3547 y Fe(Theorem)30 b(3)42 b Fk(L)l(et)31 b Fn(T)1034 3516 y Fy(4)1022 3570 y Fm(i;d)p Fd(\033)1191 3547 y Fo(2)c(S)6 b Fp(\(4)p Fn(;)14 b Fe(m)p Fn(;)g(d)p Fd(\033)r Fn(;)g Fp(\001\))p Fk(,)1870 3526 y Ff(e)1856 3547 y Fn(T)1917 3516 y Fy(4)1905 3570 y Fm(i;d)p Fd(\033)2074 3547 y Fo(2)27 b(S)6 b Fp(\(4)p Fn(;)30 b Fe(~)-64 b(m)o Fn(;)14 b(d)p Fd(\033)s Fn(;)g Fp(\001\))p Fk(,)33 b(multiplicity)g(ve) l(ctors)333 3657 y Fe(m)p Fk(,)45 b Fe(~)-64 b(m)28 b Fk(b)l(eing)i(as)g(in)g(L)l(emma)g(2.)39 b(Then)31 b(p)l(ositive)g Fn(\016)1953 3627 y Fy(4)1950 3679 y Fm(i)1990 3657 y Fp(\()p Fn(j)5 b Fp(\))30 b Fk(exist)g(such)f(that)1464 3899 y Fn(T)1525 3865 y Fy(4)1513 3920 y Fm(i;d)p Fd(\033)1677 3899 y Fp(=)1767 3795 y Fm(r)r Fy(+3)1765 3820 y Ff(X)1768 3997 y Fm(j)s Fy(=)p Fm(r)1899 3899 y Fn(\016)1939 3865 y Fy(4)1936 3920 y Fm(i)1976 3899 y Fp(\()p Fn(j)5 b Fp(\))2093 3878 y Ff(e)2079 3899 y Fn(T)2140 3865 y Fy(4)2128 3920 y Fm(j;d)p Fd(\033)2274 3899 y Fn(;)333 4145 y Fk(wher)l(e)31 b Fn(r)25 b Fp(=)e Fn(r)755 4157 y Fm(i)813 4145 y Fk(satis\014es)1596 4245 y Fn(t)1626 4257 y Fm(i)1677 4245 y Fp(=)1765 4229 y(~)1764 4245 y Fn(t)1794 4257 y Fm(r)1825 4265 y Fh(i)1879 4245 y Fn(<)1967 4229 y Fp(~)1966 4245 y Fn(t)1996 4257 y Fm(r)2027 4265 y Fh(i)2053 4257 y Fy(+1)2142 4245 y Fn(:)333 4382 y Fk(L)l(et)43 b(the)g(extende)l(d)g(p)l(artitions)h(b)l (e)f Fo(f)p Fn(t)1567 4394 y Fy(1)1604 4382 y Fn(;)14 b(:)g(:)g(:)g(;)g(t)1819 4394 y Fm(k)q Fy(+8)1943 4382 y Fo(g)43 b Fk(and)h Fo(f)2246 4367 y Fp(~)2245 4382 y Fn(t)2275 4394 y Fy(1)2311 4382 y Fn(;)14 b(:)g(:)g(:)g(;)2497 4367 y Fp(~)2496 4382 y Fn(t)2526 4394 y Fy(2)p Fm(k)q Fy(+8)2684 4382 y Fo(g)p Fk(.)78 b(Then)44 b Fn(\016)3099 4352 y Fy(4)3096 4403 y Fm(i)3136 4382 y Fp(\()p Fn(j)5 b Fp(\))p Fn(;)14 b(j)53 b Fp(=)333 4481 y Fn(r)n(;)14 b(:)g(:)g(:)g(;)g(r)21 b Fp(+)d(3)29 b Fk(ar)l(e)h(determine)l(d)h(by)f (the)g(formul\032:)610 4724 y Fn(\016)650 4690 y Fy(4)647 4744 y Fm(i)687 4724 y Fp(\()p Fn(r)r Fp(\))85 b(=)1681 4649 y Fn(T)1742 4619 y Fy(3)1730 4680 y Fm(i;d)p Fd(\033)1868 4664 y Fg(\(1\))1949 4649 y Fp(\()p Fn(t)2011 4661 y Fm(i)p Fy(+1)2123 4649 y Fp(\))2172 4628 y Ff(e)2155 4649 y Fn(C)5 b Fp(\()p Fn(r)r Fp(\))p 1032 4705 1941 4 v 1032 4796 a Fn(T)1093 4767 y Fy(3)1081 4828 y Fm(i;d)p Fd(\033)1218 4812 y Fg(\(1\))1300 4796 y Fp(\()p Fn(t)1362 4808 y Fm(i)p Fy(+1)1474 4796 y Fp(\))1523 4775 y Ff(e)1506 4796 y Fn(C)g Fp(\()p Fn(r)r Fp(\))19 b(+)1792 4775 y Ff(e)1775 4796 y Fn(C)5 b Fp(\()p Fn(r)21 b Fp(+)d(1\))h(+)f Fn(T)2249 4767 y Fy(3)2237 4828 y Fm(i;d)p Fd(\033)2374 4812 y Fg(\(1\))2456 4796 y Fp(\()p Fn(t)2518 4808 y Fm(i)p Fy(+2)2630 4796 y Fp(\))2678 4775 y Ff(e)2662 4796 y Fn(C)t Fp(\()p Fn(r)k Fp(+)c(2\))p eop %%Page: 6 6 6 5 bop 333 274 a Fi(6)2269 b(T.)23 b(Bosner)h(and)g(M.)f(Rogina)p 333 324 3095 4 v 467 521 a Fn(\016)507 487 y Fy(4)504 541 y Fm(i)544 521 y Fp(\()p Fn(r)f Fp(+)c(1\))83 b(=)1475 446 y Fn(T)1536 416 y Fy(3)1524 477 y Fm(i;d)p Fd(\033)1662 461 y Fg(\(1\))1743 446 y Fp(\()p Fn(t)1805 458 y Fm(i)p Fy(+1)1917 446 y Fp(\))1966 425 y Ff(e)1949 446 y Fn(C)5 b Fp(\()p Fn(r)r Fp(\))20 b(+)2235 425 y Ff(e)2219 446 y Fn(C)t Fp(\()p Fn(r)i Fp(+)c(1\))p 1032 502 1941 4 v 1032 593 a Fn(T)1093 564 y Fy(3)1081 625 y Fm(i;d)p Fd(\033)1218 608 y Fg(\(1\))1300 593 y Fp(\()p Fn(t)1362 605 y Fm(i)p Fy(+1)1474 593 y Fp(\))1523 572 y Ff(e)1506 593 y Fn(C)5 b Fp(\()p Fn(r)r Fp(\))19 b(+)1792 572 y Ff(e)1775 593 y Fn(C)5 b Fp(\()p Fn(r)21 b Fp(+)d(1\))h(+)f Fn(T)2249 564 y Fy(3)2237 625 y Fm(i;d)p Fd(\033)2374 608 y Fg(\(1\))2456 593 y Fp(\()p Fn(t)2518 605 y Fm(i)p Fy(+2)2630 593 y Fp(\))2678 572 y Ff(e)2662 593 y Fn(C)t Fp(\()p Fn(r)k Fp(+)c(2\))3280 521 y(\(14\))467 825 y Fn(\016)507 791 y Fy(4)504 846 y Fm(i)544 825 y Fp(\()p Fn(r)k Fp(+)c(2\))83 b(=)1517 751 y Fn(T)1578 720 y Fy(3)1566 781 y Fm(i)p Fy(+1)p Fm(;d)p Fd(\033)1788 765 y Fg(\(1\))1869 751 y Fp(\()p Fn(t)1931 763 y Fm(i)p Fy(+3)2043 751 y Fp(\))2092 730 y Ff(e)2075 751 y Fn(C)5 b Fp(\()p Fn(r)21 b Fp(+)d(4\))h(+)2504 730 y Ff(e)2488 751 y Fn(C)t Fp(\()p Fn(r)j Fp(+)c(3\))p 1032 806 2252 4 v 1032 897 a Fn(T)1093 868 y Fy(3)1081 929 y Fm(i)p Fy(+1)p Fm(;d)p Fd(\033)1302 913 y Fg(\(1\))1384 897 y Fp(\()p Fn(t)1446 909 y Fm(i)p Fy(+2)1558 897 y Fp(\))1607 876 y Ff(e)1590 897 y Fn(C)5 b Fp(\()p Fn(r)21 b Fp(+)d(2\))h(+)2019 876 y Ff(e)2003 897 y Fn(C)t Fp(\()p Fn(r)j Fp(+)c(3\))g(+)g Fn(T)2476 868 y Fy(3)2464 929 y Fm(i)p Fy(+1)p Fm(;d)p Fd(\033)2685 913 y Fg(\(1\))2767 897 y Fp(\()p Fn(t)2829 909 y Fm(i)p Fy(+3)2941 897 y Fp(\))2989 876 y Ff(e)2973 897 y Fn(C)t Fp(\()p Fn(r)k Fp(+)c(4\))467 1129 y Fn(\016)507 1095 y Fy(4)504 1150 y Fm(i)544 1129 y Fp(\()p Fn(r)k Fp(+)c(3\))83 b(=)1723 1055 y Fn(T)1784 1025 y Fy(3)1772 1086 y Fm(i)p Fy(+1)p Fm(;d)p Fd(\033)1994 1069 y Fg(\(1\))2075 1055 y Fp(\()p Fn(t)2137 1067 y Fm(i)p Fy(+3)2249 1055 y Fp(\))2298 1034 y Ff(e)2281 1055 y Fn(C)5 b Fp(\()p Fn(r)22 b Fp(+)c(4\))p 1032 1110 V 1032 1201 a Fn(T)1093 1173 y Fy(3)1081 1234 y Fm(i)p Fy(+1)p Fm(;d)p Fd(\033)1302 1217 y Fg(\(1\))1384 1201 y Fp(\()p Fn(t)1446 1213 y Fm(i)p Fy(+2)1558 1201 y Fp(\))1607 1180 y Ff(e)1590 1201 y Fn(C)5 b Fp(\()p Fn(r)21 b Fp(+)d(2\))h(+)2019 1180 y Ff(e)2003 1201 y Fn(C)t Fp(\()p Fn(r)j Fp(+)c(3\))g(+)g Fn(T)2476 1173 y Fy(3)2464 1234 y Fm(i)p Fy(+1)p Fm(;d)p Fd(\033)2685 1217 y Fg(\(1\))2767 1201 y Fp(\()p Fn(t)2829 1213 y Fm(i)p Fy(+3)2941 1201 y Fp(\))2989 1180 y Ff(e)2973 1201 y Fn(C)t Fp(\()p Fn(r)k Fp(+)c(4\))333 1399 y Fk(wher)l(e,)31 b(as)f(in)g(\(6\))1400 1520 y Ff(e)1384 1541 y Fn(C)t Fp(\()p Fn(i)p Fp(\))23 b(=)1651 1428 y Ff(Z)1697 1616 y Fm(suppor)r(t)1958 1520 y Ff(e)1944 1541 y Fn(T)2005 1506 y Fy(3)1993 1567 y Fm(i;d)p Fd(\033)2131 1551 y Fg(\(1\))2226 1541 y Fn(d\033)2316 1553 y Fy(2)2354 1541 y Fn(:)499 1778 y Fp(T)-7 b(o)18 b(use)g(this)h(Lemma)f(and)g(Theorem,) i(it)f(is)f(remained)g(to)g(calculate)g Fn(T)2694 1748 y Fy(3)2682 1800 y Fm(i)2730 1778 y Fp(\()p Fn(t)2792 1790 y Fm(i)p Fy(+1)2904 1778 y Fp(\))h(and)f Fn(T)3168 1748 y Fy(3)3156 1800 y Fm(i)3204 1778 y Fp(\()p Fn(t)3266 1790 y Fm(i)p Fy(+2)3378 1778 y Fp(\).)333 1878 y(F)-7 b(rom)27 b(form)n(ula)745 2087 y Fn(T)806 2052 y Fy(3)794 2107 y Fm(i)842 2087 y Fp(\()p Fn(x)p Fp(\))d(=)1149 2031 y(1)p 1075 2068 190 4 v 1075 2144 a Fn(C)1134 2156 y Fy(2)1171 2144 y Fp(\()p Fn(i)p Fp(\))1288 1974 y Ff(Z)1371 1994 y Fm(x)1334 2162 y(t)1359 2170 y Fh(i)1427 2087 y Fn(B)1494 2052 y Fy(2)1490 2107 y Fm(i)1531 2087 y Fp(\()p Fn(t)p Fp(\))p Fn(q)s Fp(\()p Fn(t)p Fp(\))14 b Fn(dt)20 b Fo(\000)2105 2031 y Fp(1)p 1959 2068 333 4 v 1959 2144 a Fn(C)2018 2156 y Fy(2)2056 2144 y Fp(\()p Fn(i)e Fp(+)g(1\))2316 1974 y Ff(Z)2399 1994 y Fm(x)2362 2162 y(t)2387 2170 y Fh(i)p Fg(+1)2502 2087 y Fn(B)2569 2052 y Fy(2)2565 2107 y Fm(i)p Fy(+1)2677 2087 y Fp(\()p Fn(t)p Fp(\))p Fn(q)s Fp(\()p Fn(t)p Fp(\))c Fn(dt;)333 2327 y Fp(where)656 2546 y Fn(C)715 2558 y Fy(2)753 2546 y Fp(\()p Fn(i)p Fp(\))23 b(=)957 2433 y Ff(Z)1040 2454 y Fm(t)1065 2462 y Fh(i)p Fg(+2)1003 2622 y Fm(t)1028 2630 y Fh(i)1180 2546 y Fn(B)1247 2512 y Fy(2)1243 2567 y Fm(i)1284 2546 y Fp(\()p Fn(t)p Fp(\))p Fn(q)s Fp(\()p Fn(t)p Fp(\))14 b Fn(dt)25 b Fp(=)1721 2490 y(1)p 1721 2527 42 4 v 1721 2603 a(6)1773 2546 y([\()p Fn(q)1865 2558 y Fm(i)1911 2546 y Fp(+)18 b(2)p Fn(q)2073 2558 y Fm(i)p Fy(+1)2184 2546 y Fp(\))p Fn(h)2264 2558 y Fm(i)2311 2546 y Fp(+)g(\(2)p Fn(q)2505 2558 y Fm(i)p Fy(+1)2635 2546 y Fp(+)g Fn(q)2755 2558 y Fm(i)p Fy(+2)2866 2546 y Fp(\))p Fn(h)2946 2558 y Fm(i)p Fy(+1)3058 2546 y Fp(])p Fn(;)333 2779 y Fp(it)28 b(is)g(easy)e(to)i(get)f(that)794 3005 y Fn(T)855 2971 y Fy(3)843 3025 y Fm(i)891 3005 y Fp(\()p Fn(t)953 3017 y Fm(i)p Fy(+1)1065 3005 y Fp(\))d(=)1218 2949 y Fn(h)1266 2961 y Fm(i)1294 2949 y Fp(\()p Fn(q)1363 2961 y Fm(i)1409 2949 y Fp(+)18 b(2)p Fn(q)1571 2961 y Fm(i)p Fy(+1)1683 2949 y Fp(\))p 1218 2986 497 4 v 1351 3062 a(6)p Fn(C)1452 3074 y Fy(2)1489 3062 y Fp(\()p Fn(i)p Fp(\))1725 3005 y Fn(;)97 b(T)1906 2971 y Fy(3)1894 3025 y Fm(i)1942 3005 y Fp(\()p Fn(t)2004 3017 y Fm(i)p Fy(+2)2116 3005 y Fp(\))23 b(=)2269 2949 y Fn(h)2317 2961 y Fm(i)p Fy(+2)2428 2949 y Fp(\(2)p Fn(q)2539 2961 y Fm(i)p Fy(+2)2669 2949 y Fp(+)18 b Fn(q)2789 2961 y Fm(i)p Fy(+3)2901 2949 y Fp(\))p 2269 2986 665 4 v 2414 3062 a(6)p Fn(C)2515 3074 y Fy(2)2552 3062 y Fp(\()p Fn(i)h Fp(+)f(1\))2943 3005 y Fn(:)333 3242 y Fp(No)n(w)27 b(w)n(e)g(ha)n(v)n(e)g(ev)n(erything)f(to)i(calculate)f Fn(T)1753 3211 y Fy(4)1741 3263 y Fm(i)1789 3242 y Fp(.)333 3480 y Fq(3.)112 b(Conclusion)499 3658 y Fp(In)36 b(preceeding)g (paragraph,)g(w)n(e)f(manage)g(to)h(construct)g(an)g(algorithm)f(for)h (calculating)333 3758 y(with)k(q-splines)f(as)h(a)f(linear)g(com)n (bination)g(of)h(p)r(olynomial)f(B-splines.)73 b(Moreo)n(v)n(er,)40 b(all)f(the)333 3858 y(co)r(e\016cien)n(ts)27 b(in)n(v)n(olv)n(ed,)f (are)h(p)r(ositiv)n(e,)g(guaran)n(teeing)f(a)h(stabilit)n(y)g(of)h(suc) n(h)f(an)g(algorithm.)333 4097 y Fq(References)346 4229 y Fi([1])41 b Fb(R.)26 b(Kulkarni,)f(P.)g(J.)g(La)o(urent)p Fi(,)e Fa(Q-splines)p Fi(,)g(Numerical)f(Algorithms,)f(1)j(\(1991\),)h (45{73)346 4324 y([2])41 b Fb(M.)31 b(R)n(ogina)p Fi(,)f Fa(On)g(Construction)h(of)h(F)-5 b(ourth)31 b(Or)l(der)g(Chebyshev)g (Splines)p Fi(,)g(Mathematical)e(Comm)n(unica-)595 4403 y(tions,)24 b(4)f(\(1999\),)j(83{92)346 4499 y([3])41 b Fb(M.)36 b(R)n(ogina)p Fi(,)g Fa(Basis)g(of)f(Splines)i(Asso)l(ciate) l(d)f(with)g(Some)g(Singular)g(Di\013er)l(ential)f(op)l(er)l(ators)p Fi(,)k(BIT)34 b(32,)595 4578 y(496{505,)25 b(1992.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF