И.А. Аптекарев, В.Н. Сорокин - Спектральная теория разностных операторов (1128570), страница 10

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 10)

’®£¤ ¯®±«¥¤®¢ ²¥«¼­®±²¼²®·¥ª < 'n; A'n >; ¯°¨­ ¤«¥¦ ¹¨µ £° ´¨ª³ (A); ±µ®¤¨²±¿ª ²®·ª¥ < '; A' >; ² ª¦¥ ¯°¨­ ¤«¥¦ ¹¥© £° ´¨ª³ (² ª ª ª¯® ³±«®¢¨¾ ®¯¥° ²®° § ¬ª­³²»©). ‘«¥¤®¢ ²¥«¼­®,(A + i)'n ! (A + i)';£¤¥ ' 2 D(A); ·²® ¨ ®§­ · ¥² § ¬ª­³²®±²¼ ®¡° § .‹¥¬¬ ¤®ª § ­ .

4‡ ¬ª­³²»© ±¨¬¬¥²°¨·­»© ®¯¥° ²®° ¿¢«¿¥²±¿ ± ¬®±®¯°¿¦¥­­»¬, ²®£¤ ¨ ²®«¼ª® ²®£¤ , ª®£¤ ¥£®¨­¤¥ª±» ¤¥´¥ª² ° ¢­» ­³«¾ d = 0:‚¥°­¥¬±¿ ª ¯°¨¬¥°³ ®¯¥° ²®° Ÿª®¡¨. ‚»·¨±«¨¬ ¥£® ¨­¤¥ª±» ¤¥´¥ª² . ¥¸¨¬ ³° ¢­¥­¨¥‘«¥¤±²¢¨¥ 18AQ = z Q; z 2 C :118¥¸¥­¨¥¬ (± ²®·­®±²¼¾ ¤® ·¨±«®¢®£® ¬­®¦¨²¥«¿) ¡³¤¥² ±²®«¡¥¶ ®°²®­®°¬¨°®¢ ­­»µ ¬­®£®·«¥­®¢. ‚®§¬®¦­» ¤¢ ±«³· ¿. ‹¨¡® Q 2 l2 (±«³· © (C )); ²®£¤ ¨­¤¥ª±» ¤¥´¥ª² ° ¢­» ¥¤¨­¨¶¥ d = 1: ‹¨¡® Q 62 l2 (±«³· © (D)); ²®£¤ ¨­¤¥ª±»¤¥´¥ª² ° ¢­» ­³«¾ d = 0:’ ª¨¬ ®¡° §®¬, ­ ¬¨ ¤®ª § ­ ±«¥¤³¾¹ ¿ ± ¬®±®¯°¿¦¥­­»©, ²®£¤ ’¥®°¥¬ 26 Ž¯¥° ²®° Ÿª®¡¨ A¨ ²®«¼ª® ²®£¤ , ª®£¤ ±®®²¢¥²±²¢³¾¹ ¿ ¥¬³ ¯°®¡«¥¬ ¬®¬¥­²®¢ ®¯°¥¤¥«¥­ .1194.  §­®±²­»¥ ®¯¥° ²®°» ¢²®°®£® ¯®°¿¤ª ¨ ­¥¯°¥°»¢­ ¿ ¤°®¡¼ ‘²¨«²¼¥± 4.1. „°®¡¼ ‘²¨«²¼¥± ¨ ¥¥ ½ª¢¨¢ «¥­²­»¥ ´®°¬»¥¯°¥°»¢­®© ¤°®¡¼¾ ‘²¨«²¼¥± ­¥¯°¥°»¢­³¾ ¤°®¡¼za01a1­ §»¢ ¾² ±«¥¤³¾¹³¾(4:1)a2az 1 3 ˆ­®£¤ ¥¥ ² ª¦¥ § ¯¨±»¢ ¾² ¢ ¢¨¤¥1; (4:2)1c0z +1c1 + c z + 2£¤¥a0 = c1 ; ak = c c1 ; k = 1; 2; :::0k k 1°¨­¨¬ ¿ ¢® ¢­¨¬ ­¨¥, ·²® · ±²® ¢ ¯°¨«®¦¥­¨¿µ ·¥²­»¥ ¨­¥·¥²­»¥ ª®½´´¨¶¨¥­²» ¤°®¡¨ (4.2) ¨¬¥¾² ° §«¨·­»© ±¬»±«,¤°®¡¼ (4.2) ² ª¦¥ § ¯¨±»¢ ¾² ¢ ¢¨¤¥1l0 +(4:3)1m1z +1l1 + m z + 2‚ ­¥ª®²®°®¬ ±¬»±«¥ ¤°®¡¼ ‘²¨«²¼¥± ¤ ¥² ¡®«¥¥ ¯®«­³¾ ¨­´®°¬ ¶¨¾ ® ±²¥¯¥­­®¬ °¿¤¥, ·¥¬ ° ±±¬®²°¥­­ ¿ ° ­¥¥ (±¬.120¯ ° £° ´ 1.5) ­¥¯°¥°»¢­ ¿ ¤°®¡¼ —¥¡»¸¥¢ -Ÿª®¡¨.

®¿±­¨¬ ½²® ±«¥¤³¾¹¨¬ ³²¢¥°¦¤¥­¨¥¬.—¥²­»¥ ¯®¤µ®¤¿¹¨¥ ­¥¯°¥°»¢­®© ¤°®¡¨‘²¨«²¼¥± (4.1) ±®¢¯ ¤ ¾² ± ¯®¤µ®¤¿¹¨¬¨ ­¥¯°¥°»¢­®© ¤°®¡¨ —¥¡»¸¥¢ -Ÿª®¡¨“²¢¥°¦¤¥­¨¥ 24A0z B0£¤¥(4:4)A1Az B1 z B 2 2A0 = a0;B0 = a1;An = a2na2n 1; n = 1; 2; :::Bn = a2n + a2n+1; n = 1; 2; ::: (4:5)„®ª § ²¥«¼±²¢®. Ž¡®§­ ·¨¬atk [w] = 2k 1a2k1z w«¥¬¥­² °­»¥ ¯°¥®¡° §®¢ ­¨¿ ¤ ¾²tk [w] = a2k 1 + z a2kaa2k 1 w : (4:6)2k—¥²­»¥ ¯®¤µ®¤¿¹¨¥ ­¥¯°¥°»¢­®© ¤°®¡¨ ‘²¨«²¼¥± (4.1) ¬®¦­® § ¯¨± ²¼ ª ªP2n(z ) =a0Q2n(z ) z t1 t2 :::tn 1 [ a n1 ] ;21211·²® ± ³·¥²®¬ (4.6) ¬®¦­® ¯¥°¥¨± ²¼ ª ªa0P2n(z ) =:Q2n(z )a1a2z a1a3a4z a2 a3a2n 3a2n 2z a4 a5 :::az a2n 2 21n 1“²¢¥°¦¤¥­¨¥ ¤®ª § ­®.

4‚¢¨¤³ “²¢¥°¦¤¥­¨¿ 8 ¯ ° £° ´ 1.5, ·¥²­»¥ ¯®¤µ®¤¿¹¨¥ ­¥¯°¥°»¢­®© ¤°®¡¨ ‘²¨«²¼¥± ‘«¥¤±²¢¨¥ 19a0z1a1a2 S (z ) = sz0 + zs12 + :::az 13(4:7)¿¢«¿¾²±¿ ¤¨ £®­ «¼­»¬¨ ¯¯°®ª±¨¬ ¶¨¿¬¨  ¤¥ (° ±±¬®²°¥­­»¬¨ ¢ ¯ ° £° ´¥ 1.1) ¤«¿ ±²¥¯¥­­®£® °¿¤ S (z ):Ž²¬¥²¨¬, ·²® ¯°¥®¡° §®¢ ­¨¿ ‘²¨«²¼¥± (4.5) ¢ «¨²¥° ²³°¥ ¯® ¬ ²¥¬ ²¨·¥±ª®© ´¨§¨ª¥ ² ª¦¥ ­ §»¢ ¾² ¯°¥®¡° §®¢ ­¨¿¬¨ Œ¨³°».1224.2. ¥ª³°°¥­²­»¥ ±®®²­®¸¥­¨¿ ¨ ° §­®±²­»¥§ ¤ ·¨, ±¢¿§ ­­»¥ ± ¤°®¡¼¾ ‘²¨«²¼¥± Ž¡¹¥¥ “²¢¥°¦¤¥­¨¥ 7 ¯ ° £° ´ 1.5 ¤ ¥² ²°¥µ·«¥­­»¥ °¥ª³°°¥­²­»¥ ±®®²­®¸¥­¨¿, ±¢¿§»¢ ¾¹¨¥ ¯®«¨­®¬» §­ ¬¥­ ²¥«¨ ¯®¤µ®¤¿¹¨µ ¤°®¡¥© (4.1) ¨ (4.4).

’¥ ¦¥ °¥ª³°°¥­²­»¥±®®²­®¸¥­¨¿ ¯°¨ ¤°³£¨µ ­ · «¼­»µ ¤ ­­»µ ² ª¦¥ ±¢¿§»¢ ¾² ·¨±«¨²¥«¨ ¯®¤µ®¤¿¹¨µ ¤°®¡¥©. „«¿ ¤°®¡¨ (4.1) ¨¬¥¥¬Yn+1(z ) = "n Yn(z ) anYn 1(z ); n = 1; 2; :::;(4:8)£¤¥ "n = z ¤«¿ ·¥²­»µ n ¨ "n = 1 ¤«¿ ­¥·¥²­»µ n: ’®£¤ ¤«¿n © ¯®¤µ®¤¿¹¥© Pn=Qn ­¥¯°¥°»¢­®© ¤°®¡¨ (4.1) ±¯° ¢¥¤«¨¢® (¤«¿ n = 0; 1; :::)Pn = Yn;P0 = Y0 = 0; P1 = Y1 = a0;Qn = Yn;Q0 = Y0 = 1; Q1 = Y1 = z:°¨ ½²®¬deg Q2n = n; deg Q2n 1 = n; deg Pn = deg Qn 1:€­ «®£¨·­®, °¥ª³°°¥­²­»¥ ±®®²­®¸¥­¨¿, § ¤ ¢ ¥¬»¥ ¤°®¡¼¾ (4.4), ¨¬¥¾² ¢¨¤Y~n+1(z ) = (z Bn )Y~n(z ) AnY~n 1(z ); n = 1; 2; :::;(4:9)£¤¥ An; Bn § ¤ ¾²±¿ ±®®²­®¸¥­¨¿¬¨ ‘²¨«²¼¥± (4.5). ®½²®¬³ ¤«¿ n = 0; 1; 2; ::: ¨¬¥¥¬P2n = Y~n;P0 = Y~0 := 0; P2 = Y~1 := A0 = a0;Q2n = Y~n; Q0 = Y~0 := 1; Q2 = Y~1 := z B0 = z a1:¥ª³°°¥­²­»¥ ±®®²­®¸¥­¨¿ (4.8) ¨ (4.9) § ¤ ¾² ° §­®±²­³¾§ ¤ ·³ ¢²®°®£® ¯®°¿¤ª .

‚ ®¡®§­ ·¥­¨¿µ ¤°®¡¨ (4.3) ¯°¨¢¥¤¥¬ ½ª¢¨¢ «¥­²­³¾ ±¨±²¥¬³ ° §­®±²­»µ ³° ¢­¥­¨© ¯¥°¢®£®¯®°¿¤ª .123“²¢¥°¦¤¥­¨¥ 25 (Š°¥©­)°¿¾² ±¨±²¥¬¥³±²¼ (z ) ¨ (z ) ³¤®¢«¥²¢®-(n+1 n = ln nn+1 n = mn+1zn+1;n = 1; 2; 3; :::(4:10)’®£¤ ¤«¿ ¯®¤µ®¤¿¹¨µ QPnn ­¥¯°¥°»¢­®© ¤°®¡¨ ‘²¨«²¼¥± (4.1) (²¥ ¦¥ ¯®¤µ®¤¿¹¨¥ ¨¬¥¾² ½ª¢¨¢ «¥­²­»¥ ¤°®¡¨ (4.2)¨ (4.3) ¯°¨ ±®®²¢¥²±²¢³¾¹¨µ § ¬¥­ µ fang $ fcng $ fln; mn g);±¯° ¢¥¤«¨¢®Q2n+1 = n;Q2n = n;n = 0; 1; :::;n = 1; 2; :::;0 = 1;1 = m1z;±®®²¢¥²±²¢³¾¹¨¥ ·¨±«¨²¥«¨ ¯®«³· ¾²±¿ ¯°¨0 = l0; 1 = 1 + l0 m1z:„®ª ¦¨²¥ ½²® ³²¢¥°¦¤¥­¨¥. ±±¬®²°¨¬ ¤°³£®© ° §­®±²­»© ®¯¥° ²®°, ±¢¿§ ­­»© ± ¤°®¡¼¾ ‘²¨«²¼¥± (4.1).

³±²¼ ®¯¥° ²®° A; ¤¥©±²¢³¾¹¨© ¢ l2;§ ¤ ¥²±¿ ¢ ±² ­¤ °²­®¬ ¡ §¨±¥ fek g :“¯° ¦­¥­¨¥ 10:= (0; ::; 0; 1; 0; :::); k = 0; 1; 2; :::± ¯®¬®¹¼¾ ¡¥±ª®­¥·­®© ¬ ²°¨¶»20 136a1 0 177A := 64 a2 0 1 5::: ::: :::„«¿ ½²®£® ®¯¥° ²®° ° ±±¬®²°¨¬ °¥§®«¼¢¥­²­³¾ ´³­ª¶¨¾11X(Ake0; e0) Xfk :1f (z ) := ((z I A) e0; e0) ==:k+1zz k+1k=0k=0ek124„«¿ ±¯¥ª²° «¼­»µ ¤ ­­»µ f (z ); ffk g1k=0®¯¥° ²®° A ±¯° ¢¥¤«¨¢®f (z ) = zS (z 2)f2k = sk ; f2k+1 = 0; k = 0; 1; 2; ::: (4:11)£¤¥ S (z ) ±²¥¯¥­­®© °¿¤, ¯°¥¤±² ¢«¿¾¹¨© ¤°®¡¼ ‘²¨«²¼¥± (4.7).„®ª § ²¥«¼±²¢®.

‘¤¥« ¥¬ ¢ ¤°®¡¨ ‘²¨«²¼¥± S (~z ) § ¬¥­³2z~ = z ¨ ° ±±¬®²°¨¬“²¢¥°¦¤¥­¨¥ 26zS (z 2) = za0z21a1a2az 2 1 3 “¬­®¦¥­¨¥ ¨ ¤¥«¥­¨¥ ª ¦¤®£® ½² ¦ ½²®© ­¥¯°¥°»¢­®© ¤°®¡¨ ­ z ¤ ¥²a0zS (z 2) =a1zaz z 2 ‡ ¬¥· ¿, ·²® ¢ ¯° ¢®© · ±²¨ ±²®¨² ­¥¯°¥°»¢­ ¿ ¤°®¡¼, ±®®²¢¥²±²¢³¾¹ ¿ °¥§®«¼¢¥­²­®© ´³­ª¶¨¨ ®¯¥° ²®° A (±¬. ¯ ° £° ´ 1.6), ¯®«³· ¥¬ ±¯° ¢¥¤«¨¢®±²¼ ¯¥°¢®£® ±®®²­®¸¥­¨¿¢ (4.11). ‘¯° ¢¥¤«¨¢®±²¼ ¢²®°®£® ±®®²­®¸¥­¨¿ ¢ (4.11) ±° §³¦¥ ¢»²¥ª ¥² ¨§ ¯¥°¢®£® ±®®²­®¸¥­¨¿, ¥±«¨ ¯°¨° ¢­¿²¼ ª®½´´¨¶¨¥­²» ±²¥¯¥­­»µ °¿¤®¢, ±²®¿¹¨µ ¢ ®¡¥¨µ · ±²¿µ ° ¢¥­±²¢ .

4’ ª¨¬ ®¡° §®¬, ª ±¯¥ª²° «¼­»¬ ¤ ­­»¬ ®¯¥° ²®° A ¬»² ª¦¥ ¬®¦¥¬ ®²­®±¨²¼ ±²¥¯¥­­®© °¿¤ S (z ) ¨ ¥£® ª®½´´¨¶¨¥­²» fsk g: Œ» ¨µ ¡³¤¥¬ ­ §»¢ ²¼ ±²¨«²¼¥±®¢±ª¨¬¨ ±¯¥ª²° «¼­»¬¨ ¤ ­­»¬¨ ®¯¥° ²®° A:1254.3. ¥¸¥­¨¥ ¯°¿¬®© § ¤ ·¨¤«¿ ±¯¥ª²° «¼­»µ ¤ ­­»µ ‘²¨«²¼¥± ¥¸¥­¨¥ ¯°¿¬®© § ¤ ·¨fak g ! fsk g¤ ¥²±¿ ±«¥¤³¾¹¥© ´®°¬³«®©.’¥®°¥¬ 27 …±«¨ ±²¥¯¥­­®© °¿¤1Xk; s0 = 1;S (z ) = zsk+1k=0° §« £ ¥²±¿ ¢ ¤°®¡¼ ‘²¨«²¼¥± a0S (z ) =a1za21z ²®sk = a1jX+12Xj2 =1aj22j3 =1aj :::Xjk 1 +13jk =1ajk ; k = 1; 2; :::(4:12)„®ª § ²¥«¼±²¢³ ²¥®°¥¬» ¯°¥¤¯®¸«¥¬ ±«¥¤³¾¹³¾ «¥¬¬³.(1)‹¥¬¬ 7 ³±²¼ sk ®¯°¥¤¥«¥­» ª ª (4.12), sk ¥±²¼jX+13X(1)sk = a2 ajajj =2 j =23433:::4’®£¤ sk+1 = a1Xjk +1jk+1 =2kX =0ajk ; k = 1; 2; :::+1s(1) sk :126(4:14)(4:13)„®ª § ²¥«¼±²¢® «¥¬¬».sk+1 = a1 sk +a2a1 Xi +13X:::+ a2i2 =2ai22i3 =22Xaj :::3j3=1ip 1 +1ai :::3 ±¯¨±»¢ ¿ (4.12) ¤«¿ sk+1 ¨¬¥¥¬Xjk 1 +1ajk +a2(a2+a3)a1jk =12X Xip =2aipa1jp+2 =1ajp :::+22Xj4=1jk 1 +1Xjk =1aj :::Xjk 1 +14jk =1 !ajk +::: ;·²® ± ³·¥²®¬ ®¡®§­ ·¥­¨¿ (4.13) ¤ ¥² (4.14).‹¥¬¬ ¤®ª § ­ .

4„®ª § ²¥«¼±²¢® ²¥®°¥¬».  ±±¬®²°¨¬ ±²¥¯¥­­»¥ °¿¤»:11 (1)XXsk(1)S (z ) = z k+1 ¨ S (z ) = zskk+1 ;s0 = s(1)0 = 1:k=0k=0‹¥¬¬ ± ³·¥²®¬ ¯° ¢¨« ³¬­®¦¥­¨¿ ±²¥¯¥­­»µ °¿¤®¢ ¤ ¥²:S (z ) z1 = a1S (z )S (1)(z ):Ž²ª³¤ S (z ) = z za1S (1)(z ) : (4:15)1Ž¡®§­ ·¨¬s(l) = akl+1l+2Xjl+2 =l+1¨® «¥¬¬¥ ¨¬¥¥¬Xajljl+2 +1+2jl+3 =l+1ajl :::+3Xjk+l 1 +1jk+l =l+11 (l )X(l)S (z ) = zskk+1 ;k=0ajk l ; k = 0; 1; 2; :::+s(0l) = 1:S (l) (z ) = z za 1S (l+1) (z ) :l+1127ajk +:::ˆ²¥° ²¨¢­® ¯®¤±² ¢«¿¿ ½²³ ´®°¬³«³ ¢ (4.15) ¯®«³· ¥¬ ¤«¿S (z ) ° §«®¦¥­¨¥ ¢ ¤°®¡¼ ‘²¨«²¼¥± ± ª®½´´¨¶¨¥­² ¬¨ fak g :1=S (z ) =z a1z z za 1S (z)2=1za1=a21z za3S (3)(2)1z1a1:a2az 1 3 ’¥®°¥¬ ¤®ª § ­ .

4“·¨²»¢ ¿ ±²°³ª²³°³ ±³¬¬ ¢ ¯° ¢®© · ±²¨ (4.12) ¬» ¡³¤¥¬¨µ ­ §»¢ ²¼ £¥­¥²¨·¥±ª¨¬¨ ±³¬¬ ¬¨.1284.4. ‚»·¨±«¥­¨¥ ®¯°¥¤¥«¨²¥«¥© ƒ ­ª¥«¿¨ °¥¸¥­¨¥ ®¡° ²­®© § ¤ ·¨¤«¿ ±¯¥ª²° «¼­»µ ¤ ­­»µ ‘²¨«²¼¥± …±«¨ ®¯°¥¤¥«¨²¥«¼ ƒ ­ª¥«¿ ±®±²®¨² ¨§ ¬®¬¥­²®¢, § ¤ ¢ ¥¬»µ £¥­¥²¨·¥±ª¨¬¨ ±³¬¬ ¬¨ (4.12), ²® ®­ «¥£ª® ¬®¦¥² ¡»²¼¯°¨¢¥¤¥­ ª ²°¥³£®«¼­®¬³ ¢¨¤³. °®¨««¾±²°¨°³¥¬ ½²® ­ ¯°¨¬¥°¥ ®¯°¥¤¥«¨²¥«¿ 3-£® ¯®°¿¤ª :s s s 0 1 2H3 = s1 s2 s3 =s2 s3 s4P1+1 a1aa11 j =1 jPPP1+11+1j+1a1 j =1 aja1 j =1 aj j =1 aj= a1 PPPPPP1+11+1j+11+1j+1j+1a1 j =1 aj a1 j =1 aj j =1 aj a1 j =1 aj j =1 aj j =1 aj 222222222222332322333‚»­®±¿ ¨§ ¯®±«¥¤­¥© ±²°®ª¨ a1; ¢»·¨² ¿ ¨§ ­¥¥ ¯°¥¤¯®±«¥¤­¾¾ ±²°®ª³ ¨ § ²¥¬ ¢»­®±¿ ¨§ ¯®±«¥¤­¥© ¯®«³·¥­­®© ±²°®ª¨a2; ¨§ ¯°¥¤¯®±«¥¤­¥© ±²°®ª¨ a1; ¯®«³·¨¬P1+1 a 1aa11 j =1 jPPPH3 = a21a2 1 2j =1 aj 2j =1 aj jj +1=1 aj PPP23j+11 j =1 aj j =1 aj j =1 aj 2222222223333312943344‚»·¨² ¿ ¨§ 3-© ±²°®ª¨ ¢²®°³¾, ¨§ 2-© ±²°®ª¨ ¯¥°¢³¾, ¯°®¤®«¦¨¬P1+1 a 1aa 1 1 j =1 j P2+12H3 = a1a2 0 a2 a2 j =1 aj P3+10 a3 a3 j =1 aj 223344‚»­®±¿ ¨§ 3-© ±²°®ª¨ a3; ¨§ 2-© ±²°®ª¨ a2 ¨ ¢»·¨² ¿ ¨§ ¯®«³·¥­­®© 3-© ±²°®ª¨ ¢²®°³¾, ¡³¤¥¬ ¨¬¥²¼P21 a1 a1 j =1 aj 2 2P322H3 = a1a2a3 0 1j =1 aj = a1 a2 a3a4 :0 0a4 2233°¨¬¥­¿¿ ­ «®£¨·­»¥ ° ±±³¦¤¥­¨¿ ª ®¯°¥¤¥«¨²¥«¾ ¯°®¨§¢®«¼­®£® ¯®°¿¤ª ¯®«³· ¥¬:Hn+1 = (a1a2)n(a3a4)n 1:::(a2n 3a2n 2)2(a2n 1a2n); n = 0; 1; :::’ ª¨¬ ¦¥ ®¡° §®¬ ª ²°¥³£®«¼­®¬³ ¢¨¤³ ¯°¨¢®¤¿²±¿ ®¯°¥¤¥«¨²¥«¨ s s : : : s s1 s2 : : : s n H~ n = : :2: : :3: : : : :n:+1: sn sn+1 : : : s2n 1ˆ¬¥¥¬H~ 1 = a1; H~ 2 = a21a2a3; H~ 3 = a31a22a23a4a5; :::H~ n = an1 (a2a3)n 1:::(a2n 2a2n 1):130ˆ² ª, °¥ª³°°¥­²­»¥ ´®°¬³«» ¤«¿ ®¯°¥¤¥«¨²¥«¥© Hn ¨ H~ nHn+1 = a1a2a3:::a2nHn; H1 = 1H~ n+1 = a1a2a3:::a2n+1H~ n ; H~ 0 = 1¤ ¾² °¥¸¥­¨¥ ®¡° ²­®© § ¤ ·¨fsng ! fak g:‘¯° ¢¥¤«¨¢ ’¥®°¥¬ 28 Š®½´´¨¶¨¥­²» ­¥¯°¥°»¢­®© ¤°®¡¨ ‘²¨«²¼¥± ¢»° ¦ ¾²±¿ ·¥°¥§ ª®½´´¨¶¨¥­²» ¥¥ ±²¥¯¥­­®£® °¿¤ ¯® ´®°¬³« ¬:~a2n = Hn+1 H~ n 1 ;Hn Hn~a2n 1 = H~n 1 Hn 1 ; n = 1; 2; ::: (4:16)Hn 2 HnŽ²¬¥²¨¬, ·²® ¨±¯®«¼§³¿ ¨±·¨±«¥­¨¥ £¥­¥²¨·¥±ª¨µ ±³¬¬ ¬®¦­® ­¥ ²®«¼ª® ¢»·¨±«¨²¼ ®¯°¥¤¥«¨²¥«¨ ƒ ­ª¥«¿, ­® ¨ ¤ ²¼´®°¬³«» ¤«¿ ¬¨­®°®¢ ¨µ ¯®±«¥¤­¥£® ±²®«¡¶ .“¯° ¦­¥­¨¥ 11 ³±²¼t[m; n] :=mXj1 =1Xajm+11j2 =j1 +1aj :::2Xm+n 1jn =jn 1+1ajn :„®ª ¦¨²¥, ·²®1) s2n =nXk=1( 1)k 1 s2n k t[2(n k + 1); 2k]2) t[m; n] = t[m 1; n] + am+2(n 1) t[m; n 1]3) s2n =2nYj =1aj +nXk=1( 1)k 1s2n k t[2(n k) + 1; 2k]131“¯° ¦­¥­¨¥ 12³±²¼ s0 : : :sHn;k := sn k 1 n k+1 : : :sn::::::::::::::::::s2n k 2 ; k = 0; :::; ns2n k : : : s2n 1 sn 1:::„®ª ¦¨²¥, ·²®Hn+1 = a1a2:::a2nHnHn;k = t[2(n k) + 1; 2k]Hn 2 :1321:5.

Характеристики

Список файлов лекций