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J/ Chebyshev , Prlynemiaks \n humericah Aral aS ie man problem is Wat Abprovan action 8 cfun fion . hy beffer appwa sma ton a furcKon ‘He enor sud be teducdAr a minimum. as introduce Chebyched polynomials which, coco WwW avews euch as priho gonad potynemiaks : pe lynontat app Kintaliony tumnertcall Wests tion pele The &* chobysined polynomial of fta spo wma ,clunnter by Te), & defined as Tr) = Cas (ncos’x) oy oq uid alertity Th (tos @) = tos nO L= wse 82 G@s'x Kecal Sse-Meides Theorem (cs 2 +1'simo)’ = Gene resmme (oS 06 = Reltrse Fesi BF} Aswan xecall binomral expans torr fer non = napechue (ate) = A ena ty + nia) arth 1 AGda-a) grap: aI rene a iy = on + WOenna + NOAM eb: iva 6 bite oe + Caio’ t bt Dee aoe, Reene anynie nl oe ahora. Oe (") @-r)) yr! EER er pura rer watt Se We now See k ex perssions (hay Mosne m lems °f (os-p- by applyny fic bidomecl eyjoayssvouw Gornula en (Cor@ resney” aid tien dekmp the veal” part g af. let @=Gsp and h= sind. Thorefove ; (ese Fesine)) = (ese)" + imc, ase)" gine — NCy (Goss) (Kme)™— C "Cy (ese (sinb)” nn + Ny (wse)(Eine3)4 oe acs (aso y “fing” Now ) Cesne > bef (as0+ cine)" f = (cos 8)" me Cy (ose) 7, (Gwe) + “cy(asey” "Sin4 Ofe2 , we Subshhete OF Gs a od howe, Tn&kS = cos(ne), Tom Vere we can prceed fsx NFA by apyay—p Ne- Moroes froma a QS ween earlier - We can ghtan “WG for nba by deakincs te eatabuh D Bearen Sones wnglead of usp tte jreoro’o -arppreach Now, Vat) = Cos(ne). We “place 4 orth ot fe have Tan = tos [Orsde) im GsCno +2) Recall (kook CoS (A+B) = COS ACS B —SinASing Cos (NO 48) = Cos nes —Aane snd Sinilasly ep ac—p n wal H-1 5 we have TWiGS = Gsnetas6 + Singnesad Trove fore Th @S = CosS Csne ~ &ndSnie Tas @\ = Cos@cosnga + SINGS NO Then Tar) + TyG) = AcoeeCosno Bu tese sx and Cocnd = Ink) SO, Tha) + TaG) = QeTa) FB Ta. = DTG “Te 6] , n>4 Chobyshoo's yecuvten dr Koomnuha key We oaw eat: “Tales =4 anh TQra When ned, ‘Va@s = Ja Oo h&s = a4 When M0 AgGy oe eee WOO ml GS S Oxo = 49° ake = 496 30 When 023, ue Thad > Qx1,@) - Ta@) = ax fates? 3] ee! + S44 — 6x2? —ay7 41 = Bt -34 44 the ansiens abeve chew That Thee) IS a pdlunomcal aap fae) 4 dosren A wrth te loachn toe fect being tale ie : rg he ‘Subs hutins, all Vee title eqn), we pad 7 ' jeans ~\ reli gh Sicee dx = | (as(ne)+ Cos (mo Vi=w 1 Sia Ae) : < 1b, fos (98) Gs(me) dg- [DS¥Os8 = Jee) d as(5-2y) T : | 0 | lassGonye. + 30s G-nie ] do ! | i ie 0 He i Ne Cheb yshew polyro ncerlg Tee | are axle maT, A OE (ie Gwen Weight functor ; = | J ¥* Shod tet claby she polynoneedt gy Wa fost KMD Te) 68 an inde pendertt Solution’ 9 tha ( Sevwad oMmogencei~ ocler; Vordiiony differential equitlion pikes xr) Ay ary i a xa ey ae | Proof Let y= WOO = Cos(ncos’a = ws nO Hore , XE Cos G or O = Cog y ay = ty .A =~ new(nes. lx de APS ~Sing- | = NSin (ne) | < Sone iO) | fo Sm Oy ine differentiath OD agua wre x. dx bet fe 2 nsm (ne) ean We: vost ted pat oe Lin @ oe i We apely quohent ule fpr Ai Ge Geethatioc tee Us ng&n(na) amd Ve Sin Au = was Ae = f?Ces@O), —1_ ox dO x Sine Bo Os ose. ~ It Ere S(nO i AN ee a Te Se dx ay* 
