Question

Notice that these polynomials form an orthogonal set with this inner product. Find the best 1²-13 Let P2 have the inner product given by evaluation at -5, -1, 1, and 5. Let po(t) = 2, P1(t)=t, and q(t) = 12 approximation to p(t) = t by polynomials in Span{Po.P1,9}. The best approximation to p(t) = t by polynomials in Span{Po.P2,q} is

Answer 1

Given that q (t) 12 13 12 -%+ LP,9) 12113.9 £f.9>= f(-5) 96-531861).96-1) + f(1).qw4f(5)•g(s) 2651365) + 26-1936-1) +253.5+39 2.1 0 Polt): 0 P.(U:] P(t):13 Let w spon po (t), Pill), q6t)} Pros W (t) = LP, Po> <P,P> IPJ/ Pt || ?, ? ZB16): 61)"(397 +E1) (9)+ 8 0) (2) +680513 - 0 ZP, P.)=P(-5).PL-5) 484-1), P(-1) 4P(5.). P115 ) tow).PL) .- 2504 11 P,WEZP.P.) = (-5)65)+ (-01-1)+1+1+5.5 = 25+1 +25=52 LP:9>- PL-5)965) +P601196-14P(0)4 968)+9(5):918) -21-539.1 +26-1336-1) +261) 2153).) 2504 :: Proyw Plt) t 626 t 013 The best approximation to Plt) t3 by polynomials 626 span {Po, P., q} is 52 t 13