正交多项式
正交多项式
设 g n ( x ) g_n(x) gn(x)是首项系数 a n ≠ 0 a_n\neq0 an=0的 n n n次多项式，如果多项式序列 g 0 ( x ) , g 1 ( x ) , ⋅ ⋅ ⋅ g_0(x),g_1\left(x\right),\cdotp\cdotp\cdotp g0(x),g1(x),⋅⋅⋅满足
( g j , g k ) = ∫ a b ρ ( x ) g j ( x ) g k ( x ) d x = { 0 , j ≠ k , A k > 0 , j = k ( j , k = 0 , 1 , ⋯ ) , (g_j,g_k)=\int_a^b\rho(x)g_j(x)g_k(x)\mathrm{d}x=\begin{cases}0,&j\neq k,\\A_k>0,&j=k\end{cases}(j,k=0,1,\cdots), (gj,gk)=∫abρ(x)gj(x)gk(x)dx={0,Ak>0,j=k,j=k(j,k=0,1,⋯),
则称多项式序列 g 0 ( x ) , g 1 ( x ) , ⋅ ⋅ ⋅ g_0(x),g_1(x),\cdotp\cdotp\cdotp g0(x),g1(x),⋅⋅⋅在 [ a , b ] [a,b] [a,b]上带权 ρ ( x ) \rho(x) ρ(x)正交，并称 g n ( x ) g_n(x) gn(x)是 [ a , b ] [a,b] [a,b]上带权 ρ ( x ) \rho(x) ρ(x)的 n n n次正交多项式.