직교 조건을 만족하는 다항함수

$P _{2} ( x ) = x ^{2} + ax + b$라 하면 $\displaystyle \int _{- 1} ^{1} P _{0} ( x ) P _{2} ( x ) dx = \int _{- 1} ^{1} ( x ^{2} + ax + b ) dx = \frac{2}{3} + 2 b = 0$ $\displaystyle \int _{- 1} ^{1} P _{1} ( x ) P _{2} ( x ) dx = \int _{- 1} ^{1} ( x ^{3} + ax ^{2} + bx ) dx$$\displaystyle = \frac{2}{3} a = 0$ $\therefore$ $a = 0$, $\displaystyle b = - \frac{1}{3}$, $\displaystyle P _{2} ( x ) = x ^{2} - \frac{1}{3}$ $P _{3} ( x ) = x ^{3} + cx ^{2} + dx + e$라 하면 $\displaystyle \int _{- 1} ^{1} ( x ^{3} + cx ^{2} + dx + e ) dx = \frac{2}{3} c + 2 e = 0$, $\displaystyle \frac{1}{3} c + e = 0$ $\displaystyle \int _{- 1} ^{1} ( x ^{4} + cx ^{3} + dx ^{2} + ex ) dx$$\displaystyle = \frac{2}{5} + \frac{2}{3} d = 0$, $\displaystyle d = - \frac{3}{5}$ $\displaystyle \begin{aligned} \int _{- 1} ^{1} {( x ^{2} - \frac{1}{3} )} ( x ^{3} + cx ^{2} + dx ^{} + e ) dx \\ = \int _{- 1} ^{1} {} ( x ^{5} + cx ^{4} + dx ^{3} + ex ^{2} ) dx - \frac{1}{3} \int _{- 1} ^{1} {} ( x ^{3} + cx ^{2} + dx ^{} + e ) dx \end{aligned}$ $\displaystyle = \frac{2}{5} c + \frac{2}{3} e = 0$ $\displaystyle \int _{0} ^{1} P _{3} ( x ) dx = \int _{0} ^{1} ( x ^{3} + cx ^{2} + dx + e ) dx$$\displaystyle = \frac{1}{4} + \frac{1}{3} c + \frac{1}{2} d + e$ $\displaystyle = \frac{1}{4} + \frac{1}{2} d$$\displaystyle = \frac{1}{4} - \frac{3}{10}$$\displaystyle = - \frac{1}{20}$