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### 8.9 Orthogonal Polynomials

#### 8.90 Introduction

8.901 Suppose that w(x) is a nonnegative real function of a real variable x. Let (a, b) be a fixed interval on the x-axis. Let us suppose further that, for n = 0, 1, 2, …, the integral

${\int }_{a}^{b}{x}^{n}w\left(x\right)\text{d}x$

exists and that the integral

${\int }_{a}^{b}w\left(x\right)\text{d}x$

is positive. In this case, there exists a sequence of polynomials p0(x), p1(x), …, pn(x), …, that is uniquely determined by the following conditions:

1. pn(x) is a polynomial of degree n and the coefficient of xn in this polynomial is positive.

2. The polynomials ...

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