ABSTRACT
Part (ii) is easily proved by observing that any polynomial of degree n-1 or less can be expressed as a linear combination of 4,(x), 4,(x), ...,4,-,( x). 6.2 The specification of the a,., is called the normalization. One method of normalization is to make each a,,, unity; another method sometimes used is implicitly given by
where b,,, is Kronecker's delta symbol, defined by
To prove this result, we first choose An so that +,+ ,(x)- Anx4,(x) contains no term in P+'. Then we express
The coefficients c,,, can be found by multiplying both sides of this equation by w(x)c$,(x) and integrating from a to b. In consequence of (6.01) this yields
Let x, ,x2, ..., x,, 0 d m Q n, be the distinct points in (a,b) at which +,(x) has a zero of odd multiplicity. Then in (a, b) the polynomial
has only zeros of even multiplicity. If m < n, then the orthogonal property shows that
which is a contradiction since the integrand does not change sign in (a, 6). Therefore m = n. Moreover, since the total number of zeros is n, each x, must be a simple zero. This completes the proof.
