Preliminaries

Consider the vector space of functions generated by the Legendre Polynomials. This has an infinite number of basis function but is complete in the sense that any sensibly smooth function can be written as a combination of the Legendre Polynomials. Moreover, Legendre Polynomials are orthogonal with respect to the inner product

i.e.,

for ,

>

Another important property of the Legendre Polynomials is that each Legendre Polynomial has exactly distinct zero's, with , in the interval . This can be shown as follows:

>

The first Legendre Polynomial is a constant function:

> with(orthopoly):P(0,x);

The next one has exactly one root, i.e. ,

> P(1,x);

Any Legendre Polynomial must have at least one zero in

, since

.

Assume that has exactly zero's in the interval

. Then is of opposite sign in each consecutive subinterval defined by these zero's. Then construct the polynomial

, where is chosen such that the sign of and in the first interval is the same. Then has the same zero's and the same sign in each of the subintervals defined by . Consequently,

,

But since the Legendre Polynomials form a basis,

, we also have that

, because of the orthogonality of the Legendre Polynomials.

Clearly both results contradict each other. Therefore, the assumption that the number of roots is must be false. Hence, the Legendre Polynomial must have exactly distinct zero's.

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