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May 25th

May 25th

# Orthogonal Polynomials

Problem. Approximate the function f (x) = ex on
the interval [−1, 1] by a quadratic polynomial .

The best approximation would be a polynomial p (x)
that minimizes the distance relative to the uniform
norm:

However there is no analytic way to find such a
polynomial. Another approach is to find a “least
squares” approximation that minimizes the integral
norm

The norm is induced by the inner product

Therefore is minimal if p is the
orthogonal projection of the function f on the

We should apply the Gram-Schmidt process to the
polynomials 1, x, x2 which form a basis for .
This would yield an orthogonal basis
Then

Orthogonal polynomials

P: the vector space of all polynomials with real
coefficients :
Basis for P:

Suppose that P is endowed with an inner product.
Definition. Orthogonal polynomials (relative to
the inner product) are polynomials
such that deg ( is a non zero constant )
and

Orthogonal polynomials can be obtained by applying
the Gram-Schmidt orthogonalization process
to the basis 1, x, x2, . . . :

Then are orthogonal polynomials.

Theorem (a) Orthogonal polynomials always exist.
(b) The orthogonal polynomial of a fixed degree is
unique up to scaling.
(c) A polynomial p ≠ 0 is an orthogonal
polynomial if and only if <p, q> = 0 for any
polynomial q with deg q < deg p.
(d) A polynomial p ≠ 0 is an orthogonal
polynomial if and only if for any
0 ≤ k < deg p.

Example.

Note that if m + n is odd.
Hence contains only even powers of x while
contains only odd powers of x.

are called the Legendre polynomials.

polynomials are subject to standardization.

The standardization for the Legendre polynomials is
. In particular,

Problem. Find
Let
We know that and for
0 ≤ k ≤ 3.

Thus

Legendre polynomials

Problem. Find a quadratic polynomial that is the
best least squares fit to the function f (x) = |x| on
the interval [−1, 1].

The best least squares fit is a polynomial p(x) that
minimizes the distance relative to the integral norm

over all polynomials of degree 2.

The norm is minimal if p is the orthogonal
projection of the function f on the subspace of
polynomials of degree at most 2.

The Legendre polynomials form an orthogonal basis
for . Therefore

In general,

Problem. Find a quadratic polynomial that is the
best least squares fit to the function f (x) = |x| on
the interval [−1, 1].

Solution :

Recurrent formula for the Legendre polynomials:

For example,

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