First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 5. APPROXIMATION THEORY 202
We need some definitions and theorems to continue with our quest. Let’s start with a formal
definition of orthogonal functions.
Definition 88. Functions {φ 0 ,φ 1 , ..., φ n } are orthogonal for the interval [a, b] and with respect
to the weight function w(x) if
〈φ j ,φ k 〉 =
∫ b
a
⎧
⎨0 if j ≠ k
w(x)φ j (x)φ k (x)dx =
⎩α j > 0 if j = k
where α j is some constant. If, in addition, α j =1for all j, then the functions are called
orthonormal.
How can we find an orthogonal or orthonormal basis for our vector space? Gram-Schmidt
process from linear algebra provides the answer.
Theorem 89 (Gram-Schmidt process). Given a weight function w(x), the Gram-Schmidt
process constructs a unique set of polynomials φ 0 (x),φ 1 (x), ..., φ n (x) where the degree of φ i (x)
is i, such that
⎧
⎨0 if j ≠ k
〈φ j ,φ k 〉 =
⎩1 if j = k
and the coefficient of x n in φ n (x) is positive.
Let’s discuss two orthogonal polynomials that can be obtained from the Gram-Schmidt
process using different weight functions.
Example 90 (Legendre Polynomials). If w(x) ≡ 1 and [a, b] =[−1, 1], the first four polynomials
obtained from the Gram-Schmidt process, when the process is applied to the monomials
1,x,x 2 ,x 3 , ..., are:
√
1 3
φ 0 (x) =√
2 ,φ 1(x) =
2 x, φ 2(x) = 1 √
5
2 2 (3x2 − 1),φ 3 (x) = 1 √
7
2 2 (5x3 − 3x).
Often these polynomials are written in its orthogonal form; that is, we drop the requirement
〈φ j ,φ j 〉 =1in the Gram-Schmidt process, and we scale the polynomials so that the value of
each polynomial at 1 equals 1. The first four polynomials in that form are
L 0 (x) =1,L 1 (x) =x, L 2 (x) = 3 2 x2 − 1 2 ,L 3(x) = 5 2 x3 − 3 2 x.
These are the Legendre polynomials; polynomials we first discussed in Gaussian quadrature,