Problem. Approximate the function f (x) = e^x 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
subspace P3 of quadratic polynomials.
Suppose that p0, p1, p2 is an orthogonal basis for P3. 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 p0, p1, p2,
. . .
such that deg pn = n (p0 is a non zero constant )
and <pn, pm> = 0 for n ≠ m.
Orthogonal polynomials can be obtained by applying
the Gram-Schmidt orthogonalization process
to the basis 1, x, x^2, . . . :
Then p0, p1, p2, . . . 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 <p, xk> = 0 for any
0 ≤ k < deg p.
Example.
Note that <xn, xm> = 0 if m + n is odd.
Hence p2k(x) contains only even powers of x while
p2k+1(x) contains only odd powers of x.
p0, p1, p2, . . . are called the Legendre
polynomials
Instead of normalization, the orthogonal
polynomials are subject to standardization.
The standardization for the Legendre polynomials is
Problem. Find P4(x)
Let
We know that P4(1) = 1 and <P4, xk>
= 0 for 0 ≤ k ≤ 3.
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 P3 of
polynomials of degree at most 2.
The Legendre polynomials P0, P1, P2
form an orthogonal basis
for P3. Therefore
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,
Definition. Chebyshev polynomials T0, T1,
T2, . . .
are orthogonal polynomials relative to the inner
product
with the standardization Tn(1) = 1
Remark. “T” is like in “Tschebyscheff”.
Change of variable in the integral : x = cosΦ
Theorem.
Recurrent formula:
Chebyshev polynomials
