27-02-2013, 09:25 AM
Orthogonal polynomials
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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:
kf − pk∞ = max
|x|≤1 |f (x) − p(x)|.
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
Orthogonal polynomials
P: the vector space of all polynomials with real
coefficients: p(x) = a0 + a1x + a2x2 + · · · + anxn.
Basis for P: 1, x, x2, . . . , xn, . . .
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 nonzero constant)
and hpn, pmi = 0 for n 6= m.