MATH528

MATH528 Lesson20: Approximation by trigonometric polynomials

Consider a truncation of the infinite series

f (x) = a_{0} + \sum_{n = 1}^{\infty} (a_{n} \cos (\frac{n π x}{L}) + b_{n} \sin (\frac{n π x}{L}))

that defines a trigonometric polynomial of degree $N$

F (x) = a_{0} + \sum_{n = 1}^{N} (a_{n} \cos (\frac{n π x}{L}) + b_{n} \sin (\frac{n π x}{L})) .

The square (or 2-norm) error is defined as

E = \int_{- π}^{π} {[f (x) - F (x)]}^{2} d x

For $a_{n} = A_{n}$ , $b_{n} = B_{n}$

E^{*} = \int_{- π}^{π} {[f (x)]}^{2} d x - π [2 a_{0}^{2} + \sum_{n = 1}^{N} (a_{n}^{2} + b_{n}^{2})]

In general

E ⩾ E^{*} ⩾ 0

E^{*} ⩾ 0 \Rightarrow \frac{1}{π} \int_{- π}^{π} {[f (x)]}^{2} d x ⩾ 2 a_{0}^{2} + \sum_{n = 1}^{\infty} (a_{n}^{2} + b_{n}^{2})

known as the Bessel inequality.