MATH528

MATH528 Lesson22: Orthogonal series

Solution of the Sturm-Liouville problem for $y : [a, b] \to ℝ$ defined by

{\begin{cases} {(p (x) y^{'})}^{'} + (q (x) + λ r (x)) y = 0 \\ k_{1} y (a) + k_{2} y^{'} (a) = 0 \\ l_{1} y (b) + l_{2} y^{'} (b) = 0 \end{cases} .,

(1)

with $p, q, r : [a, b] \to ℝ$ , and $λ, k_{1}, k_{2}, l_{1}, l_{2} \in ℝ$ , gives a family of orthogonal functions ${y_{0}, y_{1}, \dots}$

Definition. Given a family of functions $ℬ = {y_{0}, y_{1}, \dots}$ (a basis set), defined on $[a, b]$ , orthogonal w.r.t the scalar product

$(f, g) = \int_{a}^{b} r (x) f (x) g (x) d x,$

and a function $f : [a, b] \to ℝ$ , a convergent series of the form

$f (x) = \sum_{m = 0}^{\infty} a_{m} y_{m} (x),$

is known as the orthogonal expansion of $f$ on $ℬ$ . The scalar coefficients $a_{m}$ are known as the Fourier coefficients w.r.t. the basis set $ℬ$ , and are determined as

$a_{m} = \frac{(f, y_{m})}{(y_{m}, y_{m})} = \frac{1}{{|| y_{m} ||}^{2}} \int_{a}^{b} f (x) y_{m} (x) d x,$

or as

$a_{m} = (f, y_{m})$

for an orthonormal basis set that has $|| y_{m} || = 1$ .