MATH529

MATH529 L04: Orthogonal function series

Overview

Complex Fourier series

For $f : [- p, p] \to ℂ$ , $f, f^{'}$ piecewise continuous
$\begin{array}{c} f (x) = A_{0} + \sum_{k = 1}^{\infty} (A_{k} \cos [k x] + B_{k} \sin [k x]), \\ A_{k} = \int_{- p}^{p} f (x) \cos (k x) \in ℂ, B_{k} = \int_{- p}^{p} f (x) \sin (k x) . \end{array}$
Use $θ_{k} = k x, \cos θ_{k} = (e^{i θ_{k}} + e^{- i θ_{k}}), \sin θ_{k} = (e^{i θ_{k}} - e^{- i θ_{k}})$ to obtain
$\begin{array}{c} f (x) = \sum_{n = - \infty}^{\infty} c_{n} e^{n x}, \\ C_{k} = (A_{k} - i B_{k}), C_{- k} = (A_{k} + i B_{k}), k \in ℕ, \\ C_{n} = \int_{- p}^{p} f (x) \exp (- \frac{i n π}{p} x) . \end{array}$

Sturm-Liouville problems

Fourier series expresses $f : [- π, π]$ as a linear combination of $\sin (k x), \cos (k x)$ .
The set ${1, \cos (x), \sin (x), \cos (2 x), \sin (2 x), \dots}$ form an orthogonal basis
Question: are there other basis sets? Answer: Yes, solutions of Sturm-Liouville
Regular Sturm-Liouville problem for $y : [a, b] \to ℝ$ , $r, q, p, r^{'} \in 𝒞 [a, b]$ , $λ \in ℝ,$
$\begin{array}{c} \frac{d}{d x} [r (x) y^{'}] + [q (x) + λ p (x)] y = 0, \\ A_{1} y (a) + B_{1} y^{'} (a) = 0, A_{2} y (b) + B_{2} y^{'} (b) = 0, \\ A_{1} B_{1} \neq 0, A_{2} B_{2} \neq 0 . \end{array}$
Sturm-Liouville scalar product
${(f, g)}_{p} = \int_{a}^{b} p (x) f (x) g (x) d x$
The regular Sturm-Liouville problem has non-zero eigensolutions $(λ_{n}, y_{n})$
$\begin{array}{ll} \exists λ_{n} \in ℝ, λ_{1} < λ_{2} < \dots, λ_{n} \to \infty as n \to \infty, & \forall λ_{n}, \exists! solution y_{n}, \\ {y_{1}, y_{2}, \dots} linearly independent, & {(y_{j}, y_{k})}_{p} = δ_{j k} . \end{array}$

Non-regular Sturm-Liouville problems

Sturm-Liouville problem for $y : [a, b] \to ℝ$ , $r, q, p, r^{'} \in 𝒞 [a, b]$ , $r, p > 0$ , $λ \in ℝ,$
$\begin{array}{c} \frac{d}{d x} [r (x) y^{'}] + [q (x) + λ p (x)] y = 0, \\ A_{1} y (a) + B_{1} y^{'} (a) = 0, A_{2} y (b) + B_{2} y^{'} (b) = 0, \\ A_{1} B_{1} \neq 0, A_{2} B_{2} \neq 0 . \end{array}$
Singular Sturm-Liouville problem
$\begin{array}{l} r (a) = 0 and \end{array} A_{1} y (a) + B_{1} y^{'} (a) = 0, r (b) = 0 and A_{2} y (b) + B_{2} y^{'} (b) = 0 .$
Periodic Sturm-Liouville problem
$\begin{array}{ll} r (a) = r (b) = 0 and no {BC}^{'} s, & r (a) = r (b) and y (a) = y (b), y^{'} (a) = y^{'} (b) \end{array}$

Self-adjoint form

For $𝒙, 𝒚 \in ℝ^{m}$ $(𝒙, 𝒚) = 𝒙^{T} 𝒚$ . Consider operator $𝑨 \in ℝ^{m \times m}$
$(𝑨 𝒙, 𝒚) = 𝒙^{T} 𝑨^{T} 𝒚 = (𝒙, 𝑨^{T} 𝒚)$
If $𝑨 = 𝑨^{T}$ (symmetric) then $(𝑨 𝒙, 𝒚) = (𝒙, 𝑨 𝒚)$ , same operator is either slot.
Consider now $u, v : [a, b] \to ℝ$ , the Sturm-Liouville operator
$ℒ u = (\frac{d}{d x} [r (x) \frac{d}{d x}] + [q (x) + λ p (x)]) u$
satisfies $(ℒ u, v) = (u, ℒ v)$ and is said to be self-adjoint in the unweighted scalar product.
Any ODE $a (x) y^{''} + b (x) y^{'} + [c (x) + λ d (x)] y = 0$ can be made self-adjoint
$\frac{d}{d x} [e^{\int (b / a) d x} y^{'}] + [\frac{c}{a} e^{\int (b / a) d x} + λ \frac{d}{a} e^{\int (b / a) d x}] y = 0$

Examples: Bessel, Legendre equations

Bessel equation
$x y^{''} + y^{'} + (a^{2} x - \frac{n^{2}}{x}) y = 0 \Leftrightarrow \frac{d}{d x} (x y^{'}) + (a^{2} x - \frac{n^{2}}{x}) y = 0,$
with solutions $J_{n} (a x), Y_{n} (a x)$ .
Legendre equation
$(1 - x^{2}) y^{''} - 2 x y^{'} + n (n + 1) y = 0 \Leftrightarrow \frac{d}{d x} [(1 - x^{2}) y^{'}] + n (n + 1) y = 0,$
with solutions $P_{n} (x)$ .

Bessel, Legendre series

Similar to Fourier series $f (x)$ can be expressed as a linear combination of other basis sets, e.g., Bessel, Legendre functions

f (x) = \sum_{n = 1}^{\infty} c_{n} J (a_{n} x)

f (x) = \sum_{n = 1}^{\infty} c_{n} P_{n} (x)