Exam 11: Orthogonal Functions and Fourier Series

The function $f ( x ) = \left\{ \begin{array} { c c } x & \text { if } x < 0 \\2 - x & \text { if } x > 0\end{array} \right\}$ is Select all that apply.

The function $f ( x ) = \left\{ \begin{array} { c c } x & \text { if } x < 0 \\2 + 5 x & \text { if } x > 0\end{array} \right\}$ has a Fourier series on $[ - 2,2 ]$ that converges at $x = 1$ to

The Fourier series of the function $f ( x ) = \left\{ \begin{aligned}x & \text { if } x < 0 \\2 - x & \text { if } x > 0\end{aligned} \right\}$ on $[ - 2,2 ]$ are Select all that apply.

The square norm of the function $f ( x ) = \sin x$ on the interval $[ 0 , \pi ]$ is

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0$ is

The solution of the eigenvalue problem $r R ^ { \prime \prime } + R ^ { \prime } + r \lambda R = 0 , R ( 0 )$ is bounded, $R ^ { \prime } ( 3 ) = 0$ is $\left( J _ { 0 } ^ { t } \left( z _ { n } \right) = 0 \right)$

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ^ { \prime } ( 1 ) = 0$ is

The Fourier series of the function $f ( x ) = | x |$ on $[ - 2,2 ]$ is Select all that apply.

The function $f ( x ) = \left\{ \begin{array} { l l } 0 & \text { if } x < 0 \\1 & \text { if } x > 0\end{array} \right\}$ has a Fourier series on $[ - 2,2 ]$ that converges at $x = 0$ to

The Fourier series of an even function might Select all that apply.

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime } ( \pi ) = 0$ is

The Fourier coeficients of the function $f ( x ) = x ^ { 2 }$ on $[ - 1,1 ]$ are Select all that apply.

In order to be assured by a theorem that the Fourier Series of $f$ on $[ a , b ]$ converges to $f$ , which of the following conditions need to be satisfied? Select all that apply.

The problem $\frac { d } { d x } \left[ r ( x ) y ^ { \prime } \right] + ( q ( x ) + \lambda p ( x ) ) y = 0 , A _ { 1 } y ( a ) + B _ { 1 } y ^ { \prime } ( a ) = 0 , A _ { 2 } y ( b ) + B _ { 2 } y ^ { \prime } ( b ) = 0$ is a regular Sturm-Liouville problem under certain conditions, including Select all that apply.

The function $f ( x ) = \left\{ \begin{array} { l l } 0 & \text { if } x < 0 \\1 & \text { if } x > 0\end{array} \right\}$ has a Fourier series on $[ - 2,2 ]$ that converges at $x = 1$ to

The Fourier Series of a function $f ( x ) = x$ defined on $[ - 1,1 ]$ is $f ( x ) = a _ { 0 } / 2 + \sum _ { n - 1 } ^ { \infty } \left( a _ { n } \cos ( n \pi x ) + b _ { n } \sin ( n \pi x ) \right)$ where Select all that apply.

The function $f ( x ) = \left\{ \begin{array} { c c } x & \text { if } x < 0 \\2 + 5 x & \text { if } x > 0\end{array} \right\}$ has a Fourier series on $[ - 2,2 ]$ that converges at $x = 0$ to

The Fourier coeficients of the function $f ( x ) = x$ on $[ - 1,1 ]$ are

The Fourier Series of a function $f$ defined on $[ - p , p ]$ is $f ( x ) = a _ { 0 } / 2 + \sum _ { n - 1 } ^ { \infty } \left( a _ { n } \cos ( n \pi x / p ) + b _ { n } \sin ( n \pi x / p ) \right)$ where Select all that apply.

Which of the following differential equations are in self-adjoint form? Select all that apply.