1.
Find a vector in \(\C^3\) that is orthogonal to \((1,1,1)\) and \((1, \omega, \omega^2)\) where \(\omega = e^{2\pi i/3}\text{.}\)
Section 5.6 Exercises
Exercises Exercises
2.
Consider the sequence of functions given by \(e_j(z) = z^j\) for \(z \in \C\text{,}\) \(j \in \mathbb{Z}\) (notice that this includes the reciprocals of the monomials). Show that \((e_j)_{-\infty}^\infty\) is an orthonormal sequence in \(RL^2\) (recall the definition from the Table ).
3.
Consider the family of functions given by \(f_k(t) = e^{ikt}\) where \(t \in \R\) and the index \(k \in \R\text{.}\) (Notice that here we are dealing with an uncountable index set, and so this is not a sequence). Show that \((f_k)_{k \in \R}\) is an orthonormal system in \(TP\) (recall the definition from the Table ).
4.
Let \(\la_1, \la_2\) be elements in the complex unit disk \(\D\text{.}\) Define functions \(f_1, f_2\) by
\begin{equation*}
f_1(z) = \frac{(1 - \abs{\la_1}^2)^{1/2}}{1 - \cc{\la_1} z},
\end{equation*}
\begin{equation*}
f_2(z) = \frac{z - \la_1}{1 - \cc{\la_1}z} \cdot \frac{(1 - \abs{\la_2}^2)^{1/2}}{1 - \cc{\la_2} z}
\end{equation*}
(the first term in \(f_2\) is a special type of function called an automorphism of the disk). Show that \(f_1, f_2\) is an orthonormal system in \(RH^2\) (recall the definition from the Table ).
1
www.math3ma.com/blog/automorphisms-of-the-unit-disc5.
Let \(\alpha \in \C\) be a complex number off the unit circle, i.e. \(\abs{\alpha}\neq 1\text{.}\) Find the Fourier coefficients of the function \(f \in RL^2\) given by
\begin{equation*}
f(z) = \frac{1}{z - \alpha}
\end{equation*}
with respect to the orthonormal sequence \(e_j(z) = z^j\) where \(j \in \Z\text{.}\) (You need to consider the interior and exterior of the disk separately.)
6.
This exercise extends to infinite dimensions the Gram-Schmidt process. Let \(x_1, x_2, \ldots\) be a sequence of linearly independent vectors in an inner product space \(V\text{.}\) Define new vectors \(e_n\) by the recursive relation
\begin{align*}
e_1 \amp= x_1 / \norm{x_1};\\
f_n \amp= x_n - \sum_{j=1}^n \ip{x_n}{e_j}e_j \text{ for } n \geq 2;\\
e_n \amp= f_n / \norm{f_n}, \text{ for } n \geq 2.
\end{align*}
Show that \(f_n\) is an orthonormal sequence with the same closed linear span as \(x_n\text{.}\)
7.
The first three Legendre polynomials are given by
2
en.wikipedia.org/wiki/Legendre_polynomials
\begin{align*}
P_0(x) \amp= 1,\\
P_1(x) \amp= x,\\
P_2(x) \amp= \frac{1}{2}(3x^2 - 1).
\end{align*}
Show that these are scalar multiples of the orthonormal set generated by applying the Gram-Schmidt process to the vectors \(1, x, x^2\) in \(L^2(-1,1)\text{.}\)
8.
Find the closest point to \(x = (1,-1,1)\) in the linear span of \((1, \omega, \omega^2)\) and \((1, \omega^2, \omega)\text{,}\) where \(\omega = e^{2\pi i/3}\text{.}\)
9.
Let \(\hilbert\) be a Hilbert space, and let \(M\) be a closed linear subspace of \(\hilbert\text{;}\) that is, \(M\) is a Hilbert space with respect to restricted inner product from \(\hilbert\text{.}\) Let \((e_n)_1^\infty\) be a complete orthonormal sequence in \(M\text{.}\) Show that for any \(x \in \hilbert\text{,}\) the best approximation to \(x\) in \(M\) is
\begin{equation*}
y = \sum_{j = 1}^\infty \ip{x}{e_j} e_j.
\end{equation*}
(Compare to the formula for projecting onto a subspace in Euclidean space Theorem 1.3.1.)
10.
Prove that the standard orthonormal sequence of “coordinate vectors” is complete in \(\ell^2\text{.}\)
11.
Prove that the orthonormal sequence of mononomials \((e_j(x))_0^\infty\) with \(e_j(z) = z^j\) is complete in \(RH^2\text{.}\)
12.
Show that for a fixed \(n\text{,}\) the orthogonal complement in \(RH^2\) of the space given by
\begin{equation*}
M = \{z^n f: f \in RH^2\}
\end{equation*}
is the space of polynomials of degree less than \(n\text{.}\)
