nyquist limit is great and all, but I really don’t want to wait for all \(T\) to be able to sample all the necessary terms to solve for every \(a_{j},b_{j}\) before we can reconstruct our signal.

So, even if we got our sequence of \(\frac{1}{2B}\) length of points, we need an alternative way to reconstruct the signal as we go.

One way to reconstruction via interpolation is just to connect the dots; however, this is bad because it creates sharp corners.

• psycoacoustics

invariant subspaces are a property of operators; it is a subspace for which the operator in question on the overall space is also an operator of the subspace.

constituents

• an operator \(T \in \mathcal{L}(V)\)
• a subspace \(U \subset V\)

requirements

\(U\) is considered invariant on \(T\) if \(u \in U \implies Tu \in U\)

(i.e. \(U\) is invariant under \(T\) if \(T |_{U}\) is an operator on \(U\))

additional information

nontrivial invariant subspace

(i.e. eigenstuff)

A proof is not given yet, but \(T \in \mathcal{L}(V)\) has an invariant subspace that’s not \(V\) nor \(\{0\}\) if \(\dim V > 1\) for complex number vector spaces and \(\dim V > 2\) for real number vector spaces.

the inverse is the the opposite of an operation. As in, if you apply the inverse of an operation to the result of applying the original with the same operation it will cancel it.

That is,

\begin{equation} A * B * B^{-1} = A \end{equation}

\(B^{-1}\) is then the inverse of \(B\) for the \(*\) operation. This is operation dependent.

1. Generate a uniform number between 0 to 1.
2. Get the inverse of the standard normal density function at that value (let number be \(y\), find the \(x\) such that \(\phi(y) = x\))
3. return \(x\)