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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2001 • Question 2

Question

Math 6420 • PDE Fall 2001 • Question 2

The dilation equation is \[ \phi (x)=\sum _k c_k\phi (2x-k). \] Show that the function \[ N_2(x)= \begin {cases} x, & 0<x<1,\\ 2-x, & 1\le x<2,\\ 0, & \text {elsewhere} \end {cases} \] satisfies the dilation equation. What are the values of \(c_k\)? Are the wavelets formed from \(N_2(x)\) an orthogonal set? Why or why not?

Raw LaTeX

The dilation equation is
\[
  \phi(x)=\sum_k c_k\phi(2x-k).
\]
Show that the function
\[
N_2(x)=
\begin{cases}
x, & 0<x<1,\\
2-x, & 1\le x<2,\\
0, & \text{elsewhere}
\end{cases}
\]
satisfies the dilation equation. What are the values of \(c_k\)? Are the wavelets formed from
\(N_2(x)\) an orthogonal set? Why or why not?

Topic Tags

Hilbert Spaces and Orthogonality Linear Programming and Optimization Orthogonal and Unitary Matrices Projectors and Orthogonal Projections