PhD Preliminary Qualifying Examination: Applied Mathematics August 13, 2007

The dilation equation is \[ \phi (x)=\sum _k c_k\phi (2x-k) \] and the associated wavelet is \[ W(x)=\sum _k(-1)^k c_{1-k}\phi (2x-k). \]

1. 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 and determine the values of \(c_k\). Sketch the associated wavelet. Briefly explain whether or not \(N_2(x)\) can act as a scaling function for a multiresolution analysis?
2. Prove that if the functions \(\{\phi (x-k)\}_k\) form an orthonormal set then \[ \int _{-\infty }^{\infty }W(x)W(x-m)\,dx = \delta _{0,m}\int _{-\infty }^{\infty }\phi ^2(x)\,dx. \]
3. Prove that \[ \int _{-\infty }^{\infty } W(x)\phi (x-m)\,dx=0 \] for all \(m\).

1. Show that a test function \(\psi (x)\) is of the form \(\psi =(x\phi )'\) where \(\phi \) is a test function if and only if \[ \int _{-\infty }^{\infty }\psi (x)\,dx=0 \quad \text {and}\quad \int _0^{\infty }\psi (x)\,dx=0. \]
2. Solve the following equation in the sense of distribution: \[ x^2\frac {d\phi }{dx}=0. \]

Consider the integral equation \[ \phi (x)-\lambda \int _0^\pi \sin (x+t)\phi (t)\,dt = \cos (x)+\sin (x), \qquad 0\le x\le \pi . \]

1. Find the unique solution when \(\lambda \ne \pm 2/\pi \).
2. Use the Fredholm alternative theorem to show that there is no solution when \(\lambda =2/\pi \).
3. Use the Fredholm alternative theorem to show that when \(\lambda =-2/\pi \) there is a one-parameter family of solutions of the form \[ \phi (x)=\frac {\cos (x)+\sin (x)}{2}+\alpha (\cos (x)-\sin (x)). \]

Suppose that \(L\) is a bounded linear operator in a Hilbert space \(H\) with closed range.

1. Prove the Fredholm alternative theorem for solutions of the inhomogeneous equation \(Lu=f\) with \(u,f\in H\).
2. Suppose that \(L\) is invertible and has a complete orthonormal set of eigenfunctions \(\phi _n\), integer \(n\). Solve the inhomogeneous equation of part (a) using an eigenfunction expansion.
3. Suppose that the adjoint operator \(L^*\) has a nontrivial nullspace \(\mathcal {N}(L^*)\) spanned by the functions \(\psi _i\), \(i=1,\ldots ,m\) and \(f\) does not lie in the orthogonal complement of \(\mathcal {N}(L^*)\). Let \(L\) have a nullspace spanned by the functions \(\phi _i\), \(i=1,\ldots ,m\). Explain the construction of the smallest least squares solution.

1. By constructing the one-dimensional Green’s function solve the inhomogeneous equation \[ \frac {d^2u}{dx^2}=f(x),\qquad u(0)=\alpha ,\qquad u(1)=\beta . \]
2. Use Green’s functions to solve \[ \frac {d^2u}{dx^2}-\alpha ^2u=f(x)\quad \text {on }L^2(-\infty ,\infty ) \] with \(\alpha \) real.

Formulate and prove the Cauchy integral formula.

Solve Laplace’s equation \(\Delta \phi =0\) in the domain between the two nonconcentric circles \(|z|=1\) and \(|z-1|=5/2\), subject to the boundary conditions \(\phi =a\) on \(|z|=1\) and \(\phi =b\) on \(|z-1|=5/2\).

Evaluate the integral \[ I=\int _0^\infty \frac {x^\alpha \,dx}{(x+1)^2} \] (where \(\alpha \) is a constant parameter, so that the integral converges).

Evaluate the integral \[ \int _{-\infty }^{\infty }\frac {\sin \alpha x}{x}\,dx \] (\(\alpha \) is a constant parameter). Explain your steps.

Find the two-term asymptotic expansion of the integral \[ I(s)=\int _{-\infty }^{\infty }\frac {e^{s(ix+1)^2}}{(x+i)^2}\,dx, \qquad s\text { is real and }s\to +\infty . \]