Written Qualifying Examination in APPLIED MATHEMATICS September 11, 1995

Let \(A\) be a real \(m\times n\) matrix with adjoint \(A^*\).

1. Prove that if \(R(A)\) and \(N(A)\) denote the range and nullspace of \(A\) respectively, then \[ R(A)\oplus N(A^*)=\mathbb {R}^m. \] (\(\oplus \) means orthogonal direct sum.)
2. Using the result of (a), state and prove Fredholm’s Alternative Theorem.

Let \(A\) be a real \(m\times n\) matrix.

1. Define the Least Square Pseudo-Inverse \(A'\).
2. Show how Gaussian Elimination can be used to compute \(A'\).

Let \(S=\{\phi _1,\phi _2,\ldots ,\phi _n,\ldots \}\) be a countable orthonormal set in a Hilbert space \(\mathcal {H}\). Prove that \(S\) is a complete orthonormal set if and only if it is a maximal orthonormal set; i.e., there exists no unit vector \(\psi \) such that \(\{\psi ,\phi _1,\phi _2,\ldots ,\phi _n,\ldots \}\) is an orthonormal set.

Let \(f(x)\) be in the Hilbert space \(L^2(-\infty ,\infty )\). Construct and use a Green’s function to find the solution in \(L^2(-\infty ,\infty )\) of the equation \[ u''-\alpha ^2u=f(x),\qquad (\alpha >0). \]

Liouville’s theorem states that a bounded entire function \(f(z)\) must be a constant.

1. Prove Liouville’s theorem.
2. Use Liouville’s theorem to prove the Fundamental Theorem of Algebra: every polynomial \(P_n(z)\) of degree \(n\ge 1\) has a root.

1. State and prove Jordan’s Lemma of complex function theory.
2. Use Jordan’s Lemma and the residue theorem to prove the equation \[ \int _0^\infty \frac {\cos (x)}{x^2+a^2}\,dx=\frac {\pi e^{-a}}{2a} \] where \(a>0\).

The (generalized) Hilbert transform \(H\) of a periodic function \(f\) can be defined as follows: For any simply connected two-dimensional domain \(\Omega \), let \(\phi \) satisfy \(\nabla ^2\phi =0\) with \(\nabla \phi \cdot \mathbf {n}=f\) on \(\partial \Omega \). Then \(Hf=\nabla \phi \cdot \mathbf {t}\) on \(\partial \Omega \). (\(\mathbf {n}\) is the exterior unit normal, \(\mathbf {t}\) is the unit tangent vector.)

1. Prove that \(H(Hf)=-f\).
2. Find a relationship between the Fourier transform and the Hilbert transform, using a unit circle as the domain \(\Omega \).
3. What is the Hilbert transform of \(f(x)=a\sin (x)+b\cos (2x)\)?

Using an integral representation for the \(n\)-th order Bessel function \(J_n(x)\), find the leading order asymptotic representation of \(J_n(x)\) for large positive \(x\).

The \(z\)-transform is defined on sequences in \(\ell ^2\) by \[ U(z)=\sum _{-\infty }^{\infty }u_nz^n, \] \[ u_n=\frac {1}{2\pi i}\int _C U(z)z^{-n-1}\,dz \] where the contour \(C\) is a closed contour enclosing the origin, and integration is counterclockwise. If \(z\) is restricted to lie on the unit circle then \(U(z)=U(e^{i\theta })=f(\theta )\).

1. Find the relationship between the \(z\)-transform and the Fourier series representation of \(f(\theta )\).
2. Use the \(z\)-transform to solve the system of equations \[ \frac {du_n}{dt}=\frac {1}{2h}(u_{n+1}-u_{n-1}) \] with \(u_n(0)=\delta _{n0}\).