Applied Math Qualifying Exam August 15, 2000

1. State the Singular Value Decomposition Theorem.
2. For any \(n\times m\) real matrices \(A\) and \(B\), show the \(\operatorname {Tr}(AB^*)=\operatorname {Tr}(BA^*)\).
3. Show that \(\lVert A\rVert =\sqrt {\operatorname {Tr}(AA^*)}\) is a norm.
4. Suppose \(A\) and \(B\) are \(n\times m\) real matrices. Find the orthogonal matrix \(Q\) that minimizes \(\lVert QA-B\rVert \) using the norm of part c.

Suppose the function \(\phi (x)\) is Lipschitz continuous on the unit interval \(0<x<2\pi \) and that \(\lim _{x\to 0}\phi (x)=\phi (0^+)\) exists. Suppose \(\int _0^\infty \frac {\sin x}{x}\,dx=\alpha \) is known. Evaluate \(\lim _{k\to \infty }\int _0^{2\pi }\phi (x)\frac {\sin kx}{x}\,dx\) (with \(k\) restricted to be integer-valued).

1. Show that the functions \(\{e^{inx}\}_{n=-\infty }^{\infty }\) form an orthogonal set in \(L^2[0,2\pi ]\).
2. Suppose \(f(x)\in L^2[0,2\pi ]\). Let \(\alpha _n=\int _0^{2\pi }f(x)e^{inx}\,dx\). Prove that \(\{\alpha _n\}_{n=-\infty }^{\infty }\) is convergent in \(\ell ^2\) and also that \(\sum _{n=-\infty }^{\infty }\alpha _ne^{inx}\) is convergent in \(L^2[0,2\pi ]\). What does \(\sum _{n=-\infty }^{\infty }\alpha _ne^{inx}\) converge to?
3. Suppose \(f(x)\) and \(g(x)\) are \(2\pi \) periodic functions and square integrable on the interval \(0<x<2\pi \). State and prove the Convolution Theorem for these functions, that is, find the Fourier coefficients of the function \(h(x)=\int _0^{2\pi } f(x-t)g(t)\,dt\) in terms of the Fourier coefficients of \(f(x)\) and \(g(x)\).

Under what conditions does a solution of the boundary value problem \(u''+u=f(x)\) with \(u(0)=\alpha \), \(u'(\pi /2)=\beta \) exist?

Find the representation of the delta function and the appropriate transform pair for the differential operator \(Lu=-u''\) on the interval \(0<x<\pi \) subject to boundary conditions \(u'(0)=0\), \(u(\pi )=0\).

Evaluate \[ \int _{-\infty }^{\infty }\frac {\sin x}{x}\,dx. \]

1. Find the Fourier transform of \(\dfrac {\sin x}{x}\).
2. Suppose that the Fourier transform of the function \(f(t)\) is 0 for frequencies whose absolute value is greater than \(\pi /h\). Find the representation of \(f(t)\) in terms of sinc functions.

Solve the discretized heat equation \(u_{nt}=\frac {1}{h^2}(u_{n+1}-2u_n+u_{n-1})\) with \(-\infty <n<\infty \) and initial data \(u_n(t=0)=\sin \frac {2\pi n}{k}\) where \(k\) is a fixed integer.

Suppose \(\Omega \) is some closed, bounded domain in two spatial dimensions.

1. Under what conditions does the equation \(\nabla ^2\phi =0\) in \(\Omega \), \(\mathbf {n}\cdot \nabla \phi =f\) on \(\partial \Omega \), have a solution? Is it unique?
2. Find an integral equation for \(\phi \) on the boundary \(\partial \Omega \). Be as explicit as possible.
3. Under what conditions does this integral equation have a solution?