PhD Preliminary Qualifying Examination Applied Mathematics Wednesday January 6 2016

Let \(X\) be a Banach space, let \(z\in X\) with \(z\ne 0\), and let \(f\in X'\) with \(f\ne 0\). Consider the linear operator \(T:X\to X\) defined for \(x\in X\) by \[ Tx=f(x)z. \]

1. Briefly explain why \(T\) is compact.
2. Use Neumann series to give a condition on \(f\) and \(z\) that ensures the equation \(x+Tx=y\) admits a unique solution \(x\) for all \(y\in X\).
3. Use the Fredholm alternative to find a condition on \(f\) and \(z\) that ensures the equation \(x+Tx=y\) admits a unique solution \(x\) for all \(y\in X\).

Let \(X\) and \(Y\) be normed vector spaces. Show that a linear operator \(T:X\to Y\) is compact if and only if for every sequence \((x_n)\) with \(\lVert x_n\rVert \le 1\), the sequence \((Tx_n)\) has a convergent subsequence.

Let \(T\) be a bounded linear operator on a complex Hilbert space \(H\). Show that the operator \(I+T^*T\) is injective.

Let \(Y\) be a subspace of a Hilbert space \(H\). Show that \(Y\) is closed if and only if \(Y=Y^{\perp \perp }\).

Let \(X\) be a Banach space. Let \((f_n)\) be a sequence in \(X'\).

1. Show that \(f_n\to f\) weakly in \(X'\) implies that for all \(x\in X\), we have \(f_n(x)\to f(x)\).
2. If there is an \(f\in X'\) such that \(f_n(x)\to f(x)\) for all \(x\in X\), does \(f_n\to f\) weakly in \(X'\)? Explain your answer.

Find the sum \[ \cos x+\cos 3x+\cos 5x+\cdots +\cos 99x. \]

Prove the fundamental theorem of algebra:

Every polynomial of degree \(n(\ge 1)\) has exactly \(n\) zeros (each zero being counted according to its multiplicity).

Evaluate the integrals

1. \[ \int _{-\infty }^{\infty }\frac {\sin x}{x}\,dx, \]
2. \[ \int _0^\infty \frac {x^\alpha }{x+1}\,dx. \]

Show that Fourier transform \[ f(x)\to F(\mu )=\int _{-\infty }^{\infty } f(x)e^{i\mu x}\,dx \] preserves the inner product and the \(L_2\) norm (Plancherel’s equations): \[ \int _{-\infty }^{\infty }\bar {f}(x)g(x)\,dx = \int _{-\infty }^{\infty }\bar {F}(\mu )G(\mu )\,\frac {d\mu }{2\pi }, \] \[ \text {and}\qquad \int _{-\infty }^{\infty }\lvert f(x)\rvert ^2\,dx = \int _{-\infty }^{\infty }\lvert F(\mu )\rvert ^2\,\frac {d\mu }{2\pi }. \] (\(\overline {\phantom {x}}\) denotes complex conjugation).

Solve Laplace’s equation \[ \phi _{xx}+\phi _{yy}=0 \quad \text {in the domain}\quad D=\{z:\lvert z-1\rvert >1,\ \lvert z-2\rvert <2\} \] subject to the boundary condition \[ \phi (x,y)=a\text { when }\lvert z-1\rvert =1 \qquad \text {and}\qquad \phi (x,y)=b\text { when }\lvert z-2\rvert =2 \] (\(a\) and \(b\) are real parameters).