Department of Mathematics University of Utah Ph.D. Preliminary Examination in Real/Complex Analysis August 1998 INSTRUCTIONS

Let \(f\) and \(g\) be two complex measurable functions on a measurable space \(X\). Show that \[ \{x\in X:f(x)\ne g(x)\} \] is a measurable subset of \(X\).

Let \(\mu \) be a measure on \(X\). Let \(f_1\ge f_2\ge \cdots f_n\ge \cdots \ge 0\) be a sequence of measurable functions. Assume \(\lim _{n\to \infty }\int _X f_n\,d\mu =0\). Show that \(f_n(x)\to 0\) a.e.

Let \(T\) be a bounded linear functional on a (linear) subspace \(M\) of a Hilbert space \(H\). Prove that \(T\) has a unique norm preserving extension to a bounded linear functional on \(H\), and that this extension vanishes on \(M^\perp \).

Let \(f_n\) be a sequence of functions in \(L^\infty (\mathbb R)\) such that \[ \sup _n\left |\int _{\mathbb R} f(x)g(x)\,dx\right |<\infty \] for all \(g\in L^1(\mathbb R)\). Show that \(\sup _n\lVert f_n\rVert <\infty \).

(a) State the definition of a Fourier transform of \(f\in L^1(\mathbb R)\);

(b) State the Fourier Inversion Theorem;

(c) Show that the Fourier transform is one-to-one on \(L^1(\mathbb R)\).

Let \(f\) be holomorphic in \(\Delta -\{0\}\). Assume that \(f'\) is bounded in \(\Delta -\{0\}\). Show that \(f\) extends to a holomorphic function on \(\Delta \).

Let \(f\) be a nonconstant entire function. Show that for every real constant \(k\) there exists \(z\in \mathbb C\) such that \(\Im f(z)=k\cdot \Re f(z)\).

Using methods of complex analysis, evaluate \[ \int _0^\infty \frac {x^2\,dx}{x^4+x^2+1} \]

Show that all zeroes of \(z^4-6z-3\) lie inside the circle \(|z|=2\).

Find a biholomorphism \(\varphi \) of \(\Delta -\mathbb R^+\) onto \(\Delta \). Here \(\mathbb R^+=\{x\in \mathbb R:x\ge 0\}\). (It will suffice to give explicitly finitely many maps whose composition is equal to \(\varphi \).)