DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN ANALYSIS January 2015

If \(f_i:\mathbb R\to \mathbb R\) are measurable functions, then is \[ \{x:2\text { is a limit point of }\{f_i(x)\}_{i=1}^{\infty }\} \] measurable?

Recall that \(p\) is a limit point of a sequence \(\{a_i\}_{i=1}^{\infty }\) if there exists an increasing sequence of natural numbers \(n_i\) so that \(p=\lim _{i\to \infty }a_{n_i}\).

Show that if \(f,g\in L^1(\lambda ,\mathbb R)\), then \(\lVert f*g\rVert _1\le \lVert f\rVert _1\lVert g\rVert _1\). Recall \[ (f*g)(x)=\int f(t)g(x-t)\,d\lambda (t). \]

Recall that if \(A\) is a linear operator on Hilbert space with bounded operator norm, then it has an adjoint \(A^*\) which is the unique operator so that for all \(v,w\), \[ \langle Av,w\rangle =\langle v,A^*w\rangle . \]

(a) Let \(P:\ell ^2(\mathbb N)\to \ell ^2(\mathbb N)\) by \(P\bar v=\bar w\) where \(w_j=0\) if \(j\) is not prime and \(w_j=v_j\) otherwise. What is \(P^*\)?

(b) Let \(P:\ell ^2(\mathbb N)\to \ell ^2(\mathbb N)\) by \(P\bar v=\bar w\) where \(w_j=v_{p_j}\), where \(p_j\) is the \(j\)th prime. What is \(P^*\)?

If \(f:\mathbb R\to \mathbb C\) is in \(L^1(\lambda )\), show that the sequence of \(L^1(\lambda )\) functions \[ g_n(x)=f\left (x+\frac {1}{n}\right ) \] converges to \(f\) in \(L^1(\lambda )\).

(a) Let \(A\subset \mathbb R\). Prove that if \(\lambda (A)>0\), then there exists \(\delta _0>0\) so that for all \(0\le \delta \le \delta _0\) there exists \(a\) such that both \(a\) and \(a+\delta \) are in \(A\). Note, \(a\) is allowed to depend on \(\delta \).

(b) Show that if \(A\subset \mathbb R\) is closed under addition and subtraction, then \(\lambda (A)=0\) or \(A=\mathbb R\).

Let \(f\) be an automorphism of the unit disk \(|z|<1\). Assume that \(f\) fixes two points. Prove that \(f(z)=z\) for all \(z\) in the unit disk.

Let \(f(z)=1+3z+z^n\).

(a) Show that \(f\) has exactly one root in the unit disk \(\{z:|z|<1\}\).

(b) Show that all roots of \(f\) lie in the disk \[ \left \{z:|z|\le \frac {3+\sqrt {13}}{2}\right \}. \]

Let \(f\) be an entire function. Suppose that there exist \(A>0\), \(R>0\), and a positive integer \(k\) such that \[ |f(z)|\le A|z|^k\ln |z| \] for \(|z|\ge R\). Show that \(f\) is a polynomial of degree at most \(k\).

Let \(f\) be a holomorphic function defined on a domain \(\Omega \) in \(\mathbb C\). Show that \(f\) is either a polynomial, or there exists \(z\in \Omega \) such that \[ f^{(n)}(z)\ne 0 \] for all integers \(n\ge 0\).

Evaluate \[ \int _C \frac {1}{z^2-z}\,dz \] where \(C\) is a closed path in \(\mathbb C-[0,1]\).