Department of Mathematics University of Utah Ph.D. Preliminary Examination in Analysis August 2017

Prove or disprove: If \((X,\mathcal M,\mu )\) is a measure space, \(A_1,\ldots \) are measurable, and \[ \sum _{i=1}^{\infty }\mu (A_i)^2<\infty , \] then \[ \mu \left (\bigcap _{m=1}^{\infty }\bigcup _{i=m}^{\infty } A_i\right )=0. \]

Prove or give a counterexample: \(B_1,B_2\) are Banach spaces and \(A:B_1\to B_2\) is continuous, linear, injective, and \(A(B_1)\) is dense. Then \(A\) is a bijection.

Show that if \(v_i\in \mathcal H\) and \(v_i\) converges weakly to \(u\), then \[ \liminf _{i\to \infty }\lVert v_i\rVert \ge \lVert u\rVert , \] and that equality occurs only when there exists a subsequence \(v_{n_j}\) so that \(v_{n_j}\) converges to \(u\) in norm.

Prove that the convolution of two \(L^1(\lambda )\) functions is in \(L^1(\lambda )\).

This question considers different notions of convergence for \(\ell ^1(\mathbb N)\):

(a) Does pointwise convergence imply \(\lVert \cdot \rVert _1\) convergence? Vice versa?

(b) Does pointwise convergence imply convergence in measure? Vice versa?

Let \(f\) be a function holomorphic for \(z\) in the annulus \(r<z<R\). Show that if for some \(\rho \) with \(r<\rho <R\), \[ \int _{|z|=\rho } z^n f(z)\,dz=0 \] for all nonnegative \(n\), then \(f\) extends to a holomorphic function on the disk \(|z|<R\).

Let \(f\) be an entire function such that for each \(z\in \mathbb C\), \[ \operatorname {Re} f(z)\cdot \operatorname {Im} f(z)\ne 1. \] Prove that \(f\) is a constant.

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

Let \(f:\mathbb C\to \mathbb C\) be a function such that:

(a) \(f\) is continuous;

(b) \(f\) is holomorphic at all points \(z\) not on the real axis;

(c) if \(z\) is on the real axis then \(f(z)\) is real.

Show that \(f(\bar z)=\overline {f(z)}\) and that \(f\) is holomorphic on all of \(\mathbb C\). In words, applying \(f\) to the conjugate input gives the complex conjugate of the original output.

Determine the number of zeroes of the polynomial \[ z^5+\frac {z^3}{3}+\frac {z^2}{4}+\frac {z}{3} \] in the region \(1/2<|z|<1\).