Real and Complex Analysis Exam Aug 18, 2006

Assume that \(f\) is non-negative and \[ \int _X f\,d\mu <\infty . \] Show that for every \(\epsilon >0\) there exists a \(\delta >0\) such that if \(A\) is a measurable subset of \(X\) such that \(\mu (A)\le \delta \) then \[ \int _A f\,d\mu \le \epsilon . \]

We say that \(f_n\to f\) in measure if for all \(\epsilon >0\) there exists an \(N>0\) such that if \(n\ge N\) then \[ \mu \bigl (\{x\mid |f_n(x)-f(x)|>\epsilon \}\bigr )<\epsilon . \] Show that if \(f_n\to f\) pointwise and \(\mu (X)<\infty \) then \(f_n\to f\) in measure.

Let \(\alpha \) be an irrational number. Let \(f\) be a real valued measurable function on \(\mathbb R\) such that \(f(x)=f(x+1)=f(x+\alpha )\). Use Fourier series to show that \(f\) is constant almost everywhere.

Let \(E_1,\ldots ,E_n\) by a finite collection of disjoint measurable subsets of \(X\) and let \(M\) be a subspace of \(L^2(X)\) spanned by the functions \(\{u_1,\ldots ,u_n\}\) where each \(u_i\) is in \(L^2(X)\) and has support contained in \(E_i\). Let \(P:L^2(X)\to \mathbb R\) be the orthogonal projection. Find a function \(h:X\times X\to \mathbb R\) such that \[ Pf(x)=\int _X h(x,y)f(y)\,d\mu (y) \] for all \(f\in L^2(X)\).

The space \(C[0,1]\) of continuous functions on \([0,1]\) is a subspace of all the \(L^p\) spaces, \(L^p[0,1]\) for \(1\le p\le \infty \). Define a linear operator \(T:C[0,1]\to \mathbb R\) by \(Tf=f(1/2)\). Show that \(T\) is a bounded operator for the \(L^p\)-norm if and only if \(p=\infty \). Show that \(T\) extends to a bounded operator on all of \(L^\infty [0,1]\).

In what follows, \(D_R(z_0)\) and \(\overline {D_R}(z_0)\) denote the open and closed discs, respectively, centered at \(z_0\), with radius \(R\).

Let \(f\) be a holomorphic function on an open set containing \(\overline {D_1}(0)\). Prove that if \(z\) is in \(D_1(0)\), then \[ f(z)=\frac {1}{2\pi i}\int _\gamma \frac {f(w)}{\sin (w-z)}\,dw \] where \(z\in D_1(0)\) and \(\gamma \) traverses the unit circle once in the positive direction.

Prove that if an entire function \(f\) satisfies an inequality of the form \[ |f(z)|\le A+B\log |f(z)| \] for some positive constants \(A\) and \(B\) and for all \(z\in \mathbb C\), then \(f\) is a constant.

Suppose \(f\) is an entire function with \(f^{(n)}(0)\ge 0\) for all non-negative integers \(n\). Find a point at which \(|f(z)|\) assumes its maximum on \(\overline {D_R}(0)\) and tell what that maximum is. Justify your answer.

Suppose \(f\) is holomorphic in an open set \(\Omega \) and \(\overline {D_R}(z_0)\subset \Omega \). If \(f\) has a zero of order \(k\) at \(z_0\) and no other zeroes in \(\overline {D_R}(z_0)\), then prove that there is an \(\epsilon >0\) such that, if \(|\lambda |<\epsilon \), then \(f\) takes on \(\lambda \) as a value \(k\) times (counting multiplicity) in the disc \(D_R(z_0)\).

Suppose \(\Omega \) and \(\Lambda \) are open subsets of the complex plane. Suppose \(\Omega \) is connected and \(\Lambda \) is a simply connected, proper subset of \(\mathbb C\). If \(\{f_n\}\) is a sequence of holomorphic functions from \(\Omega \) to \(\Lambda \), prove that either \(\{f_n\}\) has a subsequence which converges uniformly on compacta to a holomorphic function on \(\Omega \) or \(\{f_n\}\) itself converges to \(\infty \) at every point of \(\Omega \).